Polynomial Growth of high Sobolev Norms of solutions to the Zakharov-Kuznetsov Equation

TL;DR

This paper proves that solutions to the 2D Zakharov-Kuznetsov equation with high regularity initial data exhibit at most polynomial growth in their high Sobolev norms over time, shedding light on energy transfer to high frequencies.

Contribution

It establishes polynomial bounds on the growth of high Sobolev norms for solutions to the 2D Zakharov-Kuznetsov equation, extending understanding of long-term behavior.

Findings

High Sobolev norms grow at most polynomially in time

Results relate to wave turbulence and energy transfer to high frequencies

Uses bilinear estimates in Bourgain's spaces

Abstract

We consider the Zakharov-Kuznetsov equation (ZK) in space dimension 2. Solutions u with initial data u\_0 $\in$ H s are known to be global if s $\ge$ 1. We prove that for any integer s $\ge$ 2, u(t) H s grows at most polynomially in t for large times t. This result is related to wave turbulence and how a solution of (ZK) can move energy to high frequencies. It is inspired by analoguous results by Staffilani on the non linear Schr{\"o}dinger Korteweg-de-Vries equation. The main ingredients are adequate bilinear estimates in the context of Bourgain's spaces.