Global well-posedness of partially periodic KP-I equation in the energy space and application

  • Tristan Robert

    Université de Cergy-Pontoise, Laboratoire AGM, 2 av. Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France
Global well-posedness of partially periodic KP-I equation in the energy space and application cover
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Abstract

In this article, we address the Cauchy problem for the KP-I equation

tu+x3ux1y2u+uxu=0\partial _{t}u + \partial _{x}^{3}u−\partial _{x}^{−1}\partial _{y}^{2}u + u\partial _{x}u = 0

for functions periodic in yy. We prove global well-posedness of this problem for any data in the energy space E={uL2(R×T),xuL2(R×T),x1yuL2(R×T)}\mathbf{E} = \{u \in L^{2}(\mathbb{R} \times \mathbb{T}),\partial _{x}u \in L^{2} (\mathbb{R} \times \mathbb{T}),\:\partial _{x}^{−1}\partial _{y}u \in L^{2}(\mathbb{R} \times \mathbb{T})\}. We then prove that the KdV line soliton, seen as a special solution of KP-I equation, is orbitally stable under this flow, as long as its speed is small enough.

Cite this article

Tristan Robert, Global well-posedness of partially periodic KP-I equation in the energy space and application. Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 7, pp. 1773–1826

DOI 10.1016/J.ANIHPC.2018.03.002