# Difference between revisions of "Korteweg-de Vries equation on R"

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* Scaling is <span class="SpellE">s_c</span> = -3/2. | * Scaling is <span class="SpellE">s_c</span> = -3/2. | ||

− | * 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 [#mKdV_on_R <span class="SpellE">mKdV</span> theory]. This is despite the failure of the key bilinear estimate [[ | + | * 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 [#mKdV_on_R <span class="SpellE">mKdV</span> theory]. This is despite the failure of the key bilinear estimate [[Bibliography#NaTkTs2001| NaTkTs2001]] |

** For s within a logarithm for s=-3/4 [<span class="SpellE">MurTao</span>-p]. | ** For s within a logarithm for s=-3/4 [<span class="SpellE">MurTao</span>-p]. | ||

− | ** Was proven for s > -3/4 [[ | + | ** Was proven for s > -3/4 [[Bibliography#KnPoVe1996|KnPoVe1996]]. |

− | ** Was proven for s > -5/8 in [[ | + | ** Was proven for s > -5/8 in [[Bibliography#KnPoVe1993b|KnPoVe1993b]]. |

− | ** Was proven for s >= 0 in [[ | + | ** Was proven for s >= 0 in [[Bibliography#Bo1993b|Bo1993b]]. |

− | ** Was proven for s > 3/4 in [[ | + | ** Was proven for s > 3/4 in [[Bibliography#KnPoVe1993|KnPoVe1993]]. |

− | ** Was proven for s > 3/2 in [[ | + | ** Was proven for s > 3/2 in [[Bibliography#BnSm1975|BnSm1975]], [[Bibliography#Ka1975|Ka1975]], [[Bibliography#Ka1979|Ka1979]], [[Bibliography#GiTs1989|GiTs1989]], [[Bibliography#Bu1980|Bu1980]]<span class="GramE">, ....</span> |

− | ** 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 [[ | + | ** 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 [[Bibliography#KnPoVe-p |KnPoVe-p]]. |

*** 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 [#mKdV_on_R corresponding result for <span class="SpellE">mKdV</span>]. | *** 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 [#mKdV_on_R corresponding result for <span class="SpellE">mKdV</span>]. | ||

− | *** Below -3/4 the solution map was known to not be C^3 [[ | + | *** Below -3/4 the solution map was known to not be C^3 [[Bibliography#Bo1993b|Bo1993b]], [[Bibliography#Bo1997|Bo1997]]; this was refined to C^2 in [[Bibliography#Tz1999b|Tz1999b]]. |

− | ** When the initial data is a real, rapidly decreasing measure one has a global smooth solution for t > 0 [[ | + | ** When the initial data is a real, rapidly decreasing measure one has a global smooth solution for t > 0 [[Bibliography#Kp1993 | Kp1993]]. Without the rapidly decreasing hypothesis one can still construct a global weak solution [[Bibliography#Ts1989|Ts1989]] |

− | * GWP in <span class="SpellE">H^s</span> for s > -3/4 (if u is real) [[ | + | * GWP in <span class="SpellE">H^s</span> for s > -3/4 (if u is real) [[Bibliography#CoKeStTkTa2003| CoKeStTkTa2003]]. |

− | ** Was proven for s > -3/10 in [[ | + | ** Was proven for s > -3/10 in [[Bibliography#CoKeStTkTa2001|CoKeStTkTa2001]] |

− | ** Was proven for s>= 0 in [[ | + | ** Was proven for s>= 0 in [[Bibliography#Bo1993b|Bo1993b]]. Global weak solutions in L^2 were constructed in [[Bibliography#Ka1983|Ka1983]], [[Bibliography#KrFa1983|KrFa1983]], and were shown to obey the expected local smoothing estimate. These weak solutions were shown to be unique in [[Bibliography#Zh1997b|Zh1997b]] |

− | ** Was proven for s>= 1 in [[ | + | ** Was proven for s>= 1 in [[Bibliography#KnPoVe1993|KnPoVe1993]]. |

− | ** Was proven for s>= 2 in [[ | + | ** Was proven for s>= 2 in [[Bibliography#BnSm1975|BnSm1975]], [[Bibliography#Ka1975|Ka1975]], [[Bibliography#Ka1979|Ka1979]]<span class="GramE">, ....</span> |

** ''Remark''<nowiki>: In the complex setting GWP fails for large data with Fourier support on the half-line [Bona/</nowiki><span class="SpellE">Winther</span>?], [<span class="SpellE">Birnir</span>]<span class="GramE">, ????</span>. This result extends to a wide class of dispersive PDE. | ** ''Remark''<nowiki>: In the complex setting GWP fails for large data with Fourier support on the half-line [Bona/</nowiki><span class="SpellE">Winther</span>?], [<span class="SpellE">Birnir</span>]<span class="GramE">, ????</span>. This result extends to a wide class of dispersive PDE. | ||

