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The relaxation limit of a homogeneous two-phase flow model: isothermal case

Huimin Yu

Abstract

This paper investigates the asymptotic behavior of a hyperbolic relaxation system designed for homogeneous two-phase flows in the limit of vanishing relaxation time. The governing equations comprise conservation laws for mixture mass and momentum, supplemented by a transport equation for the gas phase mass that includes a stiff relaxation source term. This source term drives the system toward local thermodynamic equilibrium. Under the assumptions of constant liquid density and an ideal isothermal gas phase, we demonstrate that, as the relaxation parameter \(ε\rightarrow 0\), a subsequence of solutions \((p^ε,u^ε)\) converges strongly in \(L_{\mathrm{loc}}^{1}\) to an entropy solution of the equilibrium Euler system. The proof integrates several analytical techniques: the construction of a suitable entropy pair and associated energy estimates, a transport equation approach for representing the error, commutator estimates, and the theory of compensated compactness. This work provides a rigorous justification of the relaxation limit for the homogeneous two-phase flows model.

The relaxation limit of a homogeneous two-phase flow model: isothermal case

Abstract

This paper investigates the asymptotic behavior of a hyperbolic relaxation system designed for homogeneous two-phase flows in the limit of vanishing relaxation time. The governing equations comprise conservation laws for mixture mass and momentum, supplemented by a transport equation for the gas phase mass that includes a stiff relaxation source term. This source term drives the system toward local thermodynamic equilibrium. Under the assumptions of constant liquid density and an ideal isothermal gas phase, we demonstrate that, as the relaxation parameter , a subsequence of solutions \((p^ε,u^ε)\) converges strongly in to an entropy solution of the equilibrium Euler system. The proof integrates several analytical techniques: the construction of a suitable entropy pair and associated energy estimates, a transport equation approach for representing the error, commutator estimates, and the theory of compensated compactness. This work provides a rigorous justification of the relaxation limit for the homogeneous two-phase flows model.
Paper Structure (28 sections, 13 theorems, 137 equations)

This paper contains 28 sections, 13 theorems, 137 equations.

Table of Contents

  1. Problem Formulation
  2. Background and Motivation
  3. On the Feasibility of a Rigorous Mathematical Proof
  4. Main Results of This Paper
  5. Existence of Solutions for the Relaxation Problem
  6. Uniform Estimates Independent of $\epsilon$
  7. Entropy Dissipation and Zero-Order Energy Estimates
  8. First-Order Energy Estimates
  9. Error Representation and Regularity
  10. Derivation of the Error Equation
  11. Regularity Estimates for $R^{\epsilon}$
  12. Estimate for $\frac{1}{2}\frac{d}{dt}\|S\|_{L^{2}}^{2}$.
  13. Estimate for $\frac{\epsilon}{2}\frac{d}{dt}\|T\|_{L^{2}}^{2}$.
  14. Estimate for $\delta\epsilon\frac{d}{dt}\int_{\mathbb{R}}ST\partial_{x}u\,dx$.
  15. Compensated Compactness and Convergence Proof
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $\epsilon >0$ be fixed. Assume the initial data $U_{0}^{\epsilon} = (\rho_{m0}^{\epsilon}(t,x),m_{0}^{\epsilon}(t,x),\Gamma_{0}^{\epsilon})^{T}$ satisfy Then the Cauchy problem for the relaxation system (1.1)-(1.3) admits a global weak solution satisfying Furthermore, this solution satisfies the following $\epsilon$-independent uniform estimates (the constant $C$ depends only on $T$ and the

Theorems & Definitions (33)

  • Definition 1.1: Admissible solution family for the relaxation limit
  • Remark 1.1
  • Theorem 1.1: Existence of an admissible solution family for the relaxation limit
  • Remark 1.2
  • Theorem 1.2: Relaxation limit convergence
  • Theorem 1.3: Convergence rate estimate
  • Lemma 3.1: Entropy dissipation
  • proof
  • Lemma 3.2: Zero-order energy estimate
  • proof
  • ...and 23 more