# Difference between revisions of "KP-II equation"

From DispersiveWiki

Jump to navigationJump to searchm (Tk-p should be Tk2000) |
m |
||

Line 6: | Line 6: | ||

** For s1 > -1/64 this is also in [[IsMj2001]]. | ** For s1 > -1/64 this is also in [[IsMj2001]]. | ||

* GWP for s1 > -1/78, s2 = 0 [[Tk2000]] assuming a moment condition. | * GWP for s1 > -1/78, s2 = 0 [[Tk2000]] assuming a moment condition. | ||

− | ** A similar result, with a slightly stricter constraint on s1 but no moment condition, was obtained in [[ | + | ** A similar result, with a slightly stricter constraint on s1 but no moment condition, was obtained in [[Tz2000]]. |

** For s1 = s2 ³ 0 this was proven in [[Bo1993c]], and this argument also applies to the periodic setting. Heuristically this result is indicated by the local smoothing estimates in [[Sau1993]]. | ** For s1 = s2 ³ 0 this was proven in [[Bo1993c]], and this argument also applies to the periodic setting. Heuristically this result is indicated by the local smoothing estimates in [[Sau1993]]. | ||

− | LWP for s1 > -1/3, s2 = 0 [[ | + | LWP for s1 > -1/3, s2 = 0 [[TkTz2001]], [[IsMj2001]] |

− | ** For s1 > -1/4, s2 = 0 this was shown in [[ | + | ** For s1 > -1/4, s2 = 0 this was shown in [[Tk2000b]] |

** For s1 > -e, s2 = 0 and small data this was shown in [[Tz1999]]. | ** For s1 > -e, s2 = 0 and small data this was shown in [[Tz1999]]. | ||

** For s1 = s2 ³ 0 this was proven in [[Bo1993c]], and this argument also applies to the periodic setting. | ** For s1 = s2 ³ 0 this was proven in [[Bo1993c]], and this argument also applies to the periodic setting. | ||

Line 15: | Line 15: | ||

** Related results are in [[IoNu1998]], [[IsMjStb2001]]. | ** Related results are in [[IoNu1998]], [[IsMjStb2001]]. | ||

* Weak solutions in a weighted L2 space were constructed in [[Fa1990]]. | * Weak solutions in a weighted L2 space were constructed in [[Fa1990]]. | ||

− | * For s1 < -1/3 the natural bilinear estimate fails [[ | + | * For s1 < -1/3 the natural bilinear estimate fails [[TkTz2001]]. |

* Remark: Unlike KP-I, KP-II does not admit soliton solutions. | * Remark: Unlike KP-I, KP-II does not admit soliton solutions. | ||

## Latest revision as of 01:22, 17 March 2007

The **KP-II equation** is the special case of the Kadomtsev-Petviashvili equation when the
parameter is positive.

- Scaling is s1 + 2s2 + 1/2 = 0.
- GWP for s1 > -1/14, s2 = 0 IsMj2003.
- For s1 > -1/64 this is also in IsMj2001.

- GWP for s1 > -1/78, s2 = 0 Tk2000 assuming a moment condition.

LWP for s1 > -1/3, s2 = 0 TkTz2001, IsMj2001

- Weak solutions in a weighted L2 space were constructed in Fa1990.
- For s1 < -1/3 the natural bilinear estimate fails TkTz2001.
- Remark: Unlike KP-I, KP-II does not admit soliton solutions.

The KP-II equation can be generalized to three dimensions (replace partial_yy with partial_yy + partial_zz), with s_1 regularity in the x direction and s_2 in the y,z directions. Scaling is now s_1 + 2s_2 – ½ = 0. In isotropic spaces, local well-posedness in H^s with s > 3/2 was established assuming the low frequency condition that partial_x^{-1} u is also in H^s Tz1999. Anisotropically, local well-posedness in the space s_1 > 1, s_2 > 0 was established in IsLopMj-p.