Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nonlinear stability of a composite wave to the Cauchy problem of 1-D full compressible Navier-Stokes-Allen-Cahn system

Dan Lei, Zhengzheng Chen

TL;DR

This work analyzes the large-time behavior of the one-dimensional full compressible NSAC system, which couples compressible Navier–Stokes dynamics with a diffuse-interface phase-field. The authors construct a smooth approximation of a composite wave formed by a $1$- and a $3$-rarefaction wave from the Euler Riemann problem and reformulate the NSAC Cauchy problem as a perturbation around this wave in Lagrangian coordinates. Under the regime $0<\\gamma-1\\leq \\varepsilon_0$, they establish global existence and prove convergence of the strong solution to the composite wave, even for arbitrarily large initial perturbations, with $v$ and $\\theta$ uniformly bounded and $0\\leq \\chi\\leq 1$. The analysis relies on an elementary $L^2$ energy method that incorporates the phase-field variable $\\chi$, yielding uniform-in-time bounds and higher-order estimates via a structured hierarchy of lemmas and Gronwall-type arguments, and it extends Nishida–Smoller-type results to the 1D full NSAC system. Potential extensions include variable viscosity/thermal conductivity and cases with nontrivial asymptotic phase states.

Abstract

The compressible Navier-Stokes-Allen-Cahn system models the motion of a mixture of two macroscopically immiscible viscous compressible fluids. In this paper, we are concerned with the large time behavior of solutions to the Cauchy problem of the one-dimensional full compressible Navier-Stokes-Allen-Cahn system. If the Riemann problem of the corresponding Euler system admits a solution which is a linear combination of 1-rarefaction wave and 3-rarefaction wave, we proved that a global strong solution to the compressible Navier-Stokes-Allen-Cahn system exists uniquely and converges to the above composite wave as time goes to infinity, provided that the adiabatic exponent $γ$ is closed to $1$. Here the initial perturbations except for the temperature function of the fluid, and the strength of rarefaction waves can be arbitrarily large. The proof is given by an elementary energy method that takes into account the effect of the phase field variable $χ(t,x)$ and the complexity of nonlinear waves.

Nonlinear stability of a composite wave to the Cauchy problem of 1-D full compressible Navier-Stokes-Allen-Cahn system

TL;DR

This work analyzes the large-time behavior of the one-dimensional full compressible NSAC system, which couples compressible Navier–Stokes dynamics with a diffuse-interface phase-field. The authors construct a smooth approximation of a composite wave formed by a - and a -rarefaction wave from the Euler Riemann problem and reformulate the NSAC Cauchy problem as a perturbation around this wave in Lagrangian coordinates. Under the regime , they establish global existence and prove convergence of the strong solution to the composite wave, even for arbitrarily large initial perturbations, with and uniformly bounded and . The analysis relies on an elementary energy method that incorporates the phase-field variable , yielding uniform-in-time bounds and higher-order estimates via a structured hierarchy of lemmas and Gronwall-type arguments, and it extends Nishida–Smoller-type results to the 1D full NSAC system. Potential extensions include variable viscosity/thermal conductivity and cases with nontrivial asymptotic phase states.

Abstract

The compressible Navier-Stokes-Allen-Cahn system models the motion of a mixture of two macroscopically immiscible viscous compressible fluids. In this paper, we are concerned with the large time behavior of solutions to the Cauchy problem of the one-dimensional full compressible Navier-Stokes-Allen-Cahn system. If the Riemann problem of the corresponding Euler system admits a solution which is a linear combination of 1-rarefaction wave and 3-rarefaction wave, we proved that a global strong solution to the compressible Navier-Stokes-Allen-Cahn system exists uniquely and converges to the above composite wave as time goes to infinity, provided that the adiabatic exponent is closed to . Here the initial perturbations except for the temperature function of the fluid, and the strength of rarefaction waves can be arbitrarily large. The proof is given by an elementary energy method that takes into account the effect of the phase field variable and the complexity of nonlinear waves.
Paper Structure (7 sections, 18 theorems, 149 equations)

This paper contains 7 sections, 18 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Euler equations and composite wave
  3. Main result
  4. Preliminaries
  5. Preliminaries
  6. Reformulation of the original problem
  7. Proof of the a priori estimates

Key Result

Theorem 1.1

For any given left end state $(v_-,u_-,\theta_-)$, suppose that the right end state $(v_+,u_+,\theta_+)$ is connected to the left end state $(v_-,u_-,\theta_-)$ by $(V^R,U^R,\Theta^R,S^R)(t,x)$, which is the solution to the Riemann problem of the compressible Euler system (1.20)-(1.21) given by (1.2 Then the Cauchy problem $(1.9)_1$-$(1.9)_4$, (1.13) admits a unique global smooth solution $(v,u,\t

Theorems & Definitions (23)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 3.1
  • Proposition 3.1: Local existence
  • ...and 13 more