Cross-diffusion systems with non-zero flux and moving boundary conditionsReport as inadecuate




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1 CERMICS - Centre d-Enseignement et de Recherche en Mathématiques et Calcul Scientifique 2 MATHERIALS - MATHematics for MatERIALS CERMICS - Centre d-Enseignement et de Recherche en Mathématiques et Calcul Scientifique, Inria de Paris

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 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 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 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.

Keywords : optimal control boundedness by entropy cross-diffusion PDE





Author: Athmane Bakhta - Virginie Ehrlacher -

Source: https://hal.archives-ouvertes.fr/



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