Cross-diffusion systems with non-zero-flux boundary conditions on a   moving domain
  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. 
Athmane Bakhta and Virginie Ehrlacher
math.AP