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| mso-list-template-ids:940591768 1067224872 1455844604 -1656194402 1682488634 204524340 683952088 -2102472138 -346767930 -227374296;}
| |
| @list l7:level1
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:\F0B7;
| |
| mso-level-tab-stop:.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:Symbol;}
| |
| @list l7:level2
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:o;
| |
| mso-level-tab-stop:1.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:"Courier New";
| |
| mso-bidi-font-family:"Times New Roman";}
| |
| @list l7:level3
| |
| {mso-level-tab-stop:1.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l7:level4
| |
| {mso-level-tab-stop:2.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l7:level5
| |
| {mso-level-tab-stop:2.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l7:level6
| |
| {mso-level-tab-stop:3.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l7:level7
| |
| {mso-level-tab-stop:3.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l7:level8
| |
| {mso-level-tab-stop:4.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l7:level9
| |
| {mso-level-tab-stop:4.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l8
| |
| {mso-list-id:723725180;
| |
| mso-list-type:hybrid;
| |
| mso-list-template-ids:-99083428 -220961180 -1921223632 -537343498 -1671161318 -1774003118 -756508450 297817086 -961482748 1254398216;}
| |
| @list l8:level1
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:\F0B7;
| |
| mso-level-tab-stop:.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:Symbol;}
| |
| @list l8:level2
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:o;
| |
| mso-level-tab-stop:1.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:"Courier New";
| |
| mso-bidi-font-family:"Times New Roman";}
| |
| @list l8:level3
| |
| {mso-level-tab-stop:1.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l8:level4
| |
| {mso-level-tab-stop:2.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l8:level5
| |
| {mso-level-tab-stop:2.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l8:level6
| |
| {mso-level-tab-stop:3.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l8:level7
| |
| {mso-level-tab-stop:3.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l8:level8
| |
| {mso-level-tab-stop:4.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l8:level9
| |
| {mso-level-tab-stop:4.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l9
| |
| {mso-list-id:783380878;
| |
| mso-list-type:hybrid;
| |
| mso-list-template-ids:1195124968 -1176856626 705838168 1430024758 -10969166 -2429068 -886557894 -981976734 -674716836 1739458092;}
| |
| @list l9:level1
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:\F0B7;
| |
| mso-level-tab-stop:.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:Symbol;}
| |
| @list l9:level2
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:o;
| |
| mso-level-tab-stop:1.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:"Courier New";
| |
| mso-bidi-font-family:"Times New Roman";}
| |
| @list l9:level3
| |
| {mso-level-tab-stop:1.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l9:level4
| |
| {mso-level-tab-stop:2.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l9:level5
| |
| {mso-level-tab-stop:2.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l9:level6
| |
| {mso-level-tab-stop:3.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l9:level7
| |
| {mso-level-tab-stop:3.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l9:level8
| |
| {mso-level-tab-stop:4.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l9:level9
| |
| {mso-level-tab-stop:4.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l10
| |
| {mso-list-id:876965743;
| |
| mso-list-type:hybrid;
| |
| mso-list-template-ids:-2121116484 607317988 -1849143530 1549421520 -185434022 1802422662 -893866580 781089088 141182364 -727819858;}
| |
| @list l10:level1
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:\F0B7;
| |
| mso-level-tab-stop:.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:Symbol;}
| |
| @list l10:level2
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:o;
| |
| mso-level-tab-stop:1.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:"Courier New";
| |
| mso-bidi-font-family:"Times New Roman";}
| |
| @list l10:level3
| |
| {mso-level-tab-stop:1.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l10:level4
| |
| {mso-level-tab-stop:2.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l10:level5
| |
| {mso-level-tab-stop:2.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l10:level6
| |
| {mso-level-tab-stop:3.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l10:level7
| |
| {mso-level-tab-stop:3.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l10:level8
| |
| {mso-level-tab-stop:4.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l10:level9
| |
| {mso-level-tab-stop:4.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l11
| |
| {mso-list-id:893352415;
| |
| mso-list-type:hybrid;
| |
| mso-list-template-ids:1277608332 -1485438268 -2123969482 -738684274 -806691202 535473678 -1044592642 -387403456 -1996555014 -1668088518;}
| |
| @list l11:level1
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:\F0B7;
| |
| mso-level-tab-stop:.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:Symbol;}
| |
| @list l11:level2
| |
| {mso-level-tab-stop:1.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l11:level3
| |
| {mso-level-tab-stop:1.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l11:level4
| |
| {mso-level-tab-stop:2.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l11:level5
| |
| {mso-level-tab-stop:2.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l11:level6
| |
| {mso-level-tab-stop:3.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l11:level7
| |
| {mso-level-tab-stop:3.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l11:level8
| |
| {mso-level-tab-stop:4.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l11:level9
| |
| {mso-level-tab-stop:4.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l12
| |
| {mso-list-id:897667319;
| |
| mso-list-type:hybrid;
| |
| mso-list-template-ids:-1623049588 1915678138 -2112877258 878758142 748328180 -1117737218 -780006804 1262361794 -1320548886 1498856602;}
| |
| @list l12:level1
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:\F0B7;
| |
| mso-level-tab-stop:.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:Symbol;}
| |
| @list l12:level2
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:o;
| |
| mso-level-tab-stop:1.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:"Courier New";
| |
| mso-bidi-font-family:"Times New Roman";}
| |
| @list l12:level3
| |
| {mso-level-tab-stop:1.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l12:level4
| |
| {mso-level-tab-stop:2.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l12:level5
| |
| {mso-level-tab-stop:2.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l12:level6
| |
| {mso-level-tab-stop:3.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l12:level7
| |
| {mso-level-tab-stop:3.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l12:level8
| |
| {mso-level-tab-stop:4.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l12:level9
| |
| {mso-level-tab-stop:4.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l13
| |
| {mso-list-id:1055665564;
| |
| mso-list-type:hybrid;
| |
| mso-list-template-ids:-656760952 -1884229592 -1392877956 962769610 1778543280 1357706460 1408030976 2034147620 1277690490 1036396832;}
| |
| @list l13:level1
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:\F0B7;
| |
| mso-level-tab-stop:.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:Symbol;}
| |
| @list l13:level2
| |
| {mso-level-tab-stop:1.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l13:level3
| |
| {mso-level-tab-stop:1.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l13:level4
| |
| {mso-level-tab-stop:2.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l13:level5
| |
| {mso-level-tab-stop:2.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l13:level6
| |
| {mso-level-tab-stop:3.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l13:level7
| |
| {mso-level-tab-stop:3.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l13:level8
| |
| {mso-level-tab-stop:4.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l13:level9
| |
| {mso-level-tab-stop:4.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l14
| |
| {mso-list-id:1070232000;
| |
| mso-list-type:hybrid;
| |
| mso-list-template-ids:1145869726 1757715046 -1187204508 435191548 350768320 -617735082 1850607782 1227895152 -443903892 306073588;}
| |
| @list l14:level1
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:\F0B7;
| |
| mso-level-tab-stop:.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:Symbol;}
| |
| @list l14:level2
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:o;
| |
| mso-level-tab-stop:1.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:"Courier New";
| |
| mso-bidi-font-family:"Times New Roman";}
| |
| @list l14:level3
| |
| {mso-level-tab-stop:1.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l14:level4
| |
| {mso-level-tab-stop:2.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l14:level5
| |
| {mso-level-tab-stop:2.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l14:level6
| |
| {mso-level-tab-stop:3.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l14:level7
| |
| {mso-level-tab-stop:3.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l14:level8
| |
| {mso-level-tab-stop:4.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l14:level9
| |
| {mso-level-tab-stop:4.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l15
| |
| {mso-list-id:1144271540;
| |
| mso-list-type:hybrid;
| |
| mso-list-template-ids:-660064188 -288425564 -1174784214 -1516893902 955296806 -1007663708 -197227728 504643970 -1495471548 -729751772;}
| |
| @list l15:level1
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:\F0B7;
| |
| mso-level-tab-stop:.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:Symbol;}
| |
| @list l15:level2
| |
| {mso-level-tab-stop:1.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l15:level3
| |
| {mso-level-tab-stop:1.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l15:level4
| |
| {mso-level-tab-stop:2.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l15:level5
| |
| {mso-level-tab-stop:2.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l15:level6
| |
| {mso-level-tab-stop:3.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l15:level7
| |
| {mso-level-tab-stop:3.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l15:level8
| |
| {mso-level-tab-stop:4.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l15:level9
| |
| {mso-level-tab-stop:4.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l16
| |
| {mso-list-id:1197427457;
| |
| mso-list-type:hybrid;
| |
| mso-list-template-ids:1574862944 -1950695562 -1664058574 -744472278 947057276 1142076072 -1971270006 853701290 2058129874 -1871823896;}
| |
| @list l16:level1
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:\F0B7;
| |
| mso-level-tab-stop:.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:Symbol;}
| |
| @list l16:level2
| |
| {mso-level-tab-stop:1.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l16:level3
| |
| {mso-level-tab-stop:1.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l16:level4
| |
| {mso-level-tab-stop:2.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l16:level5
| |
| {mso-level-tab-stop:2.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l16:level6
| |
| {mso-level-tab-stop:3.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l16:level7
| |
| {mso-level-tab-stop:3.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l16:level8
| |
| {mso-level-tab-stop:4.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l16:level9
| |
| {mso-level-tab-stop:4.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l17
| |
| {mso-list-id:1235817101;
| |
| mso-list-type:hybrid;
| |
| mso-list-template-ids:-484300216 1772663010 2003322996 -605260242 1213481922 -778387562 -1722508324 -633936128 -1336356512 160365994;}
| |
| @list l17:level1
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:\F0B7;
| |
| mso-level-tab-stop:.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:Symbol;}
| |
| @list l17:level2
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:o;
| |
| mso-level-tab-stop:1.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:"Courier New";
| |
| mso-bidi-font-family:"Times New Roman";}
| |
| @list l17:level3
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:\F0A7;
| |
| mso-level-tab-stop:1.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:Wingdings;}
| |
| @list l17:level4
| |
| {mso-level-tab-stop:2.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l17:level5
| |
| {mso-level-tab-stop:2.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l17:level6
| |
| {mso-level-tab-stop:3.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l17:level7
| |
| {mso-level-tab-stop:3.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l17:level8
| |
| {mso-level-tab-stop:4.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l17:level9
| |
| {mso-level-tab-stop:4.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l18
| |
| {mso-list-id:1351446674;
| |
| mso-list-type:hybrid;
| |
| mso-list-template-ids:-652204930 975195168 -839748382 -1938498170 -385849910 2144479048 1032863454 538328702 -1315785518 -1973886200;}
| |
| @list l18:level1
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:\F0B7;
| |
| mso-level-tab-stop:.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:Symbol;}
| |
| @list l18:level2
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:o;
| |
| mso-level-tab-stop:1.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:"Courier New";
| |
| mso-bidi-font-family:"Times New Roman";}
| |
| @list l18:level3
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:\F0A7;
| |
| mso-level-tab-stop:1.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:Wingdings;}
| |
| @list l18:level4
| |
| {mso-level-tab-stop:2.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l18:level5
| |
| {mso-level-tab-stop:2.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l18:level6
| |
| {mso-level-tab-stop:3.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l18:level7
| |
| {mso-level-tab-stop:3.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l18:level8
| |
| {mso-level-tab-stop:4.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l18:level9
| |
| {mso-level-tab-stop:4.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l19
| |
| {mso-list-id:1353916209;
| |
| mso-list-type:hybrid;
| |
| mso-list-template-ids:-1560374772 1093200440 223497064 1490601500 -612976162 130296844 985066278 -664612648 -2054365446 -663688072;}
| |
| @list l19:level1
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:\F0B7;
| |
| mso-level-tab-stop:.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:Symbol;}
| |
| @list l19:level2
| |
| {mso-level-tab-stop:1.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l19:level3
| |
| {mso-level-tab-stop:1.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l19:level4
| |
| {mso-level-tab-stop:2.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l19:level5
| |
| {mso-level-tab-stop:2.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l19:level6
| |
| {mso-level-tab-stop:3.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l19:level7
| |
| {mso-level-tab-stop:3.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l19:level8
| |
| {mso-level-tab-stop:4.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l19:level9
| |
| {mso-level-tab-stop:4.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l20
| |
| {mso-list-id:1418282890;
| |
| mso-list-type:hybrid;
| |
| mso-list-template-ids:-1991619790 433720286 2122108218 -515215862 -175098546 1080094448 42740232 -932559314 -1508491926 1711071164;}
| |
| @list l20:level1
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:\F0B7;
| |
| mso-level-tab-stop:.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:Symbol;}
| |
| @list l20:level2
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:o;
| |
| mso-level-tab-stop:1.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:"Courier New";
| |
| mso-bidi-font-family:"Times New Roman";}
| |
| @list l20:level3
| |
| {mso-level-tab-stop:1.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l20:level4
| |
| {mso-level-tab-stop:2.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l20:level5
| |
| {mso-level-tab-stop:2.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l20:level6
| |
| {mso-level-tab-stop:3.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l20:level7
| |
| {mso-level-tab-stop:3.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l20:level8
| |
| {mso-level-tab-stop:4.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l20:level9
| |
| {mso-level-tab-stop:4.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l21
| |
| {mso-list-id:1419864039;
| |
| mso-list-type:hybrid;
| |
| mso-list-template-ids:-1692898040 1271593640 -1801830626 -494103412 1275769334 -1705615396 302279644 1144168706 1100762864 -1614642336;}
| |
| @list l21:level1
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:\F0B7;
| |
| mso-level-tab-stop:.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:Symbol;}
| |
| @list l21:level2
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:o;
| |
| mso-level-tab-stop:1.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:"Courier New";
| |
| mso-bidi-font-family:"Times New Roman";}
| |
| @list l21:level3
| |
| {mso-level-tab-stop:1.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l21:level4
| |
| {mso-level-tab-stop:2.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l21:level5
| |
| {mso-level-tab-stop:2.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l21:level6
| |
| {mso-level-tab-stop:3.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l21:level7
| |
| {mso-level-tab-stop:3.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l21:level8
| |
| {mso-level-tab-stop:4.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l21:level9
| |
| {mso-level-tab-stop:4.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l22
| |
| {mso-list-id:1442846612;
| |
| mso-list-type:hybrid;
| |
| mso-list-template-ids:2067847166 -1019993162 -1991994912 -1273466774 129152608 952912330 894091772 1751558862 509648994 2007105630;}
