On Kato's smoothing effect for a fractional version of the Zakharov-Kuznetsov equation

TL;DR

This paper investigates the regularity properties of solutions to a fractional Zakharov-Kuznetsov equation, demonstrating a local smoothing effect of rac{ ext{alpha}}{2} derivatives in space, despite challenges posed by the non-local operator.

Contribution

It introduces a novel approach using perturbation and pseudo-differential calculus to establish local smoothing effects for a fractional PDE with non-local operators.

Findings

Solutions gain rac{ ext{alpha}}{2} derivatives in regularity locally in space.

The method overcomes difficulties from the non-local fractional Laplacian.

Regularity on certain subsets propagates with infinite speed.

Abstract

In this work we study some regularity properties associated to the initial value problem (IVP) \begin{equation}\label{main1} \left\{ \begin{array}{ll} \partial_{t}u-\partial_{x_{1}}(-\Delta)^{\alpha/2} u+u\partial_{x_{1}}u=0, \quad 0< \alpha\leq 2,& \\ u(x,0)=u_{0}(x),\quad x=(x_{1},x_{2},\dots,x_{n})\in \mathbb{R}^{n},\, n\geq 2,\quad t\in\mathbb{R},& \\ \end{array} \right. \end{equation} where $(-\Delta)^{\alpha/2}$ denotes the $n-$dimensional fractional Laplacian. We show that solutions to the IVP (0.1) with initial data in a suitable Sobolev space exhibit a local smoothing effect in the spatial variable of $\frac{\alpha}{2}$ derivatives, almost everywhere in time. One of the main difficulties that emerge when trying to obtain this regularizing effect underlies that the operator in consideration is non-local, and the property we are trying to describe is local, so new ideas are required. Nevertheless, to avoid these problems, we use a perturbation argument replacing $(-\Delta)^{\frac{\alpha}{2}}$ by $(I-\Delta)^{\frac{\alpha}{2}},$ that through the use of pseudo-differential calculus allows us to show that solutions become locally smoother by $\frac{\alpha}{2}$ of a derivative in all spatial directions. As a by-product, we use this particular smoothing effect to show that the extra regularity of the initial data on some distinguished subsets of the Euclidean space is propagated by the flow solution with infinity speed.