A note on the Cauchy problem for the 2D generalized Zakharov-Kuznetsov equations

Stéphane Vento; Francis Ribaud

A note on the Cauchy problem for the 2D generalized Zakharov-Kuznetsov equations

Analysis of PDEs 2011-11-21 v1

Authors: Stéphane Vento , Francis Ribaud

Abstract

In this note we study the generalized 2D Zakharov-Kuznetsov equations $\partial_tu+\Delta\partial_xu+u^k\partial_xu=0$ for $k\ge 2$ . By an iterative method we prove the local well-posedness of these equations in the Sobolev spaces $H^s(\mathbb{R}^2)$ for $s>1/4$ if $k=2$ , $s>5/12$ if $k=3$ and $s>1-2/k$ if $k\ge 4$ .

Keywords

cauchy problem well-posedness sobolev regularity of partial differential equations

Cite

@article{arxiv.1111.4384,
  title  = {A note on the Cauchy problem for the 2D generalized Zakharov-Kuznetsov equations},
  author = {Stéphane Vento and Francis Ribaud},
  journal= {arXiv preprint arXiv:1111.4384},
  year   = {2011}
}

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