| |
| @list l22:level1
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:\F0B7;
| |
| mso-level-tab-stop:.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:Symbol;}
| |
| @list l22:level2
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:o;
| |
| mso-level-tab-stop:1.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:"Courier New";
| |
| mso-bidi-font-family:"Times New Roman";}
| |
| @list l22:level3
| |
| {mso-level-tab-stop:1.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l22:level4
| |
| {mso-level-tab-stop:2.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l22:level5
| |
| {mso-level-tab-stop:2.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l22:level6
| |
| {mso-level-tab-stop:3.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l22:level7
| |
| {mso-level-tab-stop:3.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l22:level8
| |
| {mso-level-tab-stop:4.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l22:level9
| |
| {mso-level-tab-stop:4.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l23
| |
| {mso-list-id:1471098332;
| |
| mso-list-type:hybrid;
| |
| mso-list-template-ids:97304936 856864862 1753926598 631136496 1466871724 -664615862 -225280294 -151111078 -529001762 1021367848;}
| |
| @list l23:level1
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:\F0B7;
| |
| mso-level-tab-stop:.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:Symbol;}
| |
| @list l23:level2
| |
| {mso-level-tab-stop:1.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l23:level3
| |
| {mso-level-tab-stop:1.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l23:level4
| |
| {mso-level-tab-stop:2.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l23:level5
| |
| {mso-level-tab-stop:2.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l23:level6
| |
| {mso-level-tab-stop:3.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l23:level7
| |
| {mso-level-tab-stop:3.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l23:level8
| |
| {mso-level-tab-stop:4.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l23:level9
| |
| {mso-level-tab-stop:4.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l24
| |
| {mso-list-id:1682974503;
| |
| mso-list-type:hybrid;
| |
| mso-list-template-ids:-2116115676 192056146 1409196666 -1008037988 -580595822 -1581115924 1231294170 886992706 -1599548510 -1721435040;}
| |
| @list l24:level1
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:\F0B7;
| |
| mso-level-tab-stop:.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:Symbol;}
| |
| @list l24:level2
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:o;
| |
| mso-level-tab-stop:1.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| mso-ansi-font-size:10.0pt;
| |
| font-family:"Courier New";
| |
| mso-bidi-font-family:"Times New Roman";}
| |
| @list l24:level3
| |
| {mso-level-tab-stop:1.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l24:level4
| |
| {mso-level-tab-stop:2.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l24:level5
| |
| {mso-level-tab-stop:2.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l24:level6
| |
| {mso-level-tab-stop:3.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l24:level7
| |
| {mso-level-tab-stop:3.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l24:level8
| |
| {mso-level-tab-stop:4.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l24:level9
| |
| {mso-level-tab-stop:4.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l25
| |
| {mso-list-id:1834493652;
| |
| mso-list-type:hybrid;
| |
| mso-list-template-ids:-1554608450 67698689 67698691 67698693 67698689 67698691 67698693 67698689 67698691 67698693;}
| |
| @list l25:level1
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:\F0B7;
| |
| mso-level-tab-stop:.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| font-family:Symbol;}
| |
| @list l25:level2
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:o;
| |
| mso-level-tab-stop:1.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| font-family:"Courier New";
| |
| mso-bidi-font-family:"Times New Roman";}
| |
| @list l25:level3
| |
| {mso-level-number-format:bullet;
| |
| mso-level-text:\F0A7;
| |
| mso-level-tab-stop:1.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;
| |
| font-family:Wingdings;}
| |
| @list l25:level4
| |
| {mso-level-tab-stop:2.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l25:level5
| |
| {mso-level-tab-stop:2.5in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l25:level6
| |
| {mso-level-tab-stop:3.0in;
| |
| mso-level-number-position:left;
| |
| text-indent:-.25in;}
| |
| @list l25:level7
| |
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| <!--[if gte mso 9]><xml>
| | The ''equations of Korteweg-de Vries type'' are all nonlinear perturbations of the [[Airy equation]]. They take the general form |
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| |
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| |
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| </head>
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|
| |
|
| <body lang=EN-US link=blue vlink=blue style='tab-interval:.5in'> | | <center><math>\partial_t u + \partial_x^3 u + \partial_x P(u) = 0</math></center> |
|
| |
|
| <div class=Section1> | | where <math>u(t,x)</math> is a function of one space and one time variable, and <math>P(u)</math> is some polynomial of <math>u</math>. One can place various normalizing constants in front of the <math>u_{xxx}</math> and <math>P(u)</math> terms, but they can usually be scaled out. The function <math>u</math> and the polynomial <math>P</math> are usually assumed to be real. |
|
| |
|
| <h2 align=center style='text-align:center'>Equations of <span class=SpellE>Korteweg</span> | | The <math>x</math> variable is usually assumed to live on the real line R (so there is some decay at infinity) or on the <span class="SpellE">torus</span> T (so the data is periodic). The half-line has also been studied, as has the case of periodic data with large period. It might be interesting to look at whether the periodicity assumption can be perturbed (e.g. quasi-periodic data); it is not clear whether the phenomena we see in the periodic problem are robust under perturbations, or are number-theoretic <span class="SpellE">artefacts</span> of perfect periodicity. |
| de <span class=SpellE>Vries</span> type</h2>
| |
|
| |
|
| <div class=MsoNormal align=center style='text-align:center'>
| | == Specific equations == |
|
| |
|
| <hr size=2 width="100%" align=center>
| | Several special cases of KdV-type equations are of interest, including |
|
| |
|
| </div> | | * The [[Korteweg-de Vries equation|Korteweg-de Vries (KdV) equation]] ([[Korteweg-de Vries equation on R|on R]], [[Korteweg-de Vries equation on the half-line|on R^+]], or [[Korteweg-de Vries equation on T|on T]]), in which <math>P(u) = 3 u^2</math>. This equation is [[completely integrable]]. |
| | * The [[modified Korteweg-de Vries equation|modified Korteweg-de Vries (mKdV) equation]] ([[Modified Korteweg-de Vries on R|on R]], [[generalized Korteweg-de Vries on the half-line|on R^+]], or [[Modified Korteweg-de Vries on T|on T]]), in which <math>P(u) = \pm 2 u^3</math>. This equation is also [[completely integrable]]. |
| | * The [[generalized Korteweg-de Vries equation|generalized Korteweg-de Vries (gKdV) equation]], in which <math>P(u) = c u^{k+1}</math> for some constants c,k. The cases k=1,2 are [[KdV]] and [[mKdV]] respectively. The quartic [[gKdV-3 equation]] and the quintic (mass-critical) [[gKdV-4 equation]] are of special interest. In general, these equations are not completely integrable. |
| | * The linearized Korteweg-de Vries equation, in which <math>P(u) = c u</math> (i.e., the <math>k=0</math> case of the generalized Korteweg-de Vries (gKdV) equation). This equations is linear in <math>u</math> and can be reduce to the simple form <math>u_T + u_{XXX} = 0</math> with the change of variables <math>X=x-t</math>, <math>T=t</math>.[http://mathoverflow.net/questions/23334/what-are-the-interesting-cases-of-the-generalized-korteweg-de-vries-equation] |
|
| |
|
| <p class=MsoNormal align=center style='text-align:center'><a name=overview></a><b>Overview</b></p> | | Drift terms <math>u_x</math> can be added, but they can be subsumed into the polynomial <math>P(u)</math> or eliminated by a [[Galilean]] transformation (except in the half-line case). Indeed, one can freely insert or remove any term of the form <math>a'(t) u_x</math> by shifting the <math>x</math> variable by <math>a(t)</math>, which is especially useful for periodic higher-order <span class="SpellE">gKdV</span> equations (setting <math>a'(t)</math> equal to the mean of <math>P(u(t))</math>). |
|
| |
|
| <p>The <span class=SpellE>KdV</span> <span class=GramE>family of equations are</span>
| | The Korteweg-de Vries equation is also a member of the [[KdV hierarchy]]. One can also couple the KdV equation to other equations, creating for instance the [[nonlinear Schrodinger-Airy system]]. |
| of the form </p>
| |
|
| |
|
| <p align=center style='text-align:center'><span class=SpellE>u_t</span> + u<span
| | == History == |
| class=GramE>_{</span>xxx} + P(u)_x = 0</p>
| |
|
| |
|
| <p><span class=GramE>where</span> u(<span class=SpellE>x,t</span>) is a
| | Historically, these types of equations first arose in the study of 2D shallow wave propagation, but have since appeared as limiting cases of many dispersive models. Interestingly, the 2D shallow wave equation can also give rise to the <span class="SpellE">Boussinesq</span> or [[Cubic NLS on R|cubic NLS]] equation by making more limiting assumptions (in particular, weak nonlinearity and slowly varying amplitude). |
| function of one space and one time variable, and P(u) is some polynomial of
| |
| u. One can place various normalizing constants in front of the u<span
| |
| class=GramE>_{</span>xxx} and P(u) terms, but they can usually be scaled | |
| out. The function u and the polynomial P are usually assumed to be real. </p>
| |
|
| |
|
| <p>Historically, these types of equations first arose in the study of 2D
| | == Conservation laws, symmetries, and criticality == |
| shallow wave propagation, but have since appeared as limiting cases of many
| |
| dispersive models. Interestingly, the 2D shallow wave equation can also
| |
| give rise to the <span class=SpellE>Boussinesq</span> or <a
| |
| href="schrodinger.html#Cubic NLS on R">1D NLS-3</a> equation by making more
| |
| limiting assumptions (in particular, weak nonlinearity and slowly varying
| |
| amplitude). </p>
| |
|
| |
|
| <p>The x variable is usually assumed to live on the real line R (so there is
| | KdV-type equations on R or T always come with three conserved quantities: |
| some decay at infinity) or on the <span class=SpellE>torus</span> T (so the
| |
| data is periodic). The half-line has also been studied, as has the case
| |
| of periodic data with large period. It might be interesting to look at
| |
| whether the periodicity assumption can be perturbed (e.g. quasi-periodic data);
| |
| it is not clear whether the phenomena we see in the periodic problem are robust
| |
| under perturbations, or are number-theoretic <span class=SpellE>artefacts</span>
| |
| of perfect periodicity. </p>
| |
|
| |
|
| <p>When <span class=GramE>P(</span>u) = c u^{k+1}, then the equation is | | <center><math>Mass:=\int u dx, \| u \|_{L^2_x}^2 := \int u^2 dx, Hamiltonian:=\int u_x^2 - V(u) dx</math></center> |
| referred to as generalized <span class=SpellE>gKdV</span> of order k, or <span
| |
| class=SpellE>gKdV</span>-k. <span class=GramE>gKdV-1</span> is the
| |
| original <span class=SpellE>Korteweg</span> de <span class=SpellE>Vries</span>
| |
|
| |
|
| (<span class=SpellE>KdV</span>) equation, gKdV-2 is the modified <span
| | where <math>V</math> is a primitive of <math>P</math>. Note that the Hamiltonian is positive-definite in the [[defocussing]] cases (if u is real); thus the defocussing equations have a better chance of long-term existence. The mass has no definite sign and so is only useful in specific cases (e.g. perturbations of a <span class="SpellE">soliton</span>). |
| class=SpellE>KdV</span> (<span class=SpellE>mKdV</span>) equation. <span
| |
| class=SpellE>KdV</span> and <span class=SpellE>mKdV</span> are quite special,
| |
| being the only equations in this family which are completely <span
| |
| class=SpellE>integrable</span>. </p> | |
|
| |
|
| <p>If k is even, the sign of c is important. The c < 0 case is known
| | In general, the above three quantities are the only conserved quantities available, but the [[KdV]] and [[mKdV]] equations come with infinitely many more such conserved quantities due to their [[completely integrable]] nature. |
| as the <span class=SpellE>defocussing</span> case, while c > 0 is the <span
| |
| class=SpellE>focussing</span> case. When k is odd, the constant c can
| |
| always be scaled out, so we do not distinguish <span class=SpellE>focussing</span>
| |
|
| |
|
| and <span class=SpellE>defocussing</span> in this case. </p>
| | The [[critical]] (or scaling) regularity is |
|
| |
|
| <p>Drift terms <span class=SpellE>u_x</span> can be added, but they can be | | <center><math>s_c = \frac{1}{2} - \frac{2}{k}.</math></center> |
| subsumed into the polynomial <span class=GramE>P(</span>u) or eliminated by a <span
| |
| class=SpellE>Gallilean</span> transformation [except in the half-line case].
| |
| Indeed, one can freely insert or remove any term of the form a'(t) <span
| |
| class=SpellE>u_x</span> by shifting the x variable by <span class=GramE>a(</span>t),
| |
| which is especially useful for periodic higher-order <span class=SpellE>gKdV</span>
| |
|
| |
|
| equations (setting a'(t) equal to the mean of P(u(t))). </p> | | In particular, [[KdV]], [[mKdV]], and [[GKdV-3 equation|gKdV-3]] are [[subcritical]] with respect to <math>L^2</math>, [[GKdV-4 equation|gKdV-4]] is <math>L^2</math> [[critical]], and all the other equations are <math>L^2</math> [[supercritical]]. Generally speaking, the potential energy term <math>V(u)</math> can be pretty much ignored in the sub-critical equations, needs to be dealt with carefully in the critical equation, and can completely dominate the Hamiltonian in the super-critical equations (to the point that blowup occurs if the equation is not defocussing</span>. Note that <math>H^1</math> is always a sub-critical regularity. |
|
| |
|
| <p><span class=SpellE>KdV</span>-type equations on R or T always come with | | The [[dispersion relation]] <math>\tau = \xi^3</math> is always increasing, which means that singularities always propagate to the left. In fact, high frequencies propagate leftward at extremely high speeds, which causes a smoothing effect if there is some decay in the initial data (<math>L^2</math> will do). On the other hand, KdV-type equations have the remarkable property of supporting localized <span class="SpellE">travelling</span> wave solutions known as [[soliton]]s, which propagate to the right. It is known that solutions to the [[completely integrable]] equations (i.e. [[KdV]] and [[mKdV]] always resolve to a superposition of solitons as <math>t \rightarrow \infty</math>, but it is an interesting open question as to whether the same phenomenon occurs for the other KdV-type equations. |
| three conserved quantities: </p>
| |
|
| |
|
| <p align=center style='text-align:center'>Mass: \<span class=SpellE>int</span>
| | == Symplectic structure == |
| u <span class=SpellE>dx</span> <br>
| |
| L^2: \<span class=SpellE>int</span> u^2 <span class=SpellE>dx</span> <br>
| |
|
| |
|
| Hamiltonian: \<span class=SpellE>int</span> u_x^2 - <span class=GramE>V(</span>u)
| | A <span class="SpellE">KdV</span>-type equation can be viewed as a <span class="SpellE">symplectic</span> flow with the Hamiltonian defined above, and the <span class="SpellE">symplectic</span> form given by |
| <span class=SpellE>dx</span></p> | |
|
| |
|
| <p><span class=GramE>where</span> V is a primitive of P. Note that the | | <center><math>(u,v) := \int u \partial_x^{-1} v dx</math>.</center> |
| Hamiltonian is positive-definite in the <span class=SpellE>defocussing</span>
| |
| cases (if u is real); thus the <span class=SpellE>defocussing</span> equations
| |
| have a better chance of long-term existence. The mass has no
| |
| definite sign and so is only useful in specific cases (e.g. perturbations of a <span
| |
| class=SpellE>soliton</span>). </p>
| |
|
| |
|
| <p>In general, the above three quantities are the only conserved quantities | | Thus <math>H^{-1/2}</math> is the natural Hilbert space in which to study the <span class="SpellE">symplectic</span> geometry of these flows. Unfortunately, the [[gKdV|gKdV-k]] equations are only locally well-posed in <math>H^{-1/2}</math> when <math>k=1</math>. In fact, the [[Korteweg-de_Vries_equation| standard KdV equation]] is bi-hamiltonian. |
| available, but the <a href="#kdv"><span class=SpellE>KdV</span></a> and <a
| |
| href="#mkdv"><span class=SpellE>mKdV</span></a> equations come with infinitely
| |
| many more such conserved quantities due to their completely <span class=SpellE>integrable</span>
| |
| nature. </p>
| |
|
| |
|
| <p>The critical (or scaling) regularity is </p>
| |
|
| |
|
| <p align=center style='text-align:center'><span class=SpellE>s_c</span> = 1/2 - | | If <math>k</math> is even, the sign of <math>c</math> is important. The <math>c < 0</math> case is known as the <span class="SpellE">defocussing</span> case, while <math>c > 0</math> is the <span class="SpellE">focussing</span> case. When <math>k</math> is odd, the constant <math>c</math> can always be scaled out, so we do not distinguish <span class="SpellE">focussing</span> and <span class="SpellE">defocussing</span> in this case. |
| 2/k.</p>
| |
|
| |
|
| <p>In particular, <a href="#kdv"><span class=SpellE>KdV</span></a>, <a
| | ==Estimates== |
| href="#mkdv"><span class=SpellE>mKdV</span></a>, and gKdV-3 are <span
| |
| class=SpellE>subcritical</span> with respect to L^2, gKdV-4 is L^2 critical,
| |
| and all the other equations are L^2 supercritical. Generally speaking,
| |
| the potential energy term V(u) can be pretty much ignored in the sub-critical
| |
| equations, needs to be dealt with carefully in the critical equation, and can
| |
| completely dominate the Hamiltonian in the super-critical equations (to the
| |
| point that blowup occurs if the equation is not <span class=SpellE>defocussing</span>).
