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Low Mach Number Limit and Convergence Rates for a Compressible Two-Fluid Model with Algebraic Pressure Closure

Yang Li, Mária Lukáčová-Medviďová, Ewelina Zatorska

Abstract

We study the low Mach number limit for a viscous compressible two-fluid model with algebraic pressure closure in the three-dimensional torus $\mathbb{T}^3$. The pressure is determined implicitly through the densities of the two phases, which makes the singular limit substantially more delicate than for models with explicit pressure laws. Working in the framework of local-in-time strong solutions, we prove that, for well-prepared initial data, solutions to the rescaled compressible two-fluid system exist on a time interval independent of the Mach number and converge to the solution of the incompressible Navier--Stokes equations as the Mach number tends to zero. In addition, we establish explicit convergence rates for the densities and the velocity field. The proof relies on uniform high-order energy estimates and a relative energy argument adapted to the implicit structure of the pressure law. These results provide a rigorous justification of the low Mach number limit for the compressible two-fluid model with algebraic pressure closure.

Low Mach Number Limit and Convergence Rates for a Compressible Two-Fluid Model with Algebraic Pressure Closure

Abstract

We study the low Mach number limit for a viscous compressible two-fluid model with algebraic pressure closure in the three-dimensional torus . The pressure is determined implicitly through the densities of the two phases, which makes the singular limit substantially more delicate than for models with explicit pressure laws. Working in the framework of local-in-time strong solutions, we prove that, for well-prepared initial data, solutions to the rescaled compressible two-fluid system exist on a time interval independent of the Mach number and converge to the solution of the incompressible Navier--Stokes equations as the Mach number tends to zero. In addition, we establish explicit convergence rates for the densities and the velocity field. The proof relies on uniform high-order energy estimates and a relative energy argument adapted to the implicit structure of the pressure law. These results provide a rigorous justification of the low Mach number limit for the compressible two-fluid model with algebraic pressure closure.
Paper Structure (14 sections, 4 theorems, 108 equations)

This paper contains 14 sections, 4 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Motivation and main results
  4. Statement of the main results
  5. Proof of Theorem \ref{['Thm1']}
  6. Uniform estimates
  7. Closure of the energy estimates
  8. Contraction of the map
  9. The end of proof of Theorem \ref{['Thm1']}
  10. Proof of Theorem \ref{['Thm2']}
  11. Relative energy inequality
  12. Estimates for the remainder
  13. Convergence rates
  14. Two useful lemmas

Key Result

Theorem 2.1

Let where $\mathbf{u}_0$ satisfies $\operatorname{div} \mathbf{u}_0=0,\mathbf{u}_0\in H^{s+1}(\Omega)$ for some integer $s \geqslant 4$. Assume further that for some sufficiently small constant $\delta_0>0$ independent of $\varepsilon$. Then the following statements hold, provided $\varepsilon>0$ is sufficiently small.

Theorems & Definitions (6)

  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Lemma A.1
  • Lemma A.2