Difference between revisions of "Generalized Korteweg-de Vries equation"

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((Some) cleaning of bibliographic references)
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* 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].
 
* 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].
** The techniques are based on [[references.html#KnPoVe1993 KnPoVe1993]] and a replacement of the IVBP with a forced IVP.
+
** The techniques are based on [[Bibliography#KnPoVe1993|KnPoVe1993]] and a replacement of the IVBP with a forced IVP.
 
** 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].
 
** 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].
 
** More specific results are known for [[Korteweg-de Vries on the half-line|KdV]], [[modified Korteweg-de Vries on the half-line|mKdV]], [[gKdV-3 on the half-line|gKdV-3]], and [[gKdV-4 on the half-line|gKdV-4]].
 
** More specific results are known for [[Korteweg-de Vries on the half-line|KdV]], [[modified Korteweg-de Vries on the half-line|mKdV]], [[gKdV-3 on the half-line|gKdV-3]], and [[gKdV-4 on the half-line|gKdV-4]].
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[Thanks to <span class="SpellE">Nikolaos</span> <span class="SpellE">Tzirakis</span> for some corrections - Ed.]
 
[Thanks to <span class="SpellE">Nikolaos</span> <span class="SpellE">Tzirakis</span> for some corrections - Ed.]
  
* 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 [[references.html#KnPoVe1993 KnPoVe1993]]
+
* 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 [[Bibliography#KnPoVe1993|KnPoVe1993]]
** Was shown for s>3/2 in [[references.html#GiTs1989 GiTs1989]]
+
** Was shown for s>3/2 in [[Bibliography#GiTs1989|GiTs1989]]
** One has ill-<span class="SpellE">posedness</span> in the supercritical regime [[references.html#BirKnPoSvVe1996 BirKnPoSvVe1996]]
+
** One has ill-<span class="SpellE">posedness</span> in the supercritical regime [[Bibliography#BirKnPoSvVe1996|BirKnPoSvVe1996]]
** For small data one has scattering [[references.html#KnPoVe1993c KnPoVe1993c]].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.
+
** For small data one has scattering [[Bibliography#KnPoVe1993c|KnPoVe1993c]].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.
** <span class="SpellE">Solitons</span> are H^1-unstable [[references.html#BnSouSr1987 BnSouSr1987]]
+
** <span class="SpellE">Solitons</span> are H^1-unstable [[Bibliography#BnSouSr1987|BnSouSr1987]]
** 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 [[references.html#St1995 St1995]]
+
** 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 [[Bibliography#St1995|St1995]]
* On R with any k, <span class="SpellE">gKdV</span>-k is GWP in <span class="SpellE">H^s</span> for s >= 1 [[references.html#KnPoVe1993 KnPoVe1993]], 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.
+
* On R with any k, <span class="SpellE">gKdV</span>-k is GWP in <span class="SpellE">H^s</span> for s >= 1 [[Bibliography#KnPoVe1993|KnPoVe1993]], 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.
* 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 [[references.html#St1997b St1997b]]
+
* 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 [[Bibliography#St1997b|St1997b]]
* 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 [[references.html#KnPoVe-p3 KnPoVe-p3]], [KnPoVe-p4].
+
* 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 [[references:KnPoVe-p3 KnPoVe-p3]], [KnPoVe-p4].
** In the completely <span class="SpellE">integrable</span> cases k=1,2 this is in [[references.html#Zg1992 Zg1992]]
+
** In the completely <span class="SpellE">integrable</span> cases k=1,2 this is in [[Bibliography#Zg1992|Zg1992]]
** Also, a non-zero solution to <span class="SpellE">gKdV</span> cannot vanish on a rectangle in <span class="SpellE">spacetime</span> [[references.html#SauSc1987 SauSc1987]]; see also [[references.html#Bo1997b Bo1997b]].
+
** Also, a non-zero solution to <span class="SpellE">gKdV</span> cannot vanish on a rectangle in <span class="SpellE">spacetime</span> [[Bibliography#SauSc1987|SauSc1987]]; see also [[Bibliography#Bo1997b|Bo1997b]].
** Extensions to higher order <span class="SpellE">gKdV</span> type equations are in [[references.html#Bo1997b Bo1997b]], [KnPoVe-p5].
+
** Extensions to higher order <span class="SpellE">gKdV</span> type equations are in [[Bibliography#Bo1997b|Bo1997b]], [KnPoVe-p5].
* 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... [[references.html#CtWs1991 CtWs1991]]
+
* 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... [[Bibliography#CtWs1991|CtWs1991]]
** A similar result for k > (5+<span class="GramE">sqrt(</span>73))/4 ~ 3.39... <span class="GramE">was</span> obtained in [[references.html#PoVe1990 PoVe1990]].
+
** A similar result for k > (5+<span class="GramE">sqrt(</span>73))/4 ~ 3.39... <span class="GramE">was</span> obtained in [[Bibliography#PoVe1990|PoVe1990]].
** 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 [[references.html#AbSe1977 AbSe1977]]
+
** 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 [[Bibliography#AbSe1977|AbSe1977]]
 
* 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]
 
* 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]
** For a single <span class="SpellE">soliton</span> this is in [MtMe-p3], [<span class="SpellE">MtMe</span>-p], [[references.html#Miz2001 Miz2001]]; earlier work is in [[references.html#Bj1972 Bj1972]], [[references.html#Bn1975 Bn1975]], [[references.html#Ws1986 Ws1986]], [[references.html#PgWs1994 PgWs1994]]
+
** For a single <span class="SpellE">soliton</span> this is in [MtMe-p3], [<span class="SpellE">MtMe</span>-p], [[Bibliography#Miz2001|Miz2001]]; earlier work is in [[Bibliography#Bj1972|Bj1972]], [[Bibliography#Bn1975|Bn1975]], [[Bibliography#Ws1986|Ws1986]], [[Bibliography#PgWs1994|PgWs1994]]
* A dissipative version of <span class="SpellE">gKdV</span>-k was analyzed in [[references.html#MlRi2001 MlRi2001]]
+
* A dissipative version of <span class="SpellE">gKdV</span>-k was analyzed in [[Bibliography#MlRi2001|MlRi2001]]
  
