Well-posedness for a molecular beam epitaxy model

Abstract

We study a general molecular beam epitaxy (MBE) equation modeling the epitaxial growth of thin films. We show that, in the deterministic case, the associated Cauchy problem admits a unique smooth solution for all time, given initial data in the space $X_0=L^2(R^d)\cap {\dot{W}}^{1,4}(R^d)$ with $d=1,2$. This improves a recent result by Agélas, who established global existence in $H^3(R^d)$. Moreover, we investigate the local existence and uniqueness of solutions in the space $X_0$ for the stochastic MBE equation, with an additive noise that is white in time and regular in the space variable.