Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Weak--strong uniqueness for bi-fluid compressible system with algebraic closure

Yang Li, Mária Lukáčová-Medvid'ová, Milan Pokorný, Ewelina Zatorska

Abstract

We consider a real two-fluid system of compressible viscous fluids with a common velocity field and algebraic closure for the pressure law. The constitutive relation involves densities of both fluids through an implicit function. The existence of global-in-time finite energy weak solutions to this system is known since the work of Novotný and Pokorný [Arch. Rational Mech. Anal., 2020]. On the other hand, existence of local-in-time strong solutions is due to Piasecki and Zatorska [J. Math Fluid Mech.,, 2022]. In this paper, we establish the weak--strong uniqueness principle using the relative entropy method. In sharp contrast to the two-phase model of Baer-Nunziato type, the volume fraction of phase $+$ obeys a transport equation with an additional nonlinear term. This gives rise to troublesome terms in the relative entropy inequality. We are able to close the estimate by making an elaborate use of the structure of the system.

Weak--strong uniqueness for bi-fluid compressible system with algebraic closure

Abstract

We consider a real two-fluid system of compressible viscous fluids with a common velocity field and algebraic closure for the pressure law. The constitutive relation involves densities of both fluids through an implicit function. The existence of global-in-time finite energy weak solutions to this system is known since the work of Novotný and Pokorný [Arch. Rational Mech. Anal., 2020]. On the other hand, existence of local-in-time strong solutions is due to Piasecki and Zatorska [J. Math Fluid Mech.,, 2022]. In this paper, we establish the weak--strong uniqueness principle using the relative entropy method. In sharp contrast to the two-phase model of Baer-Nunziato type, the volume fraction of phase obeys a transport equation with an additional nonlinear term. This gives rise to troublesome terms in the relative entropy inequality. We are able to close the estimate by making an elaborate use of the structure of the system.
Paper Structure (9 sections, 7 theorems, 93 equations)

This paper contains 9 sections, 7 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Formulation of the problem
  3. Global weak solutions
  4. Local-in-time strong solutions
  5. Relative entropy inequality--preliminaries
  6. Derivation of the equations for
  7. Derivation of relative entropy inequality
  8. Relative entropy inequality for the strong solution
  9. The weak--strong uniqueness principle

Key Result

Theorem 2.1

Let $(\alpha , \varrho_-,\varrho_+,{\bf u})$ be a global weak solution to the system spec--eq1.3bi obtained in NM20, and let $(\beta , {\widetilde{\varrho}_-},{\widetilde{\varrho}_+},{\bf v})$ be the strong solution obtained in PZ1 emanating from the same initial data. Then the two solutions coincid

Theorems & Definitions (11)

  • Theorem 2.1
  • Definition 1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Lemma 3.1
  • Remark 3.2
  • Lemma 3.3
  • Theorem 5.1
  • ...and 1 more