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On the relaxation towards mechanical equilibrium for two-pressure compressible flows

Cosmin Burtea, Timothée Crin-Barat, Pierre Gonin--Joubert

TL;DR

The study addresses relaxation to mechanical equilibrium in a two-pressure, one-velocity Baer–Nunziato model. It develops two symmetrizations that expose a pressure-dissipation mechanism and yield estimates uniform in the compaction viscosity $\mu$, enabling a rigorous relaxation limit to the Kapila (one-velocity, one-pressure) model as $\mu\to0$. For fixed $\mu$, it establishes a global-in-time classical solution for small data by leveraging a pressure-dissipation structure and hypocoercivity in mass-Lagrangian coordinates; it also proves a uniform local-in-time result and analyzes ill-prepared data with a $O(\sqrt{\mu})$ convergence rate to Kapila. The results provide a solid mathematical foundation for the relaxation toward mechanical equilibrium in compressible two-fluid flows and offer a structured energy framework that may extend to higher dimensions under suitable conditions.

Abstract

We introduce a symmetrization of a one-velocity two-pressures Baer-Nunziato type model for mixtures of barotropic compressible fluids. It allows us to justify the zero compaction viscosity limit and to recover a solution of the so-called Kapila model. On the other hand, the symmetrization highlights a pressure-induced stabilization mechanism which allows us to recover a global-in-time existence result for initial data close to constant states.

On the relaxation towards mechanical equilibrium for two-pressure compressible flows

TL;DR

The study addresses relaxation to mechanical equilibrium in a two-pressure, one-velocity Baer–Nunziato model. It develops two symmetrizations that expose a pressure-dissipation mechanism and yield estimates uniform in the compaction viscosity , enabling a rigorous relaxation limit to the Kapila (one-velocity, one-pressure) model as . For fixed , it establishes a global-in-time classical solution for small data by leveraging a pressure-dissipation structure and hypocoercivity in mass-Lagrangian coordinates; it also proves a uniform local-in-time result and analyzes ill-prepared data with a convergence rate to Kapila. The results provide a solid mathematical foundation for the relaxation toward mechanical equilibrium in compressible two-fluid flows and offer a structured energy framework that may extend to higher dimensions under suitable conditions.

Abstract

We introduce a symmetrization of a one-velocity two-pressures Baer-Nunziato type model for mixtures of barotropic compressible fluids. It allows us to justify the zero compaction viscosity limit and to recover a solution of the so-called Kapila model. On the other hand, the symmetrization highlights a pressure-induced stabilization mechanism which allows us to recover a global-in-time existence result for initial data close to constant states.
Paper Structure (18 sections, 7 theorems, 87 equations)

This paper contains 18 sections, 7 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Introduction
  3. Aims of the paper
  4. Reformulation of the system
  5. Symmetrization
  6. A first symetrisation
  7. A second symmetrization
  8. Main results
  9. Uniform-in-$\mu$ local-in-time well-posedness and pressure-relaxation limit
  10. Non-uniform-in-$\mu$ global-in-time well-posedness result
  11. Proof of Theorem \ref{['thm:GWP']}
  12. Linearization
  13. A priori estimates
  14. Energy estimates
  15. Hypocoercive estimates in $L^2$
  16. ...and 3 more sections

Key Result

Proposition 2.1

Let $\mathcal{U}$ an open bounded convex set such that ${\overline {\mathcal{U}}}\subset\mathcal{O}$ and $\varphi$ as defined in def_varphi-change_of_var. There exists some $\eta>0$ such that the restriction of $\varphi$ to ${\mathcal{U}}\times\left( -\eta,\eta\right)$ is a $C^{\infty}$-diffeomorph

Theorems & Definitions (12)

  • Proposition 2.1
  • Theorem 3.1
  • Remark 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Remark 3.5
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 2 more