The Zakharov-Kuznetsov equation in weighted Sobolev spaces

TL;DR

This paper investigates the well-posedness of the two-dimensional Zakharov-Kuznetsov equation within weighted Sobolev spaces, extending understanding of solution behavior in these function spaces.

Contribution

It establishes well-posedness results for the Zakharov-Kuznetsov equation in weighted Sobolev spaces, a novel analysis in this context.

Findings

Well-posedness in weighted Sobolev spaces established

Extension of solution theory to include weights

Insights into decay and regularity of solutions

Abstract

In this work we consider the initial value problem (IVP) associated to the two dimensional Zakharov-Kuznetsov equation $$\left. \begin{array}{rl} u_t+\partial_x^3 u+\partial_x \partial_y^2 u +u \partial_x u &\hspace{-2mm}=0,\qquad\qquad (x,y)\in\mathbb R^2,\; t\in\mathbb R,\\ u(x,y,0)&\hspace{-2mm}=u_0(x,y). \end{array} \right\}$$ We study the well-posedness of the IVP in the weighted Sobolev spaces $$H^s(\mathbb R^2) \cap L^2((1+x^2+y^2)^{r} dx dy),$$ with $s,r\in\mathbb R$.