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

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.