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

Francis Ribaud (1) , Stéphane Vento (2)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 LAGA - Laboratoire Analyse, Géométrie et Applications

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 $k\ge 4$.

Mots clés: KdV-like equations; Cauchy problem

Domaines: Equations aux dérivées partielles [math.AP]

Francis Ribaud, Stéphane Vento. A note on the Cauchy problem for the 2D generalized Zakharov-Kuznetsov equations. 2011. ⟨hal-00642572⟩