Global regularity of solutions of equation modeling epitaxy thin film growth in $${\mathbb{R}^d, d = 1,2}$$ R d , d = 1 , 2

Abstract

We show existence and uniqueness of global strong solutions for any initial data \({u_0 \in H^s(\mathbb{R}^d)}\), with \({d\in \{1,2\}, s \geq 3}\), of the general equation of surface growth models arising in the context of epitaxy thin films in the presence of the coarsening process, density variations and the Ehrlich–Schwoebel effect. Up to now, the problem of existence and smoothness of global solutions of such equations remains open in \({\mathbb{R}^d, d \in \{1,2\}}\). In this article, we show that taking into account of the main physical phenomena and a better approximation of terms related to them in the mathematical model lead to a kind of ”cancelation” of nonlinear terms between them in some spaces, and from this, we obtain existence and uniqueness of global strong solutions for such equations in \({\mathbb{R}^d, d \in \{1,2\}}\).