Existence of solutions for the surface electromigration equation

TL;DR

This paper proves local well-posedness for the surface electromigration equation in Sobolev spaces, using advanced Fourier analysis techniques, and explores solutions both in pure and KdV wave backgrounds.

Contribution

It establishes the local well-posedness of the surface electromigration equation in Sobolev spaces, including in the presence of KdV solitary waves, using novel analytical methods.

Findings

Well-posedness in $H^s(\mathbb{R}^2)$ for $s>1/2$

Existence of solutions on KdV wave backgrounds

Application of bilinear estimates and smoothing properties

Abstract

We consider a model that describes electromigration in nanoconductors known as surface electromigration (SEM) equation. Our purpose here is to establish local well-posedness for the associated initial value problem in Sobolev spaces from two different points of view. In the first one, we study the pure Cauchy problem and establish local well-posedness in $H^s(\mathbb{R}^2)$, $s>1/2$. In the second one, we study the Cauchy problem on the background of a Korteweg-de Vries solitary traveling wave in a less regular space. To obtain our results we make use of the smoothing properties of solutions for the linear problem corresponding to the Zakharov-Kuznetsov equation for the latter problem. For the former problem we use bilinear estimates in Fourier restriction spaces established by Molinet and Pilod.