| |
| Note that H^1 is always a sub-critical regularity. </p>
| |
|
| |
|
| <p>The dispersion relation \<span class=SpellE>tau</span> = \xi^3 is always
| | The perturbation theory for the KdV-type equations rests on a number of [[Linear Airy estimates|linear]], [[Bilinear Airy estimates|bilinear]], [[Trilinear Airy estimates|trilinear]], or [[Multilinear Airy estimates|multilinear]] estimates for the [[Airy equation]]. These estimates involve a number of function space norms, such as the [[X^s,b spaces]]. See the page on [[Airy estimates]] for more details. |
| increasing, which means that singularities always propagate to the left.
| |
| In fact, high frequencies propagate leftward at extremely high speeds, which
| |
| causes a smoothing effect if there is some decay in the initial data (L^2 will
| |
| do). On the other hand, <span class=SpellE>KdV</span>-type equations have
| |
| the remarkable property of supporting localized <span class=SpellE>travelling</span> | |
| wave solutions known as <span class=SpellE>solitons</span>, which propagate to
| |
| the right. It is known that solutions to the completely <span
| |
| class=SpellE>integrable</span> equations (i.e. <span class=SpellE>KdV</span>
| |
|
| |
|
| and <span class=SpellE>mKdV</span>) always resolve to a superposition of <span
| | [[Category:Airy]] |
| class=SpellE>solitons</span> as t -> infinity, but it is an interesting open
| | [[Category:Equations]] |
| question as to whether the same phenomenon occurs for the other <span
| |
| class=SpellE>KdV</span>-type equations. </p>
| |
| | |
| <p>A <span class=SpellE>KdV</span>-type equation can be viewed as a <span
| |
| class=SpellE>symplectic</span> flow with the Hamiltonian defined above, and the
| |
| | |
| <span class=SpellE>symplectic</span> form given by </p>
| |
| | |
| <p align=center style='text-align:center'>{<span class=GramE>u</span>, v} := \<span
| |
| class=SpellE>int</span> u <span class=SpellE>v_x</span> <span class=SpellE>dx</span>.</p>
| |
| | |
| <p>Thus H<span class=GramE>^{</span>-1/2} is the natural Hilbert space in which
| |
| to study the <span class=SpellE>symplectic</span> geometry of these
| |
| flows. Unfortunately, the <span class=SpellE>gKdV</span>-k equations are
| |
| only locally well-posed in H<span class=GramE>^{</span>-1/2} when k=1. </p>
| |
| | |
| <div class=MsoNormal align=center style='text-align:center'>
| |
| | |
| <hr size=2 width="100%" align=center>
| |
| | |
| </div>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><a name=Airy></a><b>Airy
| |
| estimates</b></p>
| |
| | |
| <p>Solutions to the Airy equation and its perturbations are either estimated in
| |
| mixed space-time norms <span class=SpellE>L^q_t</span> <span class=SpellE>L^r_x</span>,
| |
| <span class=SpellE>L^r_x</span> <span class=SpellE>L^q_t</span>, or in X^{<span
| |
| class=SpellE>s<span class=GramE>,b</span></span>} spaces, defined by </p>
| |
| | |
| <p align=center style='text-align:center'><tt><span style='font-size:10.0pt'>||
| |
| u ||_{<span class=SpellE>s<span class=GramE>,b</span></span>} = ||
| |
| <xi>^s <tau-xi^3>^b \hat{u} ||_2.</span></tt></p>
| |
| | |
| <p>Linear space-time estimates in which the space norm is evaluated first are
| |
| known as <a href="#kdv_linear"><span class=SpellE>Strichartz</span> estimates</a>,
| |
| but these estimates only play a minor role in the theory. A more
| |
| important category of linear estimates are the smoothing estimates and maximal
| |
| function estimates. The X^{<span class=SpellE>s<span
| |
| class=GramE>,b</span></span>} spaces are used primarily for <a
| |
| href="#kdv_bilinear">bilinear estimates</a>, although more recently <a
| |
| href="#KdV_multilinear"><span class=SpellE>multilinear</span> estimates have
| |
| begun to appear</a>. These spaces and estimates first appear in the
| |
| context of the <span class=SpellE>Schrodinger</span> equation in [<a
| |
| href="references.html#Bo1993b">Bo1993b</a>], although the analogues spaces for
| |
| the wave equation appeared earlier [<a href="references.html#RaRe1982">RaRe1982</a>],
| |
| [<a href="references.html#Be1983">Be1983</a>] in the context of <span
| |
| class=SpellE>propogation</span> of singularities. See also [<a
| |
| href="references.html#Bo1993">Bo1993</a>], [<a href="references.html#KlMa1993">KlMa1993</a>].
| |
| | |
| </p>
| |
| | |
| <div class=MsoNormal align=center style='text-align:center'>
| |
| | |
| <hr size=2 width="100%" align=center>
| |
| | |
| </div>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><a name="kdv_linear"></a><b>Linear
| |
| Airy estimates</b></p> | |
| | |
| <ul type=disc>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l4 level1 lfo2;tab-stops:list .5in'>If u is in X^{0,1/2+} on <b>R</b>,
| |
| then</li>
| |
| | |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l4 level2 lfo2;tab-stops:list 1.0in'>u is in L^\<span
| |
| class=SpellE>infty_t</span> L^2_x (energy estimate)</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l4 level2 lfo2;tab-stops:list 1.0in'><span class=SpellE>D_x</span>^{1/4}
| |
| u is in L^4_t <span class=SpellE>BMO_x</span> (endpoint <span
| |
| class=SpellE>Strichartz</span>) [<a href="references.html#KnPoVe1993">KnPoVe1993</a>]</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l4 level2 lfo2;tab-stops:list 1.0in'><span class=SpellE>D_x</span>
| |
| u is in L^\<span class=SpellE>infty_x</span> L^2_t (sharp Kato smoothing
| |
| effect) [<a href="references.html#KnPoVe1993">KnPoVe1993</a>].
| |
| Earlier versions of this estimate were obtained in [<a
| |
| href="references.html#Ka1979b">Ka1979b</a>], [<a
| |
| href="references.html#KrFa1983">KrFa1983</a>].</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l4 level2 lfo2;tab-stops:list 1.0in'><span class=SpellE>D_x</span>^{-1/4}
| |
| u is in L^4_x L^\<span class=SpellE>infty_t</span> (Maximal function) [<a
| |
| href="references.html#KnPoVe1993">KnPoVe1993</a>], [<a
| |
| href="references.html#KnRu1983">KnRu1983</a>]</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l4 level2 lfo2;tab-stops:list 1.0in'><span class=SpellE>D_x</span>^{-3/4-}
| |
| u is in L^2_x L^\<span class=SpellE>infty_t</span> (L^2 maximal function)
| |
| [<a href="references.html#KnPoVe1993">KnPoVe1993</a>]</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l4 level2 lfo2;tab-stops:list 1.0in'><i>Remark</i>: Further
| |
| estimates are available by <span class=SpellE>Sobolev</span>,
| |
| differentiation, Holder, and interpolation. For instance:</li>
| |
| <ul type=square>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l4 level3 lfo2;tab-stops:list 1.5in'><span class=SpellE>D_x</span>
| |
| u is in L^2_{<span class=SpellE>x,t</span>} locally in space [<a
| |
| href="references.html#Ka1979b">Ka1979b</a>] - use Kato and Holder (can
| |
| also be proven directly by integration by parts)</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l4 level3 lfo2;tab-stops:list 1.5in'>u is in L^2_{<span
| |
| class=SpellE>x,t</span>} locally in time - use energy and Holder</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l4 level3 lfo2;tab-stops:list 1.5in'><span class=SpellE>D_x</span>^{3/4-}
| |
| u is in L^8_x L^2_t locally in time - interpolate previous with Kato</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l4 level3 lfo2;tab-stops:list 1.5in'><span class=SpellE>D_x</span>^{1/6}
| |
| u is in L^6_{<span class=SpellE>x,t</span>} - interpolate energy with
| |
| endpoint <span class=SpellE>Strichartz</span> (or Kato with maximal)</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l4 level3 lfo2;tab-stops:list 1.5in'><span class=SpellE>D_x</span><span
| |
| class=GramE>^{</span>1/8} u is in L^8_t L^4_x - interpolate energy with
| |
| endpoint <span class=SpellE>Strichartz</span>. (In particular, <span
| |
| class=SpellE>D_x</span><span class=GramE>^{</span>1/8} u is also in
| |
| L^4_{<span class=SpellE>x,t</span>}).</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l4 level3 lfo2;tab-stops:list 1.5in'>u is in L^8_{<span
| |
| class=SpellE>x,t</span>}- use previous and <span class=SpellE>Sobolev</span>
| |
| | |
| in space</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l4 level3 lfo2;tab-stops:list 1.5in'>If u is in
| |
| X^{0,1/3+}, then u is in L^4_{<span class=SpellE>x,t</span>} [<a
| |
| href="references.html#Bo1993b">Bo1993b</a>] - interpolate previous with
| |
| the trivial identity X^{0,0} = L^2</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l4 level3 lfo2;tab-stops:list 1.5in'>If u is in
| |
| X^{0,1/4+}, then <span class=SpellE>D_x</span>^{1/2} u is in L^4_x L^2_t
| |
| [<a href="references.html#Bo1993b">Bo1993b</a>] - interpolate Kato with
| |
| X^{0,0} = L^2</li>
| |
| | |
| </ul>
| |
| </ul>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l4 level1 lfo2;tab-stops:list .5in'>If u is in X^{0,1/2+} on <b>T</b>,
| |
| then</li>
| |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l4 level2 lfo2;tab-stops:list 1.0in'><span class=GramE>u</span>
| |
| is in L^\<span class=SpellE>infty_t</span> L^2_x (energy estimate).
| |
| | |
| This is also true in the large period case.</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l4 level2 lfo2;tab-stops:list 1.0in'><span class=GramE>u</span>
| |
| is in L^4_{<span class=SpellE>x,t</span>} locally in time (in fact one
| |
| only needs u in X^{0,1/3} for this) [<a href="references.html#Bo1993b">Bo1993b</a>].</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l4 level2 lfo2;tab-stops:list 1.0in'><span class=SpellE>D_x</span><span
| |
| class=GramE>^{</span>-\<span class=SpellE>eps</span>} u is in L^6_{<span
| |
| class=SpellE>x,t</span>} locally in time. [<a
| |
| href="references.html#Bo1993b">Bo1993b</a>]. It is conjectured that
| |
| this can be improved to L^8_{<span class=SpellE>x<span class=GramE>,t</span></span>}.</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l4 level2 lfo2;tab-stops:list 1.0in'><i>Remark</i>: there
| |
| is no smoothing on the circle, so one can never gain regularity.</li>
| |
| </ul>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l4 level1 lfo2;tab-stops:list .5in'>If u is in X^{0,1/2} on a
| |
| circle with large period \lambda, then</li>
| |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l4 level2 lfo2;tab-stops:list 1.0in'><span class=GramE>u</span>
| |
| is in L^4_{<span class=SpellE>x,t</span>} locally in time, with a bound
| |
| of \lambda^{0+}.</li>
| |
| | |
| <ul type=square>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l4 level3 lfo2;tab-stops:list 1.5in'>In fact, when u has frequency
| |
| N, the constant is like \lambda^{0+} (N<span class=GramE>^{</span>-1/8}
| |
| + \lambda^{-1/4}), which recovers the 1/8 smoothing effect on the line
| |
| in L^4 mentioned earlier. [<a href="references.html#CoKeStTaTk-p2">CoKeStTkTa-p2</a>]</li>
| |
| </ul>
| |
| </ul>
| |
| </ul>
| |
| | |
| <h4 align=center style='text-align:center'>
| |
| | |
| <hr size=2 width="100%" align=center>
| |
| | |
| </h4>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><a name="kdv_bilinear"></a><b>Bilinear
| |
| Airy estimates</b></p>
| |
| | |
| <ul type=disc>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l23 level1 lfo3;tab-stops:list .5in'>The key algebraic fact is</li>
| |
| </ul>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><tt><span
| |
| style='font-size:10.0pt'>\xi_1^3 + \xi_2^3 + \xi_3^3 = 3 \xi_1 \xi_2 \xi_3</span></tt>
| |
| <br>
| |
| <tt><span style='font-size:10.0pt'>(whenever \xi_1 + \xi_2 + \xi_3 = 0)</span></tt></p>
| |
| | |
| <ul type=disc>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l11 level1 lfo4;tab-stops:list .5in'>The -3/4+ estimate [<a
| |
| href="references.html#KnPoVe1996">KnPoVe1996</a>] on <b>R</b>:</li>
| |
| </ul>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><tt><span
| |
| style='font-size:10.0pt'>|| (<span class=SpellE>uv</span><span class=GramE>)_</span>x
| |
| ||_{-3/4+, -1/2+} <~ || u ||_{-3/4+, 1/2+} || v ||_{-3/4+, 1/2+}</span></tt></p>
| |
| | |
| <ul type=disc>
| |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l10 level2 lfo5;tab-stops:list 1.0in'>The above estimate
| |
| fails at the endpoint -3/4. [<a href="references.html#NaTkTs-p">NaTkTs2001</a>]</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l10 level2 lfo5;tab-stops:list 1.0in'>As a corollary of
| |
| this estimate we have the -3/8+ estimate [<a
| |
| href="references.html#CoStTk1999">CoStTk1999</a>] on <b>R</b>: If u and v
| |
| have no low frequencies ( |\xi| <~ 1 ) then</li>
| |
| | |
| </ul>
| |
| </ul>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><tt><span
| |
| style='font-size:10.0pt'>|| (<span class=SpellE>uv</span><span class=GramE>)_</span>x
| |
| ||_{0, -1/2+} <~ || u ||_{-3/8+, 1/2+} || v ||_{-3/8+, 1/2+}</span></tt></p>
| |
| | |
| <ul type=disc>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l27 level1 lfo6;tab-stops:list .5in'>The -1/2 estimate [<a
| |
| href="references.html#KnPoVe1996">KnPoVe1996</a>] on <b>T</b>: if <span
| |
| class=SpellE>u,v</span> have mean zero, then for all s >= -1/2</li>
| |
| | |
| </ul>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><tt><span
| |
| style='font-size:10.0pt'>|| (<span class=SpellE>uv</span><span class=GramE>)_</span>x
| |
| ||_{s, -1/2} <~ || u ||_{s, 1/2} || v ||_{s, 1/2}</span></tt></p>
| |
| | |
| <ul type=disc>
| |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l12 level2 lfo7;tab-stops:list 1.0in'>The above estimate
| |
| fails for s < -1/2. Also, one cannot replace 1/2, -1/2 by 1/2+,
| |
| -1/2+. [<a href="references.html#KnPoVe1996">KnPoVe1996</a>]</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l12 level2 lfo7;tab-stops:list 1.0in'>This estimate also
| |
| holds in the large period case if one is willing to lose a power of
| |
| \lambda^{0+} in the constant. [<a href="references.html#CoKeStTaTk-p2">CoKeStTkTa-p2</a>]</li>
| |
| </ul>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l12 level1 lfo7;tab-stops:list .5in'><i>Remark</i>: In principle,
| |
| a complete list of bilinear estimates could be obtained from [<a
| |
| href="references.html#Ta-p2">Ta-p2</a>].</li>
| |
| </ul>
| |
| | |
| <div class=MsoNormal align=center style='text-align:center'>
| |
| | |
| <hr size=2 width="100%" align=center>
| |
| | |
| </div>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><a
| |
| name="kdv_trilinear"></a><span class=SpellE><b>Trilinear</b></span><b> Airy
| |
| estimates</b></p>
| |
| | |
| <ul type=disc>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l3 level1 lfo8;tab-stops:list .5in'>The key algebraic fact is
