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The combined non-equilibrium diffusion and low Mach number limits of the compressible Navier-Stokes-Fourier-P1 approximation radiation model

Fucai Li, Shuxing Zhang

TL;DR

This work analyzes the combined non equilibrium diffusion and low Mach number limits for the NSF-P1 radiation model under large temperature variations and general initial data. It develops uniform-in-$\epsilon$ energy estimates by introducing an equivalent pressure and velocity to balance radiation-induced singularities, employing weighted Sobolev norms and acoustic wave decay tools to control fast modes. The authors characterize the limiting systems across scattering regimes parameterized by $\delta\in[0,2]$, showing diffusion-dominated limits when scattering is strong and a family of $I_0$-$I_1$ relations in weaker regimes, including a fully general initial data result at $\delta=0$ where $I_0$ vanishes and $I_1$ is damped. These results clarify how radiation isotropy and scattering intensity shape reduced models in radiation hydrodynamics and provide rigorous justification for the corresponding asymptotic limits.

Abstract

In this paper, we investigate the combined non-equilibrium diffusion and low Mach number limits of the compressible Navier-Stokes-Fourier-P1 (NSF-P1) model with general initial data, which arises in the radiation hydrodynamics. Compared to the classical compressible Navier-Stokes-Fourier system, the NSF-P1 model has an asymmetric singular structure caused by the radiation field. To handle these singular terms, we introduce an equivalent pressure and an equivalent velocity to balance the order of singularity and establish the uniform estimates of solutions by designating appropriate weighted norms as well as carrying out delicate energy analysis. We conclude that, for partially general initial data and the strong scattering effect, the NSF-P1 model converges to the system of low Mach number heat-conducting viscous flows coupled with a diffusion equation. We also discuss the variations of the limit equations as the scattering intensity changes. Furthermore, when the scattering effect is sufficiently weak, we can obtain the singular limits of the NSF-P1 model with fully general initial data.

The combined non-equilibrium diffusion and low Mach number limits of the compressible Navier-Stokes-Fourier-P1 approximation radiation model

TL;DR

This work analyzes the combined non equilibrium diffusion and low Mach number limits for the NSF-P1 radiation model under large temperature variations and general initial data. It develops uniform-in- energy estimates by introducing an equivalent pressure and velocity to balance radiation-induced singularities, employing weighted Sobolev norms and acoustic wave decay tools to control fast modes. The authors characterize the limiting systems across scattering regimes parameterized by , showing diffusion-dominated limits when scattering is strong and a family of - relations in weaker regimes, including a fully general initial data result at where vanishes and is damped. These results clarify how radiation isotropy and scattering intensity shape reduced models in radiation hydrodynamics and provide rigorous justification for the corresponding asymptotic limits.

Abstract

In this paper, we investigate the combined non-equilibrium diffusion and low Mach number limits of the compressible Navier-Stokes-Fourier-P1 (NSF-P1) model with general initial data, which arises in the radiation hydrodynamics. Compared to the classical compressible Navier-Stokes-Fourier system, the NSF-P1 model has an asymmetric singular structure caused by the radiation field. To handle these singular terms, we introduce an equivalent pressure and an equivalent velocity to balance the order of singularity and establish the uniform estimates of solutions by designating appropriate weighted norms as well as carrying out delicate energy analysis. We conclude that, for partially general initial data and the strong scattering effect, the NSF-P1 model converges to the system of low Mach number heat-conducting viscous flows coupled with a diffusion equation. We also discuss the variations of the limit equations as the scattering intensity changes. Furthermore, when the scattering effect is sufficiently weak, we can obtain the singular limits of the NSF-P1 model with fully general initial data.
Paper Structure (9 sections, 21 theorems, 203 equations)

This paper contains 9 sections, 21 theorems, 203 equations.

Table of Contents

  1. Introduction
  2. The model
  3. Previous results on singular limits of the models of radiation hydrodynamics
  4. Our results
  5. Preliminaries
  6. Uniform estimates
  7. Convergence for the case $\delta \in (0,2]$
  8. Proof of Theorem \ref{['Ls 0 thm']}
  9. Appendix

Key Result

Theorem 1.1

Let $\delta\in(0,2]$ and $s\geq 4$ be an integer. Assume that the initial data initial data satisfy for the three positive constants $\theta_c$, $I_c$ and $M_0$ independent of $\epsilon$, and $\theta_c$ and $I_c$ satisfying $I_c=e^{4\theta_c}$. Then there exist constants $\epsilon_0\in (0,1]$ and $T_0>0$ such that, for all $\epsilon \leq \epsilon_0$, the Cauchy problem nondim and initial data has

Theorems & Definitions (39)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.3
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • ...and 29 more