* 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. | * 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. | ||

− | * <span class="SpellE">Solitons</span> are <span class="SpellE">orbitally</span> H^1 stable [[ | + | * <span class="SpellE">Solitons</span> are <span class="SpellE">orbitally</span> H^1 stable [[Bibliography#Bj1972|Bj1972]] |

** 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] | ** 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] | ||

− | ** In L^2, orbital stability has been obtained in [[ | + | ** In L^2, orbital stability has been obtained in [[Bibliography#MeVe2003|MeVe2003]]. |

==KdV-like systems== | ==KdV-like systems== | ||

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<center><math>\partial_t u + \partial_x^3 u + a_3 \partial_x^3 v + u \partial_x u + a_1 v \partial_x v + a_2 \partial_x (uv)= 0</math><br ><math>b_1 \partial_t v + \partial_x^3 v + b_2 a_3 \partial_x^3 u + v \partial_x v + b_2 a_2 u \partial_x u + b_2 a_1 \partial_x (uv) + r \partial_x v</math></center> | <center><math>\partial_t u + \partial_x^3 u + a_3 \partial_x^3 v + u \partial_x u + a_1 v \partial_x v + a_2 \partial_x (uv)= 0</math><br ><math>b_1 \partial_t v + \partial_x^3 v + b_2 a_3 \partial_x^3 u + v \partial_x v + b_2 a_2 u \partial_x u + b_2 a_1 \partial_x (uv) + r \partial_x v</math></center> | ||

− | <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 [[ | + | <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 [[Bibliography#GeaGr1984|GeaGr1984]] to study strongly interacting pairs of weakly nonlinear long waves, and studied further in [[Bibliography#BnPoSauTm1992|BnPoSauTm1992]]. In [[Bibliography#AsCoeWgg1996|AsCoeWgg1996]] 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. |

[[Category:Equations]] | [[Category:Equations]] | ||

[[Category:Airy]] | [[Category:Airy]] |

## Revision as of 03:24, 31 July 2006

The local and global well-posedness theory for the Korteweg-de Vries equation on the real line is as follows.

- Scaling is s_c = -3/2.
- LWP in H^s for s >= -3/4 [CtCoTa-p], using a modified Miura transform and the [#mKdV_on_R mKdV theory]. This is despite the failure of the key bilinear estimate NaTkTs2001
- For s within a logarithm for s=-3/4 [MurTao-p].
- Was proven for s > -3/4 KnPoVe1996.
- Was proven for s > -5/8 in KnPoVe1993b.
- Was proven for s >= 0 in Bo1993b.
- Was proven for s > 3/4 in KnPoVe1993.
- Was proven for s > 3/2 in BnSm1975, Ka1975, Ka1979, GiTs1989, Bu1980, ....
- One has local ill-posedness(in the sense that the map is not uniformly continuous) for s < -3/4 (in the complex setting) by soliton examples KnPoVe-p.
- When the initial data is a real, rapidly decreasing measure one has a global smooth solution for t > 0 Kp1993. Without the rapidly decreasing hypothesis one can still construct a global weak solution Ts1989

- GWP in H^s for s > -3/4 (if u is real) CoKeStTkTa2003.
- Was proven for s > -3/10 in CoKeStTkTa2001
- Was proven for s>= 0 in Bo1993b. Global weak solutions in L^2 were constructed in Ka1983, KrFa1983, and were shown to obey the expected local smoothing estimate. These weak solutions were shown to be unique in Zh1997b
- Was proven for s>= 1 in KnPoVe1993.
- Was proven for s>= 2 in BnSm1975, Ka1975, Ka1979, ....
*Remark*: In the complex setting GWP fails for large data with Fourier support on the half-line [Bona/Winther?], [Birnir], ????. This result extends to a wide class of dispersive PDE.

- By use of the inverse scattering transform one can show that smooth solutions eventually resolve into solitons, that two colliding solitons emerge as (slightly phase shifted) solitons, etc.
- Solitons are orbitally H^1 stable Bj1972
- In H^s, 0 <= s < 1, the orbital stability of solitons is at most polynomial (the distance to the ground state manifold in H^s norm grows like at most O(t^{1-s+}) in time) [RaySt-p]
- In L^2, orbital stability has been obtained in MeVe2003.

## KdV-like systems

The KdV equation can also be generalized to a 2x2 system

where b_1,b_2 are positive constants and a_1,a_2,a_3,r are real constants. This system was introduced in GeaGr1984 to study strongly interacting pairs of weakly nonlinear long waves, and studied further in BnPoSauTm1992. In AsCoeWgg1996 it was shown that this system was also globally well-posed on L^2.

It is an interesting question as to whether these results can be pushed further to match the KdV theory; the apparent lack of complete integrability in this system (for generic choices of parameters b_i, a_i, r) suggests a possible difficulty.