| |
| (various permutations of)</li>
| |
| </ul>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><tt><span
| |
| style='font-size:10.0pt'>\xi_1^3 + \xi_2^3 + \xi_3^3 + \xi_4^3 = 3
| |
| (\xi_1+\xi_4) (\xi_2+\xi_4) (\xi_3+\xi_4)</span></tt> <br>
| |
| | |
| <tt><span style='font-size:10.0pt'>(whenever \xi_1 + \xi_2 + \xi_3 + \xi_4 = 0)</span></tt></p>
| |
| | |
| <ul type=disc>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l16 level1 lfo9;tab-stops:list .5in'>The 1/4 estimate [<a
| |
| href="references.html#Ta-p2">Ta-p2</a>] on <b>R</b>:</li>
| |
| </ul>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><tt><span
| |
| style='font-size:10.0pt'>|| (<span class=SpellE>uvw</span><span class=GramE>)_</span>x
| |
| ||_{1/4, -1/2+} <~ || u ||_{1/4, 1/2+} || v ||_{1/4, 1/2+} || w
| |
| ||_{1/4, 1/2+}</span></tt></p>
| |
| | |
| <p class=MsoNormal style='mso-margin-top-alt:auto;margin-right:.5in;mso-margin-bottom-alt:
| |
| auto;margin-left:.5in'>The 1/4 is sharp [<a href="references.html#KnPoVe1996">KnPoVe1996</a>].<span
| |
| style='mso-spacerun:yes'> </span>We also have</p>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><tt><span
| |
| style='font-size:10.0pt'>|| <span class=SpellE><span class=GramE>uv<u>w</u></span></span>
| |
| ||_{-1/4, -5/12+} <~ || u ||_{-1/4, 7/12+} || v ||_{-1/4, 7/12+} || w
| |
| ||_{-1/4, 7/12+}</span></tt></p>
| |
| | |
| <p class=MsoNormal style='mso-margin-top-alt:auto;margin-right:.5in;mso-margin-bottom-alt:
| |
| auto;margin-left:.5in'><span class=GramE>see</span> [<span class=SpellE>Cv</span>-p].</p>
| |
| | |
| <ul type=disc>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l15 level1 lfo10;tab-stops:list .5in'>The 1/2 estimate [<a
| |
| href="references.html#CoKeStTaTk-p3">CoKeStTkTa-p3</a>] on <b>T</b>: if <span
| |
| class=SpellE>u,v,w</span> have mean zero, then</li>
| |
| | |
| </ul>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><tt><span
| |
| style='font-size:10.0pt'>|| (<span class=SpellE>uvw</span><span class=GramE>)_</span>x
| |
| ||_{1/2, -1/2} <~ || u ||_{1/2, 1/2*} || v ||_{1/2, 1/2*} || w
| |
| ||_{1/2, 1/2*}</span></tt></p>
| |
| | |
| <p class=MsoNormal style='mso-margin-top-alt:auto;margin-right:.5in;mso-margin-bottom-alt:
| |
| auto;margin-left:.5in'>The 1/2 is sharp [<a href="references.html#KnPoVe1996">KnPoVe1996</a>].</p>
| |
| | |
| <ul type=disc>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l28 level1 lfo11;tab-stops:list .5in'><i>Remark</i>: the <span
| |
| class=SpellE>trilinear</span> estimate always needs one more derivative of
| |
| regularity than the bilinear estimate; this is consistent with the heuristics
| |
| from the Miura transform from <span class=SpellE>mKdV</span> to <span
| |
| class=SpellE>KdV</span>.</li>
| |
| </ul>
| |
| | |
| <div class=MsoNormal align=center style='text-align:center'>
| |
| | |
| <hr size=2 width="100%" align=center>
| |
| | |
| </div>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><a
| |
| name="KdV_multilinear"></a><span class=SpellE><b>Multilinear</b></span><b> Airy
| |
| estimates</b></p>
| |
| | |
| <ul type=disc>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l13 level1 lfo12;tab-stops:list .5in'>We have the <span
| |
| class=SpellE>quintilinear</span> estimate on <b>R</b>: [<a
| |
| href="references.html#CoKeStTaTk-p2">CoKeStTkTa-p2</a>]</li>
| |
| | |
| </ul>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><tt><span
| |
| style='font-size:10.0pt'>\<span class=SpellE>int</span> u^3 v^2 <span
| |
| class=SpellE>dx</span> <span class=SpellE>dt</span> <~ || u ||<span
| |
| class=GramE>_{</span>1/4+,1/2+}^3 || v ||_{-3/4+,1/2+}^2</span></tt></p>
| |
| | |
| <ul type=disc>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l5 level1 lfo13;tab-stops:list .5in'>The analogue for this on <b>T</b>
| |
| | |
| is: [<a href="references.html#CoKeStTaTk-p2">CoKeStTkTa-p2</a>, <a
| |
| href="references.html#CoKeStTaTk-p3">CoKeStTkTa-p3</a>]</li>
| |
| </ul>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><tt><span
| |
| style='font-size:10.0pt'>\<span class=SpellE>int</span> u^3 v^2 <span
| |
| class=SpellE>dx</span> <span class=SpellE>dt</span> <~ || u ||<span
| |
| class=GramE>_{</span>1/2,1/2*}^3 || v ||_{-1/2,1/2*}^2</span></tt></p>
| |
| | |
| <p class=MsoNormal style='mso-margin-top-alt:auto;margin-right:.5in;mso-margin-bottom-alt:
| |
| auto;margin-left:.5in'>In fact, this estimate also holds for large period, but
| |
| a loss of lambda<span class=GramE>^{</span>0+}.</p>
| |
| | |
| <div class=MsoNormal align=center style='text-align:center'>
| |
| | |
| <hr size=2 width="100%" align=center>
| |
| | |
| </div>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><a name=kdv></a><b>The
| |
| <span class=SpellE>KdV</span> equation</b></p>
| |
| | |
| <p>The <span class=SpellE>KdV</span> equation is </p>
| |
| | |
| <p align=center style='text-align:center'><span class=SpellE>u_t</span> + <span
| |
| class=SpellE>u_xxx</span> + u <span class=SpellE>u_x</span> = 0.</p>
| |
| | |
| <p>It is completely <span class=SpellE>integrable</span>, and has infinitely
| |
| many conserved quantities. Indeed, for each non-negative integer k, there
| |
| is a conserved quantity which is roughly equivalent to the <span class=SpellE>H^k</span>
| |
| norm of u. </p>
| |
| | |
| <p>The <span class=SpellE>KdV</span> equation has been studied on the <a
| |
| href="#kdv_on_R">line</a>, the <a href="#kdv_on_T">circle</a>, and the <a
| |
| href="#KdV_on_R+">half-line</a>. </p>
| |
| | |
| <div class=MsoNormal align=center style='text-align:center'>
| |
| | |
| <hr size=2 width="100%" align=center>
| |
| | |
| </div>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><a name="kdv_on_R"></a><span
| |
| class=SpellE><b>KdV</b></span><b> on R</b></p>
| |
| | |
| <ul type=disc>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l17 level1 lfo14;tab-stops:list .5in'>Scaling is <span
| |
| class=SpellE>s_c</span> = -3/2.</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l17 level1 lfo14;tab-stops:list .5in'>LWP in <span class=SpellE>H^s</span>
| |
| for s >= -3/4 [<span class=SpellE>CtCoTa</span>-p], using a modified
| |
| Miura transform and the <a href="#mKdV_on_R"><span class=SpellE>mKdV</span>
| |
| theory</a>. This is despite the failure of the key bilinear estimate
| |
| [<a href="references.html#NaTkTs-p">NaTkTs2001</a>]</li>
| |
| | |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l17 level2 lfo14;tab-stops:list 1.0in'>For s within a
| |
| logarithm for s=-3/4 [<span class=SpellE>MurTao</span>-p].</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l17 level2 lfo14;tab-stops:list 1.0in'>Was proven for s
| |
| > -3/4 [<a href="references.html#KnPoVe1996">KnPoVe1996</a>].</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l17 level2 lfo14;tab-stops:list 1.0in'>Was proven for s
| |
| > -5/8 in [<a href="references.html#KnPoVe1993b">KnPoVe1993b</a>].</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l17 level2 lfo14;tab-stops:list 1.0in'>Was proven for s
| |
| >= 0 in [<a href="references.html#Bo1993b">Bo1993b</a>].</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l17 level2 lfo14;tab-stops:list 1.0in'>Was proven for s
| |
| > 3/4 in [<a href="references.html#KnPoVe1993">KnPoVe1993</a>].</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l17 level2 lfo14;tab-stops:list 1.0in'>Was proven for s
| |
| > 3/2 in [<a href="references.html#BnSm1975">BnSm1975</a>], [<a
| |
| href="references.html#Ka1975">Ka1975</a>], [<a
| |
| href="references.html#Ka1979">Ka1979</a>], [<a
| |
| href="references.html#GiTs1989">GiTs1989</a>], [<a
| |
| href="references.html#Bu1980">Bu1980</a>]<span class=GramE>, ....</span></li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l17 level2 lfo14;tab-stops:list 1.0in'>One has local ill-<span
| |
| class=SpellE><span class=GramE>posedness</span></span><span class=GramE>(</span>in
| |
| the sense that the map is not uniformly continuous) for s < -3/4 (in
| |
| the complex setting) by <span class=SpellE>soliton</span> examples [<a
| |
| href="references.html#KnPoVe-p"><span class=SpellE>KnPoVe</span>-p</a>].</li>
| |
| <ul type=square>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l17 level3 lfo14;tab-stops:list 1.5in'>For real <span
| |
| class=SpellE>KdV</span> this has also been established in [<span
| |
| class=SpellE>CtCoTa</span>-p], by the Miura transform and the <a
| |
| href="#mKdV_on_R">corresponding result for <span class=SpellE>mKdV</span></a>.</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l17 level3 lfo14;tab-stops:list 1.5in'>Below -3/4 the
| |
| solution map was known to not be C^3 [<a href="references.html#Bo1993b">Bo1993b</a>],
| |
| [<a href="references.html#Bo1997">Bo1997</a>]; this was refined to C^2
| |
| in [<a href="references.html#Tz1999b">Tz1999b</a>].</li>
| |
| </ul>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l17 level2 lfo14;tab-stops:list 1.0in'>When the initial
| |
| data is a real, rapidly decreasing measure one has a global smooth
| |
| solution for t > 0 [<a href="referencs.html#Kp1993">Kp1993</a>].
| |
| | |
| Without the rapidly decreasing hypothesis one can still construct a
| |
| global weak solution [<a href="references.html#Ts1989">Ts1989</a>]</li>
| |
| </ul>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l17 level1 lfo14;tab-stops:list .5in'>GWP in <span class=SpellE>H^s</span>
| |
| for s > -3/4 (if u is real) [<a href="references.html#CoKeStTaTk2003">CoKeStTkTa2003</a>].</li>
| |
| | |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l17 level2 lfo14;tab-stops:list 1.0in'>Was proven for s
| |
| > -3/10 in [<a href="references.html#CoKeStTkTa2001">CoKeStTkTa2001</a>]</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l17 level2 lfo14;tab-stops:list 1.0in'>Was proven for
| |
| s>= 0 in [<a href="references.html#Bo1993b">Bo1993b</a>]. Global
| |
| weak solutions in L^2 were constructed in [<a
| |
| href="references.html#Ka1983">Ka1983</a>], [<a
| |
| href="references.html#KrFa1983">KrFa1983</a>], and were shown to obey the
| |
| expected local smoothing estimate. These weak solutions were shown
| |
| to be unique in [<a href="references.html#Zh1997b">Zh1997b</a>]</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l17 level2 lfo14;tab-stops:list 1.0in'>Was proven for
| |
| s>= 1 in [<a href="references.html#KnPoVe1993">KnPoVe1993</a>].</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l17 level2 lfo14;tab-stops:list 1.0in'>Was proven for
| |
| s>= 2 in [<a href="references.html#BnSm1975">BnSm1975</a>], [<a
| |
| href="references.html#Ka1975">Ka1975</a>], [<a
| |
| href="references.html#Ka1979">Ka1979</a>]<span class=GramE>, ....</span></li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l17 level2 lfo14;tab-stops:list 1.0in'><i>Remark</i>: In
| |
| the complex setting GWP fails for large data with Fourier support on the
| |
| half-line [Bona/<span class=SpellE>Winther</span>?], [<span class=SpellE>Birnir</span>]<span
| |
| class=GramE>, ????</span>. This result extends to a wide class of
| |
| dispersive PDE.</li>
| |
| </ul>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l17 level1 lfo14;tab-stops:list .5in'>By use of the inverse
| |
| scattering transform one can show that smooth solutions eventually resolve
| |
| into <span class=SpellE>solitons</span>, that two colliding <span
| |
| class=SpellE>solitons</span> emerge as (slightly phase shifted) <span
| |
| class=SpellE>solitons</span>, etc.</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l17 level1 lfo14;tab-stops:list .5in'><span class=SpellE>Solitons</span>
| |
| are <span class=SpellE>orbitally</span> H^1 stable [<a
| |
| href="references.html#Bj1972">Bj1972</a>]</li>
| |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l17 level2 lfo14;tab-stops:list 1.0in'>In <span
| |
| class=SpellE>H^s</span>, 0 <= s < 1, the orbital stability of <span
| |
| class=SpellE>solitons</span> is at most polynomial (the distance to the
| |
| ground state manifold in <span class=SpellE>H^s</span> norm grows like at
| |
| most O(t^{1-s+}) in time) [<span class=SpellE>RaySt</span>-p]</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l17 level2 lfo14;tab-stops:list 1.0in'>In L^2, orbital
| |
| stability has been obtained in [<a href="references.html#MeVe2003">MeVe2003</a>].</li>
| |
| </ul>
| |
| </ul>
| |
| | |
| <p class=MsoNormal>The <span class=SpellE>KdV</span> equation can also be
| |
| generalized to a 2x2 system </p>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><span class=SpellE><tt><span
| |
| style='font-size:10.0pt'>u_t</span></tt></span><tt><span style='font-size:10.0pt'>
| |
| | |
| + <span class=SpellE>u_xxx</span> + a_3 <span class=SpellE>v_xxx</span> + u <span
| |
| class=SpellE>u_x</span> + a_1 v <span class=SpellE>v_x</span> + a_2 (<span
| |
| class=SpellE>uv</span><span class=GramE>)_</span>x = 0</span></tt> <br>
| |
| | |
| <tt><span style='font-size:10.0pt'>b_1 <span class=SpellE>v_t</span> + <span
| |
| class=SpellE>v_xxx</span> + b_2 a_3 <span class=SpellE>u_xxx</span> + v <span
| |
| class=SpellE>v_x</span> + b_2 a_2 u <span class=SpellE>u_x</span> + b_2 a_1 (<span
| |
| class=SpellE>uv</span>)_x + r <span class=SpellE>v_x</span></span></tt></p>
| |
| | |
| <p class=MsoNormal><span class=GramE>where</span> b_1,b_2 are positive
| |
| constants and a_1,a_2,a_3,r are real constants. This system was
| |
| introduced in [<a href="references.html#GeaGr1984">GeaGr1984</a>] to study
| |
| strongly interacting pairs of weakly nonlinear long waves, and studied further
| |
| in [<a href="references.html#BnPoSauTm1992">BnPoSauTm1992</a>]. In [<a
| |
| href="references.html#AsCoeWgg1996">AsCoeWgg1996</a>] it was shown that this
| |
| system was also globally well-posed on L^2. <br>
| |
| It is an interesting question as to whether these results can be pushed further
| |
| to match the <span class=SpellE>KdV</span> theory; the apparent lack of
| |
| complete <span class=SpellE>integrability</span> in this system (for generic
| |
| choices of parameters <span class=SpellE>b_i</span>, <span class=SpellE>a_i</span>,
| |
| | |
| <span class=GramE>r</span>) suggests a possible difficulty. </p>
| |
| | |
| <div class=MsoNormal align=center style='text-align:center'>
| |
| | |
| <hr size=2 width="100%" align=center>
| |
| | |
| </div>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><a name="KdV_on_R+"></a><span
| |
| class=SpellE><b>KdV</b></span><b> on R^+</b></p>
| |
| | |
