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

Huo, Xiaokai; Jüngel, Ansgar; Tzavaras, Athanasios

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

Files

ms-cahnhilliard.pdf (468.86 KB)

Type

Article

Authors

Huo, Xiaokai
Jüngel, Ansgar
Tzavaras, Athanasios

KAUST Department

Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division
Applied Mathematics and Computational Science Program

Date

2023-06-07

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 H2(Ω) bounds for the densities, which cannot be obtained from the energy or entropy inequalities alone.

Citation

Huo, X., Jüngel, A., & Tzavaras, A. E. (2023). Existence and weak–strong uniqueness for Maxwell–Stefan–Cahn–Hilliard systems. Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire. https://doi.org/10.4171/aihpc/89

Acknowledgements

XH and AJ acknowledge partial support from the Austrian Science Fund (FWF), grants P33010, W1245, and F65. AET acknowledges support from baseline funds of the King Abdullah University of Science and Technology (KAUST). This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, ERC Advance grant number 101018153.

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