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Existence analysis of a stationary compressible fluid model for heat-conducting and chemically reacting mixtures

Miroslav Buliček, Ansgar Jüngel, Milan Pokorný, Nicola Zamponi

Abstract

The existence of large-data weak solutions to a steady compressible Navier-Stokes-Fourier system for chemically reacting fluid mixtures is proved. General free energies are considered satisfying some structural assumptions, with a pressure containing a $γ$-power law. The model is thermodynamically consistent and contains the Maxwell-Stefan cross-diffusion equations in the Fick-Onsager form as a special case. Compared to previous works, a very general model class is analyzed, including cross-diffusion effects, temperature gradients, compressible fluids, and different molar masses. A priori estimates are derived from the entropy balance and the total energy balance. The compactness for the total mass density follows from an estimate for the pressure in $L^p$ with $p>1$, the effective viscous flux identity, and uniform bounds related to Feireisl's oscillations defect measure. These bounds rely heavily on the convexity of the free energy and the strong convergence of the relative chemical potentials.

Existence analysis of a stationary compressible fluid model for heat-conducting and chemically reacting mixtures

Abstract

The existence of large-data weak solutions to a steady compressible Navier-Stokes-Fourier system for chemically reacting fluid mixtures is proved. General free energies are considered satisfying some structural assumptions, with a pressure containing a -power law. The model is thermodynamically consistent and contains the Maxwell-Stefan cross-diffusion equations in the Fick-Onsager form as a special case. Compared to previous works, a very general model class is analyzed, including cross-diffusion effects, temperature gradients, compressible fluids, and different molar masses. A priori estimates are derived from the entropy balance and the total energy balance. The compactness for the total mass density follows from an estimate for the pressure in with , the effective viscous flux identity, and uniform bounds related to Feireisl's oscillations defect measure. These bounds rely heavily on the convexity of the free energy and the strong convergence of the relative chemical potentials.

Paper Structure

This paper contains 44 sections, 21 theorems, 319 equations, 1 table.

Table of Contents

  1. Introduction
  2. Balance equations
  3. Notation
  4. Constitutive equations
  5. Hypotheses
  6. Solution concept and main result
  7. Key ideas of the proof
  8. State of the art and originality of the paper
  9. Auxiliary results
  10. A priori estimates for smooth solutions
  11. Weak sequential compactness for smooth solutions
  12. Convergence results based on a priori estimates
  13. Effective viscous flux
  14. Estimates based on the convexity of the free energy
  15. Renormalized continuity equation
  16. ...and 29 more sections

Key Result

Theorem \oldthetheorem

Let Hypotheses (H1)--(H7) hold. Let $\beta > 2/3$ and $\gamma > 3/2$. Then there exists a renormalized variational entropy solution to 1.massbal--1.J, 1.r--1.p. Moreover, if $\beta >1$ and $\gamma > 5/3$, then the solution is also a renormalized weak solution.

Theorems & Definitions (45)

  • Remark \oldthetheorem: Discussion of the hypotheses
  • Remark \oldthetheorem: Example of a free energy
  • Remark \oldthetheorem: Example of reaction terms
  • Definition \oldthetheorem: Weak and variational entropy solutions
  • Theorem \oldthetheorem: Large-data existence of solutions
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • ...and 35 more