* 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 [[references.html#St1997b St1997b]]
+
* 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 [[Bibliography#St1997b|St1997b]]
* On T with k >= 3, <span class="SpellE">gKdV</span>-k is LWP for s >= 1/2 [[references.html#CoKeStTaTk-p3 CoKeStTkTa-p3]]
+
* On T with k >= 3, <span class="SpellE">gKdV</span>-k is LWP for s >= 1/2 [[references:CoKeStTaTk-p3 CoKeStTkTa-p3]]
** Was shown for s >= 1 in [[references.html#St1997c St1997c]]
+
** Was shown for s >= 1 in [[Bibliography#St1997c|St1997c]]
** Analytic well-<span class="SpellE">posedness</span> fails for s < 1/2 [[references.html#CoKeStTaTk-p3 CoKeStTkTa-p3]], [[references.html#KnPoVe1996 KnPoVe1996]]
+
** Analytic well-<span class="SpellE">posedness</span> fails for s < 1/2 [[references:CoKeStTaTk-p3 CoKeStTkTa-p3]], [[Bibliography#KnPoVe1996|KnPoVe1996]]
** For arbitrary smooth non-<span class="SpellE">linearities</span>, weak H^1 solutions were constructed in [[references.html#Bo1993b Bo1993]].
+
** For arbitrary smooth non-<span class="SpellE">linearities</span>, weak H^1 solutions were constructed in [[references:Bo1993b Bo1993]].
* 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 [[references.html#St1997c St1997c]]
+
* 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 [[Bibliography#St1997c|St1997c]]
** The estimates in [[references.html#CoKeStTaTk-p3 CoKeStTkTa-p3]] suggest that this is improvable to 13/14 - 2/7k, but this has only been proven in the sub-critical case k=3 [[references.html#CoKeStTaTk-p3 CoKeStTkTa-p3]]. In the critical and super-critical cases there are some low-frequency issues which may require the techniques in [[references.html#KeTa-p <span class="SpellE">KeTa</span>-p]].
+
** The estimates in [[references:CoKeStTaTk-p3 CoKeStTkTa-p3]] suggest that this is improvable to 13/14 - 2/7k, but this has only been proven in the sub-critical case k=3 [[references:CoKeStTaTk-p3 CoKeStTkTa-p3]]. In the critical and super-critical cases there are some low-frequency issues which may require the techniques in [[references:KeTa-p <span class="SpellE">KeTa</span>-p]].
  
 
[[Category:Equations]]
 
[[Category:Equations]]

Revision as of 19:38, 28 July 2006

Half-line theory

The gKdV Cauchy-boundary problem on the half-line is

u_t + u_{xxx} + u_x + u^k u_x = 0; u(x,0) = u_0(x); u(0,t) = h(t)

The sign of u_{xxx} is important (it makes the influence of the boundary x=0 mostly negligible), the sign of u u_x is not. The drift term u_x is convenient for technical reasons; it is not known whether it is truly necessary.

  • LWP is known for initial data in H^s and boundary data in H^{(s+1)/3} when s > 3/4 [CoKn-p].
    • The techniques are based on KnPoVe1993 and a replacement of the IVBP with a forced IVP.
    • This has been improved to s >= s_c = 1/2 - 2/k when k > 4 [CoKe-p].
    • More specific results are known for KdV, mKdV, gKdV-3, and gKdV-4.

Miscellaneous gKdV results

[Thanks to Nikolaos Tzirakis for some corrections - Ed.]

  • On R with k > 4, gKdV-k is LWP down to scaling: s >= s_c = 1/2 - 2/k KnPoVe1993
    • Was shown for s>3/2 in GiTs1989
    • One has ill-posedness in the supercritical regime BirKnPoSvVe1996
    • For small data one has scattering KnPoVe1993c.Note that one cannot have scattering in L^2 except in the critical case k=4 because one can scale solitons to be arbitrarily small in the non-critical cases.
    • Solitons are H^1-unstable BnSouSr1987
    • If one considers an arbitrary smooth non-linearity (not necessarily a power) then one has LWP for small data in H^s, s > 1/2 St1995
  • On R with any k, gKdV-k is GWP in H^s for s >= 1 KnPoVe1993, 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.
  • On R with any k, gKdV-k has the H^s norm growing like t^{(s-1)+} in time for any integer s >= 1 St1997b
  • On R with any non-linearity, a non-zero solution to gKdV cannot be supported on the half-line R^+ (or R^-) for two different times references:KnPoVe-p3 KnPoVe-p3, [KnPoVe-p4].
    • In the completely integrable cases k=1,2 this is in Zg1992
    • Also, a non-zero solution to gKdV cannot vanish on a rectangle in spacetime SauSc1987; see also Bo1997b.
    • Extensions to higher order gKdV type equations are in Bo1997b, [KnPoVe-p5].
  • On R with non-integer k, one has decay of O(t^{-1/3}) in L^\infty for small decaying data if k > (19 - sqrt(57))/4 ~ 2.8625... CtWs1991
    • A similar result for k > (5+sqrt(73))/4 ~ 3.39... was obtained in PoVe1990.
    • 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 AbSe1977
  • In the L^2 subcritical case 0 < k < 4, multisoliton solutions are asymptotically H^1-stable [MtMeTsa-p]
  • A dissipative version of gKdV-k was analyzed in MlRi2001