On the Cauchy problems associated to a ZK-KP-type family equations with a transversal fractional dispersion

TL;DR

This paper investigates the well-posedness and ill-posedness of a family of ZK-KP-type equations with fractional dispersion in anisotropic Sobolev spaces, focusing on the effects of the fractional derivative parameter.

Contribution

It provides a detailed analysis of the well-posedness thresholds for ZK-KP-type equations with fractional dispersion, highlighting the role of the fractional order in solution behavior.

Findings

Identifies conditions for well-posedness in anisotropic Sobolev spaces.

Establishes ill-posedness results for certain fractional orders.

Analyzes the influence of fractional dispersion on solution regularity.

Abstract

In this paper we examine the well-posedness and ill-posedeness of the Cauchy problems associated to a family equations of ZK-KP-type \[ \begin{cases} u_{t}=u_{xxx}-\mathscr{H}D_{x}^{\alpha}u_{yy}+uu_{x}, \cr u(0)=\psi \in Z \end{cases} \] in anisotropic Sobolev spaces, where $1\le \alpha \le 1$, $\mathscr{H}$ is the Hilbert transform and $D_{x}^{\alpha}$ is the fractional derivative, both with respect to $x$.