| <ul type=disc>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l29 level1 lfo15;tab-stops:list .5in'>The <span class=SpellE>KdV</span>
| |
| | |
| Cauchy-boundary problem on the half-line is</li>
| |
| </ul>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><span class=SpellE>u_t</span>
| |
| + u<span class=GramE>_{</span>xxx} + <span class=SpellE>u_x</span> + u <span
| |
| class=SpellE>u_x</span> = 0; u(x,0) = u_0(x); u(0,t) = h(t)</p>
| |
| | |
| <p class=MsoNormal style='margin-left:.5in'>The sign of u<span class=GramE>_{</span>xxx}
| |
| is important (it makes the influence of the boundary x=0 mostly negligible),
| |
| the sign of u <span class=SpellE>u_x</span> is not. The drift term <span
| |
| class=SpellE>u_x</span> appears naturally from the derivation of <span
| |
| class=SpellE>KdV</span> from fluid mechanics. (On R, this drift term can
| |
| be eliminated by a <span class=SpellE>Gallilean</span> transform, but this is
| |
| not available on the half-line). </p>
| |
| | |
| <ul type=disc>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l8 level1 lfo16;tab-stops:list .5in'>Because one is restricted to
| |
| the half-line, it becomes a little tricky to use the Fourier
| |
| transform. One approach is to use the Fourier-<span class=SpellE>Laplace</span>
| |
| transform instead.</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l8 level1 lfo16;tab-stops:list .5in'>Some compatibility conditions
| |
| between u_0 and h are needed. The higher the regularity, the more
| |
| compatibility conditions are needed. If the initial data u_0 is in <span
| |
| class=SpellE>H^s</span>, then by scaling heuristics the natural space for
| |
| h is in H<span class=GramE>^{</span>(s+1)/3}. (Remember that time
| |
| has dimensions <i>length</i>^3).</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l8 level1 lfo16;tab-stops:list .5in'>LWP is known for initial
| |
| data in <span class=SpellE>H^s</span> and boundary data in H<span
| |
| class=GramE>^{</span>(s+1)/3} for s >= 0 [<span class=SpellE>CoKe</span>-p],
| |
| assuming compatibility. The drift term may be omitted because of the
| |
| time localization.</li>
| |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l8 level2 lfo16;tab-stops:list 1.0in'>For s > 3/4 this
| |
| was proven in [<a href="references.html#BnSuZh-p"><span class=SpellE>BnSuZh</span>-p</a>]
| |
| (assuming that there is a drift term).</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l8 level2 lfo16;tab-stops:list 1.0in'>Was proven for data
| |
| in sufficiently weighted H^1 spaces in [<a href="references.html#Fa1983">Fa1983</a>].</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l8 level2 lfo16;tab-stops:list 1.0in'>From the real line
| |
| theory one might expect to lower this to -3/4, but there appear to be
| |
| technical difficulties with this.</li>
| |
| </ul>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l8 level1 lfo16;tab-stops:list .5in'>GWP is known for initial
| |
| data in L^2 and boundary data in H<span class=GramE>^{</span>7/12},
| |
| assuming compatibility.</li>
| |
| <ul type=circle>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l8 level2 lfo16;tab-stops:list 1.0in'>for initial data in
| |
| H^1 and boundary data in H^{5/6}_loc this was proven in [<a
| |
| href="references.html#BnSuZh-p"><span class=SpellE>BnSuZh</span>-p</a>]</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l8 level2 lfo16;tab-stops:list 1.0in'>Was proven for smooth
| |
| data in [<a href="references.html#BnWi1983">BnWi1983</a>]</li>
| |
| </ul>
| |
| </ul>
| |
| | |
| <div class=MsoNormal align=center style='text-align:center'>
| |
| | |
| <hr size=2 width="100%" align=center>
| |
| | |
| </div>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><a name="kdv_on_T"></a><span
| |
| class=SpellE><b>KdV</b></span><b> on T</b></p>
| |
| | |
| <ul type=disc>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l6 level1 lfo17;tab-stops:list .5in'>Scaling is <span
| |
| class=SpellE>s_c</span> = -3/2.</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l6 level1 lfo17;tab-stops:list .5in'>C^0 LWP in <span
| |
| class=SpellE>H^s</span> for s >= -1, assuming u is real [<span
| |
| class=SpellE>KpTp</span>-p]</li>
| |
| | |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l6 level2 lfo17;tab-stops:list 1.0in'>C^0 LWP in <span
| |
| class=SpellE>H^s</span> for s >= -5/8 follows (at least in principle)
| |
| from work on the <span class=SpellE>mKdV</span> equation by [Takaoka and <span
| |
| class=SpellE>Tsutsumi</span>?]</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l6 level2 lfo17;tab-stops:list 1.0in'>Analytic LWP in <span
| |
| class=SpellE>H^s</span> for s >= -1/2, in the complex case [<a
| |
| href="references.html#KnPoVe1996">KnPoVe1996</a>]. In addition to
| |
| the usual bilinear estimate, one needs a linear estimate to keep the
| |
| solution in <span class=SpellE>H^s</span> for t>0.</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l6 level2 lfo17;tab-stops:list 1.0in'>Analytic LWP was
| |
| proven for s >= 0 in [<a href="references.html#Bo1993b">Bo1993b</a>].</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l6 level2 lfo17;tab-stops:list 1.0in'>Analytic ill <span
| |
| class=SpellE>posedness</span> at s<-1/2, even in the real case [<a
| |
| href="references.html#Bo1997">Bo1997</a>]</li>
| |
| <ul type=square>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l6 level3 lfo17;tab-stops:list 1.5in'>This has been
| |
| refined to failure of uniform continuity at s<-1/2 [<span
| |
| class=SpellE>CtCoTa</span>-p]</li>
| |
| </ul>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l6 level2 lfo17;tab-stops:list 1.0in'>Remark: s=-1/2 is the
| |
| <span class=SpellE>symplectic</span> regularity, and so the machinery of
| |
| infinite-dimensional <span class=SpellE>symplectic</span> geometry
| |
| applies once one has a continuous flow, although there are some
| |
| technicalities involving approximating <span class=SpellE>KdV</span> by a
| |
| suitable <span class=SpellE>symplectic</span> finite-dimensional
| |
| flow. In particular one has <span class=SpellE>symplectic</span>
| |
| | |
| non-squeezing [CoKeStTkTa-p9], [<a href="references.html#Bo1999">Bo1999</a>].</li>
| |
| </ul>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l6 level1 lfo17;tab-stops:list .5in'>C^0 GWP in <span
| |
| class=SpellE>H^s</span> for s >= -1, in the real case [<span
| |
| class=SpellE>KpTp</span>-p].</li>
| |
| <ul type=circle>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l6 level2 lfo17;tab-stops:list 1.0in'>Analytic GWP in <span
| |
| class=SpellE>H^s</span> in the real case for s >= -1/2 [<a
| |
| href="references.html#CoKeStTaTk-p2">CoKeStTkTa-p2</a>]; see also [<a
| |
| href="references.html#CoKeStTaTk-p3">CoKeStTkTa-p3</a>].</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l6 level2 lfo17;tab-stops:list 1.0in'>A short proof for the
| |
| s > -3/10 case is in [<a href="references.html#CoKeStTaTk-p2a">CoKeStTkTa-p2a</a>]</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l6 level2 lfo17;tab-stops:list 1.0in'>Was proven for s
| |
| >= 0 in [<a href="references.html#Bo1993b">Bo1993b</a>].</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l6 level2 lfo17;tab-stops:list 1.0in'>GWP for real initial
| |
| data which are measures of small norm [<a href="references.html#Bo1997">Bo1997</a>]
| |
| <span class=GramE>The</span> small norm restriction is presumably
| |
| technical.</li>
| |
| <ul type=square>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l6 level3 lfo17;tab-stops:list 1.5in'><i>Remark</i>:
| |
| measures have the same scaling as H<span class=GramE>^{</span>-1/2}, but
| |
| neither space includes the other. (Measures are in H<span
| |
| class=GramE>^{</span>-1/2-\eps} though).</li>
| |
| | |
| </ul>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l6 level2 lfo17;tab-stops:list 1.0in'>One has GWP for real
| |
| random data whose Fourier coefficients decay like 1/|k| (times a Gaussian
| |
| random variable) [<a href="references.html#Bo1995c">Bo1995c</a>].
| |
| Indeed one has an invariant measure.</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l6 level2 lfo17;tab-stops:list 1.0in'><span class=SpellE>Solitons</span>
| |
| are asymptotically H^1 stable [MtMe-p3], [<span class=SpellE>MtMe</span>-p].<span
| |
| style='mso-spacerun:yes'> </span>Indeed, the solution decouples into a
| |
| finite sum of <span class=SpellE>solitons</span> plus dispersive
| |
| radiation [<a href="references.html#EckShr1988">EckShr1988</a>]</li>
| |
| | |
| </ul>
| |
| </ul>
| |
| | |
| <div class=MsoNormal align=center style='text-align:center'>
| |
| | |
| <hr size=2 width="100%" align=center>
| |
| | |
| </div>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><a name=mkdv></a><b>The
| |
| modified <span class=SpellE>KdV</span> equation</b></p>
| |
| | |
| <p>The (<span class=SpellE>defocussing</span>) <span class=SpellE>mKdV</span>
| |
| | |
| equation is </p>
| |
| | |
| <p align=center style='text-align:center'><span class=SpellE>u_t</span> + <span
| |
| class=SpellE>u_xxx</span> = 6 u^2 <span class=SpellE>u_x</span>.</p>
| |
| | |
| <p>It is completely <span class=SpellE>integrable</span>, and has infinitely
| |
| many conserved quantities. Indeed, for each non-negative integer k, there
| |
| is a conserved quantity which is roughly equivalent to the <span class=SpellE>H^k</span>
| |
| | |
| norm of u. This equation has been studied on the <a href="#mKdV_on_R">line</a>,
| |
| <a href="#mKdV_on_T">circle</a>, and <a href="#gKdV_on_R+">half-line</a>. </p>
| |
| | |
| <p>The <i>Miura transformation</i> v = <span class=SpellE>u_x</span> + u^2
| |
| transforms a solution of <span class=SpellE>defocussing</span> <span
| |
| class=SpellE>mKdV</span> to a solution of <a href="#kdv"><span class=SpellE>KdV</span></a>
| |
| | |
| </p>
| |
| | |
| <p align=center style='text-align:center'><span class=SpellE>v_t</span> + <span
| |
| class=SpellE>v_xxx</span> = 6 v <span class=SpellE>v_x</span>.</p>
| |
| | |
| <p>Thus one expects the LWP and GWP theory for <span class=SpellE>mKdV</span>
| |
| to be one derivative higher than that for <span class=SpellE>KdV</span>. </p>
| |
| | |
| <p>The <span class=SpellE>focussing</span> <span class=SpellE>mKdV</span> </p>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><span class=SpellE>u_t</span>
| |
| + <span class=SpellE>u_xxx</span> = - 6 u^2 <span class=SpellE>u_x</span></p>
| |
| | |
| <p><span class=GramE>is</span> very similar, except that the Miura transform is
| |
| now v = <span class=SpellE>u_x</span> + <span class=SpellE>i</span> u^2.
| |
| This transforms <span class=SpellE>focussing</span> <span class=SpellE>mKdV</span>
| |
| to <i>complex-valued</i> <span class=SpellE>KdV</span>, which is a slightly
| |
| less tractable equation. (However, the transformed solution v is still
| |
| real in the highest order term, so in principle the real-valued theory carries
| |
| over to this case). </p>
| |
| | |
| <p>The Miura transformation can be generalized. If v and w solve the
| |
| system </p>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><span class=SpellE>v_t</span>
| |
| + <span class=SpellE>v_xxx</span> = 6(v^2 + w) <span class=SpellE>v_x</span> <br>
| |
| <span class=SpellE>w_t</span> + <span class=SpellE>w_xxx</span> = 6(v^2 + w) <span
| |
| class=SpellE>w_x</span></p>
| |
| | |
| <p class=MsoNormal>Then u = v^2 + <span class=SpellE>v_x</span> + w is a
| |
| solution of <span class=SpellE>KdV</span>. In particular, if a and b are
| |
| constants and v solves </p>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><span class=SpellE>v_t</span>
| |
| + <span class=SpellE>v_xxx</span> = 6(a^2 v^2 + <span class=SpellE><span
| |
| class=GramE>bv</span></span>) <span class=SpellE>v_x</span></p>
| |
| | |
| <p class=MsoNormal><span class=GramE>then</span> u = a^2 v^2 + <span
| |
| class=SpellE>av_x</span> + <span class=SpellE>bv</span> solves <span
| |
| class=SpellE>KdV</span> (this is the <i>Gardener transform</i>). </p>
| |
| | |
| <div class=MsoNormal align=center style='text-align:center'>
| |
| | |
| <hr size=2 width="100%" align=center>
| |
| | |
| </div>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><a name="mKdV_on_R"></a><span
| |
| class=SpellE><span class=GramE><b>mKdV</b></span></span><b> on R and R^+</b></p>
| |
| | |
| <ul type=disc>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l18 level1 lfo18;tab-stops:list .5in'>Scaling is <span
| |
| class=SpellE>s_c</span> = -1/2.</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l18 level1 lfo18;tab-stops:list .5in'>LWP in <span class=SpellE>H^s</span>
| |
| for s >= 1/4 [<a href="references.html#KnPoVe1993">KnPoVe1993</a>]</li>
| |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l18 level2 lfo18;tab-stops:list 1.0in'>Was shown for
| |
| s>3/2 in [<a href="references.html#GiTs1989">GiTs1989</a>]</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l18 level2 lfo18;tab-stops:list 1.0in'>This is sharp in the
| |
| <span class=SpellE>focussing</span> case [<a
| |
| href="references.html#KnPoVe-p"><span class=SpellE>KnPoVe</span>-p</a>],
| |
| in the sense that the solution map is no longer uniformly continuous for
| |
| s < 1/4.</li>
| |
| <ul type=square>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l18 level3 lfo18;tab-stops:list 1.5in'>This has been
| |
| extended to the <span class=SpellE>defocussing</span> case in [<span
| |
| class=SpellE>CtCoTa</span>-p], by a high-frequency approximation of <span
| |
| class=SpellE>mKdV</span> by <a href="schrodinger.html#Cubic NLS on R">NLS</a>.
| |
| | |
| (This high frequency approximation has also been utilized in [<a
| |
| href="references.html#Sch1998">Sch1998</a>]).</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l18 level3 lfo18;tab-stops:list 1.5in'>Below 1/4 the
| |
| solution map was known to not be C^3 in [<a
| |
| href="references.html#Bo1993b">Bo1993b</a>], [<a
| |
| href="references.html#Bo1997">Bo1997</a>].</li>
| |
| </ul>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l18 level2 lfo18;tab-stops:list 1.0in'>The same result has
| |
| also been established for the half-line [<span class=SpellE>CoKe</span>-p],
| |
| assuming boundary data is in H<span class=GramE>^{</span>(s+1)/3} of
| |
| course.</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l18 level2 lfo18;tab-stops:list 1.0in'>Global weak
| |
| solutions in L^2 were constructed in [<a href="references.html#Ka1983">Ka1983</a>].
