The high-friction limit in Euler-Korteweg equations for fluid mixtures is analyzed. The convergence of the solutions towards the zeroth-order limiting system and the first-order correction is shown, assuming suitable uniform bounds. Three results are proved: The first-order correction system is shown to be of Maxwell-Stefan type and its diffusive part is parabolic in the sense of Petrovskii. The high-friction limit towards the first-order Chapman-Enskog approximate system is proved in the weak-strong solution context for general Euler-Korteweg systems. Finally, the limit towards the zeroth-order system is shown for smooth solutions in the isentropic case and for weak-strong solutions in the Euler-Korteweg case. These results include the case of constant capillarities and multicomponent quantum hydrodynamic models.

No abstract

We develop a new finite difference scheme for the Maxwell-Stefan diffusion system. The scheme is conservative, energy stable and positivity-preserving. These nice properties stem from a variational structure and are proved by reformulating the finite difference scheme into an equivalent optimization problem. The solution to the scheme emerges as the minimizer of the optimization problem, and as a consequence energy stability and positivity-preserving properties are obtained.

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 to 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.

No abstract