Cross-diffusion systems with non-zero-flux boundary conditions on a moving domain

TL;DR

This paper develops and analyzes a multi-species cross-diffusion model on a moving domain with non-zero-flux boundary conditions, proving existence, optimal control solutions, and long-term convergence, supported by numerical tests.

Contribution

It introduces a novel cross-diffusion system with non-zero-flux boundary conditions on a moving domain and establishes existence, optimal control, and convergence results.

Findings

Existence of a global weak solution is proven.

Optimal control solutions are established under certain conditions.

Concentrations converge to constant profiles over time.

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 mod- eling of a Physical Vapor Deposition process. Using the boundedness by entropy method, we prove the existence of a global weak solution to the obtained system. In addition, existence of a solution to an optimal control 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 dif- ferent 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.