On the fractional Zakharov-Kuznetsov equation

Speaker: Argenis Mendez 

Pontificia Universidad Católica de Valparaíso

Date: August 16, 2022 at 12 Santiago time

Abstract: In this talk, we will present some new results related to the regularity properties of the initial value problem (IVP) for the equation
\begin{equation}\label{eq1}
\left\{
\begin{array}{ll}
\partial_{t}u-\partial_{x_{1}}(-\Delta)^{\alpha/2} u+u\partial_{x_{1}}u=0, \quad 0< \alpha< 2, & \\
u(x,0)=u_{0}(x),x=(x_{1},x_{2},\dots,x_{n})\in \mathbb{R}^{n},n\geq 2,& t\in\mathbb{R}, \\
\end{array}
\right.
\end{equation}
where $(-\Delta)^{\alpha/2}$ denotes the $n-$dimensional fractional Laplacian.

In the case that \(\alpha=2,\) the equation is known as the Zakharov-Kuznetsov-(ZK) equation, Zakharov and Kuznetsov proposed it as a model to describe the propagation of ion-sound waves in magnetic fields in dimension n=3.

A property that enjoys the solutions of the ZK equation is Kato’s smoothing effect. Roughly speaking, the solution to the initial value problem is, locally, one derivative smoother (in all directions) in comparison to the initial data.

The goal of this talk is to show that despite the non-local character of the operator \((-\Delta)^{\frac{\alpha}{2}}\), the solution of the equation (IVP) is locally smoother. It becomes \(\frac{\alpha}{2}-\) smoother in all directions.

As a byproduct, we show the applicability of this result in establishing the propagation of localized regularity of the solutions of (IVP) in a suitable Sobolev space.

Venue: Online via Zoom / Sala de seminarios John Von Neumann, Beauchef 851,  piso 7
Chair: Claudio Muñoz