| |
| Thus in L^2 one has global existence but no uniform continuity.</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l18 level2 lfo18;tab-stops:list 1.0in'>Uniqueness is also
| |
| known when the initial data lies in the weighted space <x>^{3/8}
| |
| u_0 in L^2 [<a href="references.html#GiTs1989">GiTs1989</a>]</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l18 level2 lfo18;tab-stops:list 1.0in'>LWP has also been
| |
| demonstrated when <xi>^s hat(u_0) lies in L^{r’} for 4/3 <
| |
| | |
| r <= 2 and s >= ½ - 1/2r [Gr-p4]</li>
| |
| </ul>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l18 level1 lfo18;tab-stops:list .5in'>GWP in <span class=SpellE>H^s</span>
| |
| for s > 1/4 [<a href="references.html#CoKeStTaTk-p2">CoKeStTkTa-p2</a>],
| |
| via the <span class=SpellE>KdV</span> theory and the Miura transform, for
| |
| both the <span class=SpellE>focussing</span> and <span class=SpellE>defocussing</span>
| |
| | |
| cases.</li>
| |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l18 level2 lfo18;tab-stops:list 1.0in'>Was proven for
| |
| s>3/5 in [<a href="references.html#FoLiPo1999">FoLiPo1999</a>]</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l18 level2 lfo18;tab-stops:list 1.0in'>Is implicit for s
| |
| >= 1 from [<a href="references.html#KnPoVe1993">KnPoVe1993</a>]</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l18 level2 lfo18;tab-stops:list 1.0in'>On the half-line GWP
| |
| is known when s >= 1 and the boundary data is in H^{11/12}, assuming
| |
| compatibility and small L^2 norm [<span class=SpellE>CoKe</span>-p]</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l18 level2 lfo18;tab-stops:list 1.0in'>GWP for smooth data
| |
| can also be achieved from inverse scattering methods [<span class=SpellE>BdmFsShp</span>-p];
| |
| the same approach also works on an interval [<span class=SpellE>BdmShp</span>-p].</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l18 level2 lfo18;tab-stops:list 1.0in'><span class=SpellE>Solitions</span>
| |
| are asymptotically H^1 stable [MtMe-p3], [<span class=SpellE>MtMe</span>-p]</li>
| |
| | |
| </ul>
| |
| </ul>
| |
| | |
| <div class=MsoNormal align=center style='text-align:center'>
| |
| | |
| <hr size=2 width="100%" align=center>
| |
| | |
| </div>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><a name="mKdV_on_T"></a><span
| |
| class=SpellE><span class=GramE><b>mKdV</b></span></span><b> on T</b></p>
| |
| | |
| <ul type=disc>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l7 level1 lfo19;tab-stops:list .5in'>Scaling is <span
| |
| class=SpellE>s_c</span> = -1/2.</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l7 level1 lfo19;tab-stops:list .5in'>C^0 LWP in L^2 in the
| |
| defocusing case [KpTp-p2]</li>
| |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l7 level2 lfo19;tab-stops:list 1.0in'>C^0 LWP in <span
| |
| class=SpellE>H^s</span> for s > 3/8 [Takaoka and <span class=SpellE>Tsutsumi</span>?]<span
| |
| style='mso-spacerun:yes'> </span>Note one has to gauge away a nonlinear
| |
| resonance term before one can apply iteration methods.</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l7 level2 lfo19;tab-stops:list 1.0in'>Analytic LWP in <span
| |
| class=SpellE>H^s</span> for s >= 1/2, in both focusing and defocusing
| |
| cases [<a href="references.html#KnPoVe1993">KnPoVe1993</a>], [<a
| |
| href="references.html#Bo1993b">Bo1993b</a>].</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l7 level2 lfo19;tab-stops:list 1.0in'>This is sharp in the
| |
| sense of analytic well-<span class=SpellE>posedness</span> [<a
| |
| href="references.html#KnPoVe1996">KnPoVe1996</a>] or uniform well-<span
| |
| class=SpellE>posedness</span> [<span class=SpellE>CtCoTa</span>-p]</li>
| |
| | |
| </ul>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l7 level1 lfo19;tab-stops:list .5in'>C^0 GWP in L^2 in the
| |
| defocusing case [KpTp-p2]</li>
| |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l7 level2 lfo19;tab-stops:list 1.0in'>Analytic GWP in <span
| |
| class=SpellE>H^s</span> for s >= 1/2<span class=GramE> [</span><a
| |
| href="references.html#CoKeStTaTk-p2">CoKeStTkTa-p2</a>], via the <span
| |
| class=SpellE>KdV</span> theory and the Miura transform, for both the <span
| |
| class=SpellE>focussing</span> and <span class=SpellE>defocussing</span>
| |
| | |
| cases.</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l7 level2 lfo19;tab-stops:list 1.0in'>Was proven for s
| |
| >= 1 in [<a href="references.html#KnPoVe1993">KnPoVe1993</a>], [<a
| |
| href="references.html#Bo1993b">Bo1993b</a>].</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l7 level2 lfo19;tab-stops:list 1.0in'>One has GWP for
| |
| random data whose Fourier coefficients decay like 1/|k| (times a Gaussian
| |
| random variable) [<a href="references.html#Bo1995c">Bo1995c</a>].
| |
| Indeed one has an invariant measure. Note that such data barely
| |
| fails to be in H<span class=GramE>^{</span>1/2}, however one can modify
| |
| the local well-<span class=SpellE>posedness</span> theory to go below
| |
| H^{1/2} provided that one has a decay like O(|k|^{-1+\eps}) on the
| |
| Fourier coefficients (which is indeed the case almost surely).</li>
| |
| | |
| </ul>
| |
| </ul>
| |
| | |
| <div class=MsoNormal align=center style='text-align:center'>
| |
| | |
| <hr size=2 width="100%" align=center>
| |
| | |
| </div>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><a name="gKdV_3_on_R"></a><span
| |
| class=GramE><b>gKdV_3</b></span><b> on R and R^+</b></p>
| |
| | |
| <ul type=disc>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l30 level1 lfo20;tab-stops:list .5in'>Scaling is <span
| |
| class=SpellE>s_c</span> = -1/6.</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l30 level1 lfo20;tab-stops:list .5in'>LWP in <span class=SpellE>H^s</span>
| |
| for s > -1/6 [Gr-p3]</li>
| |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l30 level2 lfo20;tab-stops:list 1.0in'>Was shown for
| |
| s>=1/12 [<a href="references.html#KnPoVe1993">KnPoVe1993</a>]</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l30 level2 lfo20;tab-stops:list 1.0in'>Was shown for
| |
| s>3/2 in [<a href="references.html#GiTs1989">GiTs1989</a>]</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l30 level2 lfo20;tab-stops:list 1.0in'>The result s >=
| |
| 1/12 has also been established for the half-line [<span class=SpellE>CoKe</span>-p],
| |
| assuming boundary data is in H<span class=GramE>^{</span>(s+1)/3} of
| |
| course..</li>
| |
| </ul>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l30 level1 lfo20;tab-stops:list .5in'>GWP in <span class=SpellE>H^s</span>
| |
| | |
| for s >= 0 [Gr-p3]</li>
| |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l30 level2 lfo20;tab-stops:list 1.0in'>For s>=1 this is
| |
| in [<a href="references.html#KnPoVe1993">KnPoVe1993</a>]</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l30 level2 lfo20;tab-stops:list 1.0in'>Presumably one can
| |
| use either the Fourier truncation method or the "I-method" to
| |
| go below L^2. Even though the equation is not completely <span
| |
| class=SpellE>integrable</span>, the one-dimensional nature of the
| |
| equation suggests that "correction term" techniques will also
| |
| be quite effective.</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l30 level2 lfo20;tab-stops:list 1.0in'>On the half-line GWP
| |
| is known when s >= 1 and the boundary data is in H^{5/4}, assuming
| |
| compatibility and small L^2 norm [<span class=SpellE>CoKe</span>-p]</li>
| |
| </ul>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l30 level1 lfo20;tab-stops:list .5in'><span class=SpellE>Solitons</span>
| |
| are H^1-stable [<a href="references.html#CaLo1982">CaLo1982</a>], [<a
| |
| href="references.html#Ws1986">Ws1986</a>], [<a
| |
| href="references.html#BnSouSr1987">BnSouSr1987</a>] and asymptotically H^1
| |
| stable [MtMe-p3], [<span class=SpellE>MtMe</span>-p]</li>
| |
| | |
| </ul>
| |
| | |
| <div class=MsoNormal align=center style='text-align:center'>
| |
| | |
| <hr size=2 width="100%" align=center>
| |
| | |
| </div>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><a name="gKdV_3_on_T"></a><span
| |
| class=GramE><b>gKdV_3</b></span><b> on T</b></p>
| |
| | |
| <ul type=disc>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l24 level1 lfo21;tab-stops:list .5in'>Scaling is <span
| |
| class=SpellE>s_c</span> = -1/6.</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l24 level1 lfo21;tab-stops:list .5in'>LWP in <span class=SpellE>H^s</span>
| |
| for s>=1/2 [<a href="references.html#CoKeStTaTk-p3">CoKeStTkTa-p3</a>]</li>
| |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l24 level2 lfo21;tab-stops:list 1.0in'>Was shown for s
| |
| >= 1 in [<a href="references.html#St1997c">St1997c</a>]</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l24 level2 lfo21;tab-stops:list 1.0in'>One has analytic
| |
| ill-<span class=SpellE>posedness</span> for s<1/2 [<a
| |
| href="references.html#CoKeStTaTk-p3">CoKeStTkTa-p3</a>] by a modification
| |
| of the example in [<a href="references.html#KnPoVe1996">KnPoVe1996</a>].</li>
| |
| </ul>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l24 level1 lfo21;tab-stops:list .5in'>GWP in <span class=SpellE>H^s</span>
| |
| | |
| for s>5/6 [<a href="references.html#CoKeStTaTk-p3">CoKeStTkTa-p3</a>]</li>
| |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l24 level2 lfo21;tab-stops:list 1.0in'>Was shown for s
| |
| >= 1 in [<a href="references.html#St1997c">St1997c</a>]</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l24 level2 lfo21;tab-stops:list 1.0in'>This result may well
| |
| be improvable by the "damping correction term" method in<span
| |
| class=GramE> [</span><a href="references.html#CoKeStTaTk-p2">CoKeStTkTa-p2</a>].</li>
| |
| | |
| </ul>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l24 level1 lfo21;tab-stops:list .5in'><i>Remark</i>: For this equation
| |
| it is convenient to make a "gauge transformation'' to subtract off
| |
| the mean of <span class=GramE>P(</span>u).</li>
| |
| </ul>
| |
| | |
| <div class=MsoNormal align=center style='text-align:center'>
| |
| | |
| <hr size=2 width="100%" align=center>
| |
| | |
| </div>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><a name="gKdV_4_on_R"></a><span
| |
| class=GramE><b>gKdV_4</b></span><b> on R and R^+</b></p>
| |
| | |
| <p class=MsoNormal>(Thanks to Felipe <span class=SpellE>Linares</span> for help
| |
| with the references here - Ed.)<span style='mso-spacerun:yes'> </span>A good
| |
| survey for the results here is in [Tz-p2].</p>
| |
| | |
| <ul type=disc>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l22 level1 lfo22;tab-stops:list .5in'>Scaling is <span
| |
| class=SpellE>s_c</span> = 0 (i.e. L^2-critical).</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l22 level1 lfo22;tab-stops:list .5in'>LWP in <span class=SpellE>H^s</span>
| |
| for s >= 0 [<a href="references.html#KnPoVe1993">KnPoVe1993</a>]</li>
| |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l22 level2 lfo22;tab-stops:list 1.0in'>Was shown for
| |
| s>3/2 in [<a href="references.html#GiTs1989">GiTs1989</a>]</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l22 level2 lfo22;tab-stops:list 1.0in'>The same result s
| |
| >= 0 has also been established for the half-line [<span class=SpellE>CoKe</span>-p],
| |
| assuming boundary data is in H<span class=GramE>^{</span>(s+1)/3} of
| |
| course..</li>
| |
| </ul>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l22 level1 lfo22;tab-stops:list .5in'>GWP in <span class=SpellE>H^s</span>
| |
| for s > 3/4 in both the focusing and defocusing cases, though one must
| |
| of course have smaller L^2 mass than the ground state in the focusing case
| |
| [<span class=SpellE>FoLiPo</span>-p].</li>
| |
| | |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l22 level2 lfo22;tab-stops:list 1.0in'>For s >= 1 and
| |
| the defocusing case this is in [<a href="references.html#KnPoVe1993">KnPoVe1993</a>]</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l22 level2 lfo22;tab-stops:list 1.0in'>Blowup has recently
| |
| been shown for the <span class=SpellE>focussing</span> case for data
| |
| close to a ground state with negative energy [Me-p]. In such
| |
| a case the blowup profile must approach the ground state (modulo <span
| |
| class=SpellE>scalings</span> and translations), see [MtMe-p4], [<a
| |
| href="references.html#MtMe2001">MtMe2001</a>]. Also, the blow up
| |
| rate in H^1 must be strictly faster than t<span class=GramE>^{</span>-1/3}
| |
| [MtMe-p4], which is the rate suggested by scaling.</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l22 level2 lfo22;tab-stops:list 1.0in'>Explicit
| |
| self-similar blow-up solutions have been constructed [<span class=SpellE>BnWe</span>-p]
| |
| but these are not in L^2.</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l22 level2 lfo22;tab-stops:list 1.0in'>GWP for small L^2
| |
| data in either case [<a href="references.html#KnPoVe1993">KnPoVe1993</a>].
| |
| In the <span class=SpellE>focussing</span> case we have GWP whenever the
| |
| L^2 norm is strictly smaller than that of the ground state Q (thanks to
| |
| Weinstein's sharp <span class=SpellE>Gagliardo-Nirenberg</span>
| |
| | |
| inequality). It seems like a reasonable (but difficult) conjecture
| |
| to have GWP for large L^2 data in the defocusing case.</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l22 level2 lfo22;tab-stops:list 1.0in'>On the half-line GWP
| |
| is known when s >= 1 and the boundary data is in H^{11/12}, assuming
| |
| compatibility and small L^2 norm [<span class=SpellE>CoKe</span>-p]</li>
| |
| </ul>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l22 level1 lfo22;tab-stops:list .5in'><span class=SpellE>Solitons</span>
| |
| are H^1-unstable [<a href="references.html#MtMe2001">MtMe2001</a>].
