Maximal mixed parabolic-hyperbolic regularity for the full equations of multicomponent fluid dynamics

TL;DR

This paper establishes maximal mixed parabolic-hyperbolic regularity for a complex fluid dynamics PDE system, proving short-time existence of strong solutions and their non-degeneracy under certain conditions.

Contribution

It introduces the concept of optimal mixed regularity for the full multicomponent fluid equations and proves short-time existence and non-degeneracy of solutions.

Findings

Proved short-time existence of strong solutions.

Established non-degeneracy of solutions under growth conditions.

Developed a special constitutive model for illustration.

Abstract

We consider a Navier-Stokes-Fick-Onsager-Fourier system of PDEs describing mass, energy and momentum balance in a Newtonian fluid with composite molecular structure. For the resulting parabolic-hyperbolic system, we introduce the notion of optimal regularity of mixed type, and we prove the short-time existence of strong solutions for a typical initial boundary-value-problem. By means of a partial maximum principle, we moreover show that such a solution cannot degenerate in finite time due to blow-up or vanishing of the temperature or the partial mass densities. This second result is however only valid under certain growth conditions on the phenomenological coefficients. In order to obtain some illustration of the theory, we set up a special constitutive model for volume-additive mixtures.