Local Existence of Strong Solutions to the 3D Zakharov-Kuznestov Equation in a Bounded Domain

TL;DR

This paper establishes the local existence of strong solutions for the 3D Zakharov-Kuznestov equation in bounded domains using regularization and perturbation techniques, applicable in dimensions 2 and 3.

Contribution

It proves the local existence of strong solutions in bounded domains for the 3D ZK equation, employing a novel regularization and singular perturbation approach.

Findings

Existence of strong solutions in 2D and 3D for short time intervals.

Boundedness of solutions independent of regularization parameter.

Method applicable to nonlinear PDEs in bounded domains.

Abstract

We consider here the local existence of strong solutions for the Zakharov-Kuznestov (ZK) equation posed in a limited domain (0,1)_{x}\times(-pi /2, pi /2)^d, d=1,2. We prove that in space dimensions 2 and 3, there exists a strong solution on a short time interval, whose length only depends on the given data. We use the parabolic regularization of the ZK equation to derive the global and local bounds independent of epsilon for various norms of the solution. In particular, we derive the local bound of the nonlinear term by a singular perturbation argument. Then we can pass to the limit and hence deduce the local existence of strong solutions.