| |
| | |
| However, small H^1 perturbations of a <span class=SpellE>soliton</span>
| |
| must asymptotically converge weakly to some rescaled <span class=SpellE>soliton</span>
| |
| shape provided that the H^1 norm stays comparable to 1 [<a
| |
| href="references.html#MtMe-p"><span class=SpellE>MtMe</span>-p</a>].</li>
| |
| </ul>
| |
| | |
| <div class=MsoNormal align=center style='text-align:center'>
| |
| | |
| <hr size=2 width="100%" align=center>
| |
| | |
| </div>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><a name="gKdV_4_on_T"></a><span
| |
| class=GramE><b>gKdV_4</b></span><b> on T</b></p>
| |
| | |
| <ul type=disc>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l14 level1 lfo23;tab-stops:list .5in'>Scaling is <span
| |
| class=SpellE>s_c</span> = 0.</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l14 level1 lfo23;tab-stops:list .5in'>LWP in <span class=SpellE>H^s</span>
| |
| for s>=1/2 [<a href="references.html#CoKeStTaTk-p3">CoKeStTkTa-p3</a>]</li>
| |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l14 level2 lfo23;tab-stops:list 1.0in'>Was shown for s
| |
| >= 1 in [<a href="references.html#St1997c">St1997c</a>]</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l14 level2 lfo23;tab-stops:list 1.0in'>Analytic well-<span
| |
| class=SpellE>posedness</span> fails for s < 1/2; this is essentially
| |
| in [<a href="references.html#KnPoVe1996">KnPoVe1996</a>]</li>
| |
| </ul>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l14 level1 lfo23;tab-stops:list .5in'>GWP in <span class=SpellE>H^s</span>
| |
| for s>=1 [<a href="references.html#St1997c">St1997c</a>]</li>
| |
| | |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l14 level2 lfo23;tab-stops:list 1.0in'>This is almost
| |
| certainly improvable by the techniques in [<a
| |
| href="references.html#CoKeStTaTk-p3">CoKeStTkTa-p3</a>], probably to s
| |
| > 6/7. There are some low-frequency issues which may require the
| |
| techniques in [<a href="references.html#KeTa-p"><span class=SpellE>KeTa</span>-p</a>].</li>
| |
| </ul>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l14 level1 lfo23;tab-stops:list .5in'><i>Remark</i>: For this
| |
| equation it is convenient to make a "gauge transformation'' to
| |
| subtract off the mean of <span class=GramE>P(</span>u).</li>
| |
| | |
| </ul>
| |
| | |
| <div class=MsoNormal align=center style='text-align:center'>
| |
| | |
| <hr size=2 width="100%" align=center>
| |
| | |
| </div>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><a name="gKdV_on_R+"></a><span
| |
| class=SpellE><span class=GramE><b>gKdV</b></span></span><b> on R^+</b></p>
| |
| | |
| <ul type=disc>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l19 level1 lfo24;tab-stops:list .5in'>The <span class=SpellE>gKdV</span>
| |
| | |
| Cauchy-boundary problem on the half-line is</li>
| |
| </ul>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><span class=SpellE>u_t</span>
| |
| + u<span class=GramE>_{</span>xxx} + <span class=SpellE>u_x</span> + <span
| |
| class=SpellE>u^k</span> <span class=SpellE>u_x</span> = 0; u(x,0) =
| |
| u_0(x); u(0,t) = h(t)</p>
| |
| | |
| <p class=MsoNormal style='margin-left:.5in'>The sign of u<span class=GramE>_{</span>xxx}
| |
| is important (it makes the influence of the boundary x=0 mostly negligible),
| |
| the sign of u <span class=SpellE>u_x</span> is not. The drift term <span
| |
| class=SpellE>u_x</span> is convenient for technical reasons; it is not known
| |
| whether it is truly necessary. </p>
| |
| | |
| <ul type=disc>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l9 level1 lfo25;tab-stops:list .5in'>LWP is known for initial
| |
| data in <span class=SpellE>H^s</span> and boundary data in H<span
| |
| class=GramE>^{</span>(s+1)/3} when s > 3/4 [<span class=SpellE>CoKn</span>-p].</li>
| |
| | |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l9 level2 lfo25;tab-stops:list 1.0in'>The techniques are
| |
| based on [<a href="references.html#KnPoVe1993">KnPoVe1993</a>] and a
| |
| replacement of the IVBP with a forced IVP.</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l9 level2 lfo25;tab-stops:list 1.0in'>This has been
| |
| improved to s >= <span class=SpellE>s_c</span> = 1/2 - 2/k when k >
| |
| 4 [<span class=SpellE>CoKe</span>-p].</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l9 level2 lfo25;tab-stops:list 1.0in'>For <a
| |
| href="#KdV_on_R+"><span class=SpellE>KdV</span></a>, <a href="#mKdV_on_R"><span
| |
| class=SpellE>mKdV</span></a>, <a href="#gKdV_3_on_R">gKdV-3</a> , and <a
| |
| href="#gKdV_4_on_R">gKdV-4</a> see the corresponding sections on this
| |
| page.</li>
| |
| </ul>
| |
| </ul>
| |
| | |
| <div class=MsoNormal align=center style='text-align:center'>
| |
| | |
| <hr size=2 width="100%" align=center>
| |
| | |
| </div>
| |
| | |
| <p align=center style='text-align:center'><a name="Schrodinger_Airy"></a><b>Nonlinear
| |
| <span class=SpellE>Schrodinger</span>-Airy equation<o:p></o:p></b></p>
| |
| | |
| <p>The equation</p>
| |
| | |
| <p align=center style='text-align:center'><span class=SpellE>u_t</span> + <span
| |
| class=SpellE>i</span> c <span class=SpellE>u_xx</span> + <span class=SpellE>u_xxx</span>
| |
| | |
| = <span class=SpellE>i</span> gamma |u|^2 u + delta |u|^2 <span class=SpellE>u_x</span>
| |
| + epsilon u^2 <span class=SpellE><u>u</u>_x</span></p>
| |
| | |
| <p><span class=GramE>on</span> R is a combination of the <a
| |
| href="schrodinger.html#Cubic_NLS_on_R">cubic NLS equation</a> , the <a
| |
| href="schrodinger.html#dnls-3_on_R">derivative cubic NLS equation</a>, <a
| |
| href="#mKdV_on_R">complex <span class=SpellE>mKdV</span></a>, and a cubic
| |
| nonlinear Airy equation.<span style='mso-spacerun:yes'> </span>This equation
| |
| is a general model for <span class=SpellE>propogation</span> of pulses in an
| |
| optical fiber [<a href="references.html#Kod1985">Kod1985</a>], [<a
| |
| href="references.html#HasKod1987">HasKod1987</a>]</p>
| |
| | |
| <p style='margin-left:.5in;text-indent:-.25in;mso-list:l25 level1 lfo26;
| |
| tab-stops:list .5in'><![if !supportLists]><span style='font-family:Symbol;
| |
| mso-fareast-font-family:Symbol;mso-bidi-font-family:Symbol'><span
| |
| style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>
| |
| </span></span></span><![endif]>When c=delta=epsilon = 0, scaling is s=-1.<span
| |
| style='mso-spacerun:yes'> </span>When c=gamma=0, scaling is –1/2.</p>
| |
| | |
| <p style='margin-left:.5in;text-indent:-.25in;mso-list:l25 level1 lfo26;
| |
| tab-stops:list .5in'><![if !supportLists]><span style='font-family:Symbol;
| |
| mso-fareast-font-family:Symbol;mso-bidi-font-family:Symbol'><span
| |
| style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>
| |
| </span></span></span><![endif]>LWP is known when s >= ¼ [<a
| |
| href="references.html#St1997d">St1997d</a>]</p>
| |
| | |
| <p style='margin-left:1.0in;text-indent:-.25in;mso-list:l25 level2 lfo26;
| |
| tab-stops:list 1.0in'><![if !supportLists]><span style='font-family:"Courier New";
| |
| mso-fareast-font-family:"Courier New"'><span style='mso-list:Ignore'>o<span
| |
| style='font:7.0pt "Times New Roman"'> </span></span></span><![endif]>For
| |
| s > ¾ this is in [<a href="references.html#Lau1997">Lau1997</a>], [<a
| |
| href="references.html#Lau2001">Lau2001</a>]</p>
| |
| | |
| <p style='margin-left:1.0in;text-indent:-.25in;mso-list:l25 level2 lfo26;
| |
| tab-stops:list 1.0in'><![if !supportLists]><span style='font-family:"Courier New";
| |
| mso-fareast-font-family:"Courier New"'><span style='mso-list:Ignore'>o<span
| |
| style='font:7.0pt "Times New Roman"'> </span></span></span><![endif]>The
| |
| s>=1/4 result is also known when c is a time-dependent function [Cv2002],
| |
| [CvLi2003]</p>
| |
| | |
| <p style='margin-left:1.0in;text-indent:-.25in;mso-list:l25 level2 lfo26;
| |
| tab-stops:list 1.0in'><![if !supportLists]><span style='font-family:"Courier New";
| |
| mso-fareast-font-family:"Courier New"'><span style='mso-list:Ignore'>o<span
| |
| style='font:7.0pt "Times New Roman"'> </span></span></span><![endif]>For
| |
| s < -1/4 and delta or epsilon non-zero, the solution map is not C^3 [<span
| |
| class=SpellE>CvLi</span>-p]</p>
| |
| | |
| <p style='margin-left:1.0in;text-indent:-.25in;mso-list:l25 level2 lfo26;
| |
| tab-stops:list 1.0in'><![if !supportLists]><span style='font-family:"Courier New";
| |
| mso-fareast-font-family:"Courier New"'><span style='mso-list:Ignore'>o<span
| |
| style='font:7.0pt "Times New Roman"'> </span></span></span><![endif]>When
| |
| delta = epsilon = 0 LWP is known for s > -1/4 [<a
| |
| href="references.html#Cv2004">Cv2004</a>]</p>
| |
| | |
| <p style='margin-left:1.5in;text-indent:-.25in;mso-list:l25 level3 lfo26;
| |
| tab-stops:list 1.5in'><![if !supportLists]><span style='font-family:Wingdings;
| |
| mso-fareast-font-family:Wingdings;mso-bidi-font-family:Wingdings'><span
| |
| style='mso-list:Ignore'>§<span style='font:7.0pt "Times New Roman"'>
| |
| </span></span></span><![endif]>For s < -1/4 the solution map is not C^3 [<span
| |
| class=SpellE>CvLi</span>-p]</p>
| |
| | |
| <div class=MsoNormal align=center style='text-align:center'>
| |
| | |
| <hr size=2 width="100%" align=center>
| |
| | |
| </div>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><a name=gKdV></a><b>Miscellaneous
| |
| | |
| <span class=SpellE>gKdV</span> results</b></p>
| |
| | |
| <p class=MsoNormal>[Thanks to <span class=SpellE>Nikolaos</span> <span
| |
| class=SpellE>Tzirakis</span> for some corrections - Ed.] </p>
| |
| | |
| <ul type=disc>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l2 level1 lfo27;tab-stops:list .5in'>On R with k > 4, <span
| |
| class=SpellE>gKdV</span>-k is LWP down to scaling: s >= <span
| |
| class=SpellE>s_c</span> = 1/2 - 2/k [<a href="references.html#KnPoVe1993">KnPoVe1993</a>]</li>
| |
| | |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l2 level2 lfo27;tab-stops:list 1.0in'>Was shown for
| |
| s>3/2 in [<a href="references.html#GiTs1989">GiTs1989</a>]</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l2 level2 lfo27;tab-stops:list 1.0in'>One has ill-<span
| |
| class=SpellE>posedness</span> in the supercritical regime [<a
| |
| href="references.html#BirKnPoSvVe1996">BirKnPoSvVe1996</a>]</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l2 level2 lfo27;tab-stops:list 1.0in'>For small data one
| |
| has scattering [<a href="references.html#KnPoVe1993c">KnPoVe1993c</a>].<span
| |
| style='mso-spacerun:yes'> </span>Note that one cannot have scattering in
| |
| L^2 except in the critical case k=4 because one can scale <span
| |
| class=SpellE>solitons</span> to be arbitrarily small in the non-critical
| |
| cases.</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l2 level2 lfo27;tab-stops:list 1.0in'><span class=SpellE>Solitons</span>
| |
| are H^1-unstable [<a href="references.html#BnSouSr1987">BnSouSr1987</a>]</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l2 level2 lfo27;tab-stops:list 1.0in'>If one considers an
| |
| arbitrary smooth non-linearity (not necessarily a power) then one has LWP
| |
| for small data in <span class=SpellE>H^s</span>, s > 1/2 [<a
| |
| href="references.html#St1995">St1995</a>]</li>
| |
| | |
| </ul>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l2 level1 lfo27;tab-stops:list .5in'>On R with any k, <span
| |
| class=SpellE>gKdV</span>-k is GWP in <span class=SpellE>H^s</span> for s
| |
| >= 1 [<a href="references.html#KnPoVe1993">KnPoVe1993</a>], though for
| |
| k >= 4 one needs the L^2 norm to be small; global weak solutions were
| |
| constructed much earlier, with the same smallness assumption when k >=
| |
| 4. This should be improvable below H^1 for all k.</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l2 level1 lfo27;tab-stops:list .5in'>On R with any k, <span
| |
| class=SpellE>gKdV</span>-k has the <span class=SpellE>H^s</span> norm
| |
| growing like t^{(s-1)+} in time for any integer s >= 1 [<a
| |
| href="references.html#St1997b">St1997b</a>]</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l2 level1 lfo27;tab-stops:list .5in'>On R with any non-linearity,
| |
| a non-zero solution to <span class=SpellE>gKdV</span> cannot be supported
| |
| on the half-line R^+ (or R^-) for two different times [<a
| |
| href="references.html#KnPoVe-p3">KnPoVe-p3</a>], [KnPoVe-p4].</li>
| |
| | |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l2 level2 lfo27;tab-stops:list 1.0in'>In the completely <span
| |
| class=SpellE>integrable</span> cases k=1,2 this is in [<a
| |
| href="references.html#Zg1992">Zg1992</a>]</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l2 level2 lfo27;tab-stops:list 1.0in'>Also, a non-zero
| |
| solution to <span class=SpellE>gKdV</span> cannot vanish on a rectangle
| |
| in <span class=SpellE>spacetime</span> [<a
| |
| href="references.html#SauSc1987">SauSc1987</a>]; see also [<a
| |
| href="references.html#Bo1997b">Bo1997b</a>].</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l2 level2 lfo27;tab-stops:list 1.0in'>Extensions to higher
| |
| order <span class=SpellE>gKdV</span> type equations are in [<a
| |
| href="references.html#Bo1997b">Bo1997b</a>], [KnPoVe-p5].</li>
| |
| </ul>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l2 level1 lfo27;tab-stops:list .5in'>On R with non-integer k, one
| |
| has decay of <span class=GramE>O(</span>t^{-1/3}) in L^\<span
| |
| class=SpellE>infty</span> for small decaying data if k > (19 - <span
| |
| class=SpellE>sqrt</span>(57))/4 ~ 2.8625... [<a
| |
| href="references.html#CtWs1991">CtWs1991</a>]</li>
| |
| | |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l2 level2 lfo27;tab-stops:list 1.0in'>A similar result for
| |
| k > (5+<span class=GramE>sqrt(</span>73))/4 ~ 3.39... <span
| |
| class=GramE>was</span> obtained in [<a href="references.html#PoVe1990">PoVe1990</a>].</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l2 level2 lfo27;tab-stops:list 1.0in'>When k=2 solutions
| |
| decay like O(t^{-1/3}), and when k=1 solutions decay generically like
| |
| O(t^{-2/3}) but like O( (t/log t)^{-2/3}) for exceptional data [<a
| |
| href="references.html#AbSe1977">AbSe1977</a>]</li>
| |
| | |
| </ul>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l2 level1 lfo27;tab-stops:list .5in'>In the L^2 <span
| |
| class=SpellE>subcritical</span> case 0 < k < 4, <span class=SpellE>multisoliton</span>
| |
| solutions are asymptotically H^1-stable [<span class=SpellE>MtMeTsa</span>-p]</li>
| |
| | |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l2 level2 lfo27;tab-stops:list 1.0in'>For a single <span
| |
| class=SpellE>soliton</span> this is in [MtMe-p3], [<span class=SpellE>MtMe</span>-p],
| |
| [<a href="references.html#Miz2001">Miz2001</a>]; earlier work is in [<a
| |
| href="references.html#Bj1972">Bj1972</a>], [<a
| |
| href="references.html#Bn1975">Bn1975</a>], [<a
| |
| href="references.html#Ws1986">Ws1986</a>], [<a
| |
| href="references.html#PgWs1994">PgWs1994</a>]</li>
| |
| | |
| </ul>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l2 level1 lfo27;tab-stops:list .5in'>A dissipative version of <span
| |
| class=SpellE>gKdV</span>-k was analyzed in [<a
| |
| href="references.html#MlRi2001">MlRi2001</a>]</li>
| |
| </ul>
| |
| | |
| <ul type=disc>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l20 level1 lfo28;tab-stops:list .5in'>On T with any k, <span
| |
| class=SpellE>gKdV</span>-k has the <span class=SpellE>H^s</span> norm
| |
| growing like t^{2(s-1)+} in time for any integer s >= 1 [<a
| |
| href="references.html#St1997b">St1997b</a>]</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l20 level1 lfo28;tab-stops:list .5in'>On T with k >= 3, <span
| |
| class=SpellE>gKdV</span>-k is LWP for s >= 1/2 [<a
| |
| href="references.html#CoKeStTaTk-p3">CoKeStTkTa-p3</a>]</li>
| |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l20 level2 lfo28;tab-stops:list 1.0in'>Was shown for s
| |
| >= 1 in [<a href="references.html#St1997c">St1997c</a>]</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l20 level2 lfo28;tab-stops:list 1.0in'>Analytic well-<span
| |
| class=SpellE>posedness</span> fails for s < 1/2 [<a
| |
| href="references.html#CoKeStTaTk-p3">CoKeStTkTa-p3</a>], [<a
| |
| href="references.html#KnPoVe1996">KnPoVe1996</a>]</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l20 level2 lfo28;tab-stops:list 1.0in'>For arbitrary smooth
| |
| non-<span class=SpellE>linearities</span>, weak H^1 solutions were
| |
| constructed in [<a href="references.html#Bo1993b">Bo1993</a>].</li>
| |
| | |
| </ul>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l20 level1 lfo28;tab-stops:list .5in'>On T with k >= 3, <span
| |
| class=SpellE>gKdV</span>-k is GWP for s >= 1 except in the <span
| |
| class=SpellE>focussing</span> case [<a href="references.html#St1997c">St1997c</a>]</li>
| |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l20 level2 lfo28;tab-stops:list 1.0in'>The estimates in [<a
| |
| href="references.html#CoKeStTaTk-p3">CoKeStTkTa-p3</a>] suggest that this
| |
| is improvable to 13/14 - 2/7k, but this has only been proven in the
| |
| sub-critical case k=3 [<a href="references.html#CoKeStTaTk-p3">CoKeStTkTa-p3</a>].
