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

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

2011-11-18

arxiv.org/abs/1111.4384v1

Mathematics

Analysis of PDEs

Scientific paper

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 $k\ge 4$.

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