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

  • Xiaokai Huo

    Technische Universität Wien, Austria
  • Ansgar Jüngel

    Technische Universität Wien, Austria
  • Athanasios E. Tzavaras

    King Abdullah University of Science and Technology, Thuwal, Saudi Arabia
Existence and weak–strong uniqueness for Maxwell–Stefan–Cahn–Hilliard systems cover
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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 nonconvex part of the energy is assumed to have a bounded Hessian. 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 bounds for the densities, which cannot be obtained from the energy or entropy inequalities alone.

Cite this article

Xiaokai Huo, Ansgar Jüngel, Athanasios E. Tzavaras, Existence and weak–strong uniqueness for Maxwell–Stefan–Cahn–Hilliard systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 41 (2024), no. 4, pp. 797–852

DOI 10.4171/AIHPC/89