Well-posedness for the Kadomtsev-Petviashvili II equation

By Hideo Takaoka. Advances in Differential Equations, 2000

72 citations recorded by the source corpus.

We study the global well-posedness of the Cauchy problem for the KP II equation. We prove the global well-posedness in the inhomogeneous-homogeneous anisotropic Sobolev spaces $H_{x,y}^{-1/78+\epsilon,0}\cap H_{x,y}^{-17/144,0}$. Though we require the use of the homogeneous Sobolev space of negative index, we obtain the global well-posedness below $L^2$.

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1ae3488a0e164ba86b425258a6e5cfe6adef43f2

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In Mathematical Physics

Citations in the corpus

LOCAL AND GLOBAL CAUCHY PROBLEMS FOR THE KADOMTSEV–PETVIASHVILI (KP–II) EQUATION IN SOBOLEV SPACES OF NEGATIVE INDICES (2001, background): “In [8], appart from the former result, Takaoka obtains local well-posedness for initial data in Sobolev anisotropic spaces with an additional condition which allows to reduce the index s1 to s1 > − (1)2 .”
Sharp Global well-posedness for KdV and modified KdV on $\R$ and $\T$ (2001, unknown): “Subsequently, Bourgain’s high/low trick has been applied to prove global well-posedness below the natural regularity of the conserved quantity in various settings [17], [24], [25], [43], [41], [29], including KdV [15] on the line.”
Sharp global well-posedness for KdV and modified KdV on ℝ and (2003, methodology): “Subsequently, Bourgain’s high/low method has been applied to prove global well-posedness below the natural regularity of the conserved quantity in various settings [20], [50], [ 48 ], [34], including KdV [18] on the line.”
Global well-posedness for the KP-I equation (2002, methodology): “Local and global well-posedness for the KP-II equation with data below L 2 were obtained in [7, 14 ,15,17].”
Well-posedness for the Kadomtsev-Petviashvili II equation and generalisations (2006, background): “) In [14] Takaoka shows local well-posedness for s1 > − 1 2 , s2 = 0, but only if the additional low frequency condition |Dx| − 12+εu0 ∈ L 2 (with suitably chosen ε) is imposed on the initial value.”
Well-posedness and scattering for the KP-II equation in a critical space (2007, background): “Takaoka [19] in the homogeneous spaces and by the first author [7] in the inhomogeneous spaces.”
Global well-posedness for the KP-BBM equations (2004, methodology): “Local and global well-posedness for the KP-II equation with data below L2 were obtained in [7,14,15,17].”