| |
| | |
| In the critical and super-critical cases there are some low-frequency
| |
| issues which may require the techniques in [<a
| |
| href="references.html#KeTa-p"><span class=SpellE>KeTa</span>-p</a>].</li>
| |
| </ul>
| |
| </ul>
| |
| | |
| <div class=MsoNormal align=center style='text-align:center'>
| |
| | |
| <hr size=2 width="100%" align=center>
| |
| | |
| </div>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><a name=hierarchy></a><b>The
| |
| | |
| <span class=SpellE>KdV</span> Hierarchy</b></p>
| |
| | |
| <p>The <span class=SpellE>KdV</span> equation </p>
| |
| | |
| <p align=center style='text-align:center'><span class=SpellE>V_t</span> + <span
| |
| class=SpellE>V_xxx</span> = 6 <span class=SpellE>V_x</span></p>
| |
| | |
| <p><span class=GramE>can</span> be rewritten in the Lax Pair form </p>
| |
| | |
| <p align=center style='text-align:center'><span class=SpellE>L_t</span> = [L,
| |
| P]</p>
| |
| | |
| <p><span class=GramE>where</span> L is the second-order operator </p>
| |
| | |
| <p align=center style='text-align:center'>L = -D^2 + V</p>
| |
| | |
| <p>(D = d/<span class=SpellE>dx</span>) and P is the third-order <span
| |
| class=SpellE>antiselfadjoint</span> operator </p>
| |
| | |
| <p align=center style='text-align:center'>P = 4D^3 + 3(DV + VD).</p>
| |
| | |
| <p>(<span class=GramE>note</span> that P consists of the <span class=SpellE>zeroth</span>
| |
| | |
| order and higher terms of the formal power series expansion of 4i L^{3/2}). </p>
| |
| | |
| <p>One can replace P with other fractional powers of L. For instance, the
| |
| <span class=SpellE>zeroth</span> order and higher terms of 4i L<span
| |
| class=GramE>^{</span>5/2} are </p>
| |
| | |
| <p align=center style='text-align:center'>P = 4D^5 + 5(D^3 V + V D^3) - 5/4 (D <span
| |
| class=SpellE>V_xx</span> + <span class=SpellE>V_xx</span> D) + 15/4 (D V^2 +
| |
| V^2 D)</p>
| |
| | |
| <p><span class=GramE>and</span> the Lax pair equation becomes </p>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><span class=SpellE>V_t</span>
| |
| + <span class=SpellE>u_xxxxx</span> = (5 V_x^2 + 10 V <span class=SpellE>V_xx</span>
| |
| + 10 V^3<span class=GramE>)_</span>x</p>
| |
| | |
| <p><span class=GramE>with</span> Hamiltonian </p>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><span class=GramE>H(</span>V)
| |
| = \<span class=SpellE>int</span> V_xx^2 - 5 V^2 <span class=SpellE>V_xx</span>
| |
| - 5 V^4.</p>
| |
| | |
| <p>These flows all commute with each <span class=GramE>other,</span> and their
| |
| Hamiltonians are conserved by all the flows simultaneously. </p>
| |
| | |
| <p>The <span class=SpellE>KdV</span> <span class=GramE>hierarchy are</span>
| |
| examples of higher order water wave models; a general formulation is</p>
| |
| | |
| <p align=center style='text-align:center'><span class=SpellE>u_t</span> + <span
| |
| class=SpellE>partial_x</span><span class=GramE>^{</span>2j+1} u = P(u, <span
| |
| class=SpellE>u_x</span>, ..., <span class=SpellE>partial_x</span>^{2j} u)</p>
| |
| | |
| <p><span class=GramE>where</span> u is real-valued and P is a polynomial with
| |
| no constant or linear terms; thus <span class=SpellE>KdV</span> and <span
| |
| class=SpellE>gKdV</span> correspond to j=1, and the higher order equations in
| |
| the hierarchy correspond to j=2,3,etc.<span style='mso-spacerun:yes'>
| |
| </span>LWP for these equations in high regularity <span class=SpellE>Sobolev</span>
| |
| spaces is in [<a href="references.html#KnPoVe1994">KnPoVe1994</a>], and
| |
| independently by <span class=SpellE><span class=GramE>Cai</span></span> (ref?);
| |
| see also [<a href="references.html#CrKpSr1992">CrKpSr1992</a>].<span
| |
| style='mso-spacerun:yes'> </span>The case j=2 was studied by <span
| |
| class=SpellE>Choi</span> (ref?).<span style='mso-spacerun:yes'> </span>The
| |
| non-scalar diagonal case was treated in [<a href="references.html#KnSt1997">KnSt1997</a>];
| |
| the periodic case was studied in [Bo-p3].<span style='mso-spacerun:yes'>
| |
| | |
| </span>Note in the periodic case it is possible to have ill-<span class=SpellE>posedness</span>
| |
| for every regularity, for instance <span class=SpellE>u_t</span> + <span
| |
| class=SpellE>u_xxx</span> = u^2 u_x^2 is ill-posed in every <span class=SpellE>H^s</span>
| |
| [Bo-p3]</p>
| |
| | |
| <p><o:p> </o:p></p>
| |
| | |
| <div class=MsoNormal align=center style='text-align:center'>
| |
| | |
| <hr size=2 width="100%" align=center>
| |
| | |
| </div>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><a name=Benjamin-Ono></a><b>Benjamin-Ono
| |
| equation</b></p>
| |
| | |
| <p class=MsoNormal>[Thanks to <span class=SpellE>Nikolay</span> <span
| |
| class=SpellE>Tzvetkov</span> and Felipe <span class=SpellE>Linares</span> for
| |
| help with this section - Ed] </p>
| |
| | |
| <p>The <i>generalized Benjamin-Ono equation</i> <span class=SpellE>BO_a</span>
| |
| is the scalar equation </p>
| |
| | |
| <p class=MsoNormal align=center style='text-align:center'><span class=SpellE>u_t</span>
| |
| + <span class=SpellE>D_x</span><span class=GramE>^{</span>1+a} <span
| |
| class=SpellE>u_x</span> + <span class=SpellE>uu_x</span> = 0</p>
| |
| | |
| <p class=MsoNormal><span class=GramE>where</span> <span class=SpellE>D_x</span>
| |
| = <span class=SpellE>sqrt</span>{-Delta} is the positive differentiation
| |
| operator. When a=1 this is <a href="#kdv"><span class=SpellE>KdV</span></a>;
| |
| when a=0 this is the Benjamin-Ono equation (BO) [<a
| |
| href="references.html#Bj1967">Bj1967</a>], [<a href="references.html#On1975">On1975</a>],
| |
| which models one-dimensional internal waves in deep water. Both of these
| |
| equations are completely <span class=SpellE>integrable</span> (see e.g. [<a
| |
| href="references.html#AbFs1983">AbFs1983</a>], [<a
| |
| href="references.html#CoiWic1990">CoiWic1990</a>]), though the intermediate
| |
| cases 0 < a < 1 are not. </p>
| |
| | |
| <p>When a=0, scaling is s = -1/2, and the following results are known: </p>
| |
| | |
| <ul type=disc>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l26 level1 lfo29;tab-stops:list .5in'>LWP in <span class=SpellE>H^s</span>
| |
| for s >= 1 [Ta-p]</li>
| |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l26 level2 lfo29;tab-stops:list 1.0in'>For s >= 9/8 this
| |
| is in [<span class=SpellE>KnKoe</span>-p]</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l26 level2 lfo29;tab-stops:list 1.0in'>For s >= 5/4 this
| |
| is in [<span class=SpellE>KocTz</span>-p]</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l26 level2 lfo29;tab-stops:list 1.0in'>For s >= 3/2 this
| |
| is in [<a href="references.html#Po1991">Po1991</a>]</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l26 level2 lfo29;tab-stops:list 1.0in'>For s > 3/2 this
| |
| is in [<a href="references.html#Io1986">Io1986</a>]</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l26 level2 lfo29;tab-stops:list 1.0in'>For s > 3 this is
| |
| in [<a href="references.html#Sau1979">Sau1979</a>]</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l26 level2 lfo29;tab-stops:list 1.0in'>For no value of s is
| |
| the solution map uniformly continuous [KocTz-p2]</li>
| |
| <ul type=square>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l26 level3 lfo29;tab-stops:list 1.5in'>For s < -1/2
| |
| this is in [<span class=SpellE>BiLi</span>-p]</li>
| |
| | |
| </ul>
| |
| </ul>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l26 level1 lfo29;tab-stops:list .5in'>Global weak solutions exist
| |
| for L^2 data [<a href="references.html#Sau1979">Sau1979</a>], [<a
| |
| href="references.html#GiVl1989b">GiVl1989b</a>], [<a
| |
| href="references.html#GiVl1991">GiVl1991</a>], [<a
| |
| href="references.html#Tom1990">Tom1990</a>]</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l26 level1 lfo29;tab-stops:list .5in'>Global well-<span
| |
| class=SpellE>posedness</span> in <span class=SpellE>H^s</span> for s >=
| |
| 1 [Ta-p]</li>
| |
| | |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l26 level2 lfo29;tab-stops:list 1.0in'>For s >= 3/2 this
| |
| is in [<a href="references.html#Po1991">Po1991</a>]</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l26 level2 lfo29;tab-stops:list 1.0in'>For smooth solutions
| |
| this is in [<a href="references.html#Sau1979">Sau1979</a>]</li>
| |
| </ul>
| |
| </ul>
| |
| | |
| <p class=MsoNormal>When 0 < a < 1, scaling is s = -1/2 - <span
| |
| class=GramE>a,</span> and the following results are known: </p>
| |
| | |
| <ul type=disc>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l1 level1 lfo30;tab-stops:list .5in'>LWP in <span class=SpellE>H^s</span>
| |
| is known for s > 9/8 – 3a/8 [<span class=SpellE>KnKoe</span>-p]</li>
| |
| | |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l1 level2 lfo30;tab-stops:list 1.0in'>For s >= 3/4 (2-a)
| |
| this is in [<a href="references.html#KnPoVe1994b">KnPoVe1994b</a>]</li>
| |
| </ul>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l1 level1 lfo30;tab-stops:list .5in'>GWP is known when s >=
| |
| (a+1)/2 when a > 4/5, from the conservation of the Hamiltonian [<a
| |
| href="references.html#KnPoVe1994b">KnPoVe1994b</a>]</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l1 level1 lfo30;tab-stops:list .5in'>The LWP results are obtained
| |
| by energy methods; it is known that pure iteration methods cannot work [<a
| |
| href="references.html#MlSauTz2001">MlSauTz2001</a>]</li>
| |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l1 level2 lfo30;tab-stops:list 1.0in'>However, this can be
| |
| salvaged by combining the <span class=SpellE>H^s</span> norm || f ||_{<span
| |
| class=SpellE>H^s</span>} with a weighted <span class=SpellE>Sobolev</span>
| |
| space, namely || <span class=SpellE>xf</span> ||_{H^{s - 2s_*}}, where
| |
| s_* = (a+1)/2 is the energy regularity.[CoKnSt-p4]</li>
| |
| | |
| </ul>
| |
| </ul>
| |
| | |
| <p class=MsoNormal>One can replace the quadratic non-linearity <span
| |
| class=SpellE>uu_x</span> by higher powers u<span class=GramE>^{</span>k-1} <span
| |
| class=SpellE>u_x</span>, in analogy with <span class=SpellE>KdV</span> and <span
| |
| class=SpellE>gKdV</span>, giving rise to the <span class=SpellE>gBO</span>-k
| |
| equations (let us take a=0 for sake of discussion).<span
| |
| style='mso-spacerun:yes'> </span>The scaling exponent is 1/2 - 1<span
| |
| class=GramE>/(</span>k-1).</p>
| |
| | |
| <ul type=disc>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l21 level1 lfo31;tab-stops:list .5in'>For k=3, one has GWP for
| |
| large data in H^1 [<span class=SpellE>KnKoe</span>-p] and LWP for small
| |
| data in <span class=SpellE>H^s</span>, s > ½ [<span class=SpellE>MlRi</span>-p]</li>
| |
| <ul type=circle>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l21 level2 lfo31;tab-stops:list 1.0in'>For small data in <span
| |
| class=SpellE>H^s</span>, s>1, LWP was obtained in [<a
| |
| href="references.html#KnPoVe1994b">KnPoVe1994b</a>]</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l21 level2 lfo31;tab-stops:list 1.0in'>With the addition of
| |
| a small viscosity term, GWP can also be obtained in H^1 by complete <span
| |
| class=SpellE>integrability</span> methods in [FsLu2000], with <span
| |
| class=SpellE>asymptotics</span> under the additional assumption that the
| |
| initial data is in L^1.</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
| |
| auto;mso-list:l21 level2 lfo31;tab-stops:list 1.0in'>For s < ½, the
| |
| solution map is not C^3 [<span class=SpellE>MlRi</span>-p]</li>
| |
| </ul>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l21 level1 lfo31;tab-stops:list .5in'>For k=4, LWP for small data
| |
| in <span class=SpellE>H^s</span>, s > 5/6 was obtained in [<a
| |
| href="references.html#KnPoVe1994b">KnPoVe1994b</a>].</li>
| |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l21 level1 lfo31;tab-stops:list .5in'>For k>4, LWP for small
| |
| data in <span class=SpellE>H^s</span>, s >=3/4 was obtained in [<a
| |
| href="references.html#KnPoVe1994b">KnPoVe1994b</a>].</li>
| |
| | |
| <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
| |
| mso-list:l21 level1 lfo31;tab-stops:list .5in'>For any k >= 3 and s
| |
| < 1/2 - 1/k the solution map is not uniformly continuous [<span
| |
| class=SpellE>BiLi</span>-p]</li>
| |
| </ul>
| |
| | |
| <p class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto'>The
| |
| <span class=SpellE>KdV</span>-Benjamin Ono (<span class=SpellE>KdV</span>-BO)
| |
| equation is formed by combining the linear parts of the <span class=SpellE>KdV</span>
| |
| | |
| and Benjamin-Ono equations together.<span style='mso-spacerun:yes'> </span>It
| |
| is globally well-posed in L^2 [<a href="references.html#Li1999">Li1999</a>],
| |
| and locally well-posed in H<span class=GramE>^{</span>-3/4+} [<span
| |
| class=SpellE>KozOgTns</span>] (see also [<span class=SpellE>HuoGuo</span>-p]
| |
| where H^{-1/8+} is obtained).<span style='mso-spacerun:yes'> </span>Similarly
| |
| one can generalize the non-linearity to be k-linear, generating for instance
| |
| the modified <span class=SpellE>KdV</span>-BO equation, which is locally
| |
| well-posed in H<span class=GramE>^{</span>1/4+} [<span class=SpellE>HuoGuo</span>-p].<span
| |
| style='mso-spacerun:yes'> </span>For general <span class=SpellE>gKdV-gBO</span>
| |
| | |
| equations one has local well-<span class=SpellE><span class=GramE>posedness</span></span><span
| |
| class=GramE><span style='mso-spacerun:yes'> </span>in</span> H^3 and above [<a
| |
| href="references.html#GuoTan1992">GuoTan1992</a>].<span
| |
| style='mso-spacerun:yes'> </span>One can also add damping terms <span
| |
| class=SpellE>Hu_x</span> to the equation; this arises as a model for ion-acoustic
| |
| waves of finite amplitude with linear Landau damping [<a
| |
| href="references.html#OttSud1982">OttSud1982</a>].</p>
| |
| | |
| </div>
| |
| | |
| </body>
| |
| | |
| </html>
| |