Cross-diffusion systems with non-zero flux and moving boundary conditions

¹ Université Paris-Est, CERMICS(ENPC), Marne-la-Vallée, France
² Université Paris-Est, CERMICS (ENPC) and INRIA (Matherials team-project), 77455 Marne-la-Vallée, France

^* Corresponding author: virginie.ehrlacher@enpc.fr

Received: 12 January 2017
Revised: 30 August 2017
Accepted: 5 October 2017

Abstract

We propose and analyze a one-dimensional multi-species cross-diffusion system with non-zero-flux boundary conditions on a moving domain, motivated by the modeling of a Physical Vapor Deposition process. Using the boundedness by entropy method introduced and developped in [5, 16], we prove the existence of a global weak solution to the obtained system. In addition, existence of a solution to an optimization problem defined on the fluxes is established under the assumption that the solution to the considered cross-diffusion system is unique. Lastly, we prove that in the case when the imposed external fluxes are constant and positive and the entropy density is defined as a classical logarithmic entropy, the concentrations of the different species converge in the long-time limit to constant profiles at a rate inversely proportional to time. These theoretical results are illustrated by numerical tests.