The weak-strong uniqueness for Maxwell--Stefan systems and some generalized systems is proved. The corresponding parabolic cross-diffusion equations are considered in a bounded domain with no-flux boundary conditions. The key points of the proofs are various inequalities for the relative entropy associated with the systems and the analysis of the spectrum of a quadratic form capturing the frictional dissipation. The latter task is complicated by the singular nature of the diffusion matrix. This difficulty is addressed by proving its positive definiteness on a subspace and using the Bott--Duffin matrix inverse. The generalized Maxwell--Stefan systems are shown to cover several known cross-diffusion systems for the description of tumor growth and physical vapor deposition processes.
|Original language||English (US)|
|Number of pages||38|
|Journal||SIAM Journal on Mathematical Analysis|
|State||Published - May 26 2022|
Bibliographical noteKAUST Repository Item: Exported on 2022-05-30
Acknowledgements: The work of the authors was supported by the European Research Council (ERC)under the European Union's Horizon 2020 research and innovation programme, ERC Advancedgrant NEURO- MORPH, 101018153. The work of the second author was partially supported by theAustrian Science Fund (FWF) grants P30000, P33010, F65, W1245.
ASJC Scopus subject areas
- Computational Mathematics
- Applied Mathematics