WIAS Preprint No. 2658, (2019)

Mass transport in multicomponent compressible fluids: Local and global well-posedness in classes of strong solutions for general class-one models


Authors

  • Bothe, Dieter
  • Druet, Pierre-Étienne
    ORCID: 0000-0001-5303-0500

2010 Mathematics Subject Classification

  • 35M33 35Q30 76N10, 35D35, 35B65, 35B35, 35K57, 35Q35, 35Q79, 76R50, 80A17, 80A32, 92E20

Keywords

  • Multicomponent flow, fluid mixture, compressible fluid, diffusion, reactive fluid, well-posedness analysis, strong solutions

DOI

10.20347/WIAS.PREPRINT.2658

Abstract

We consider a system of partial differential equations describing mass transport in a multicomponent isothermal compressible fluid. The diffusion fluxes obey the Fick-Onsager or Maxwell- Stefan closure approach. Mechanical forces result into one single convective mixture velocity, the barycentric one, which obeys the Navier-Stokes equations. The thermodynamic pressure is defined by the Gibbs-Duhem equation. Chemical potentials and pressure are derived from a thermodynamic potential, the Helmholtz free energy, with a bulk density allowed to be a general convex function of the mass densities of the constituents. The resulting PDEs are of mixed parabolic-hyperbolic type. We prove two theoretical results concerning the well-posedness of the model in classes of strong solutions: 1. The solution always exists and is unique for short-times and 2. If the initial data are sufficiently near to an equilibrium solution, the well-posedness is valid on arbitrary large, but finite time intervals. Both results rely on a contraction principle valid for systems of mixed type that behave like the compressible Navier- Stokes equations. The linearised parabolic part of the operator possesses the self map property with respect to some closed ball in the state space, while being contractive in a lower order norm only. In this paper, we implement these ideas by means of precise a priori estimates in spaces of exact regularity.

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