Weak-strong uniqueness for Maxwell-Stefan systems

Abstract

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.

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

“…We refer to [21,Lemma 17] for the proof. The property for the kernel follows from ker D BD (c) = ker P L (c) = L ⊥ (c).…”

“…The uniqueness of strong solutions to Maxwell-Stefan systems has been shown in [18,22], and uniqueness results for weak solutions in a very special case can be found in [8]. A weak-strong uniqueness result for Maxwell-Stefan systems was proved in [21]. Concerning uniqueness results for fourth-order equations, we refer to [9] for single-species Cahn-Hilliard equations, [24] for single-species thin-film equations, and [15] for the quantum drift-diffusion equations.…”

Asymptotic derivation of multicomponent compressible flows with heat conduction and mass diffusion

“…For the second term we need to invert system (3.4), in order to determine ρ i u i . Following the analysis of [14] for the inversion of the Maxwell-Stefan problem, we invert (3.4) using the so-called Bott-Duffin inverse. The need to introduce the Bott-Duffin inverse arises from the fact that the desired inversion has to respect the constraint i ρ i u i = 0, i.e.…”

“…In our context, we are interested in the constrained inversion M x = d, x ∈ L, where L = ran(M ) and thus L ⊥ = ker(M ). In order to be compatible with the notation of [14], we set…”

“…Existence analysis for heat-conducting multicomponent flows is provided in [18,25,17] while the Chapman-Enskog procedure is analyzed in the isothermal context in [11,13]. Finally, we refer to [14] for introducing the use of the Bott-Duffin inverse to invert the Maxwell-Stefan system (1.12); the latter provides an important ingredient for inverting the algebraic system and comparing the entropic structure of the Type-I model with the usual entropic structure of hyperbolic-parabolic systems in the format discussed in [16,6].…”