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arXiv · 2205.06478

Existence and weak-strong uniqueness for Maxwell-Stefan-Cahn-Hilliard systems

Xiaokai Huo·Ansgar Jüngel·Athanasios E. Tzavaras
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Abstract

A Maxwell-Stefan system for fluid mixtures with driving forces depending on Cahn-Hilliard-type chemical potentials is analyzed. The corresponding parabolic cross-diffusion equations contain fourth-order derivatives and are considered in a bounded domain with no-flux boundary conditions. The main difficulty of the analysis is the degeneracy of the diffusion matrix, which is overcome by proving the positive definiteness of the matrix on a subspace and using the Bott--Duffin matrix inverse. The global existence of weak solutions and a weak-strong uniqueness property are shown by a careful combination of (relative) energy and entropy estimates, yielding $H^2(\Omega)$ bounds for the densities, which cannot be obtained from the energy or entropy inequalities alone.

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Xiaokai Huo, Ansgar Jüngel, Athanasios E. Tzavaras. 2022-05-13. Existence and weak-strong uniqueness for Maxwell-Stefan-Cahn-Hilliard systems. https://arxiv.org/abs/2205.06478

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