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

TL;DR

This paper proves the global existence of strong solutions to the 3D Zakharov-Kuznestov equation in a bounded domain, introducing anisotropic estimation techniques that could apply to other nonlinear PDEs.

Contribution

It establishes the first known global existence result for strong solutions of the 3D ZK equation and presents a novel anisotropic estimation method for nonlinear PDEs.

Findings

Existence of solutions in 3D for all time T

Application of anisotropic estimation to cancel nonlinear terms

Potential applicability to other nonlinear equations

Abstract

We consider the Zakharov-Kuznestov (ZK) equation posed in a limited domain (0,1)_{x}\times(-\pi /2, \pi /2)^d, d=1,2 supplemented with suitable boundary conditions. We prove that there exists a solution u \in \mathcal C ([0, T]; H^1(\dom)) to the initial and boundary value problem for the ZK equation in both dimensions 2 and 3 for every T>0. To the best of our knowledge, this is the first result of the global existence of strong solutions for the ZK equation in 3D. More importantly, the idea behind the application of anisotropic estimation to cancel the nonlinear term, we believe, is not only suited for this model but can also be applied to other nonlinear equations with similar structures. At the same time, the uniqueness of solutions is still open in 2D and 3D due to the partially hyperbolic feature of the model.