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Vanishing viscosity limit to the planar rarefaction wave with vacuum for 3-D full compressible Navier-Stokes equations with temperature-dependent transport coefficients

Meichen Hou, Lingjun Liu, Shu Wang, Lingda Xu

Abstract

In this paper, we construct a family of global-in-time solutions of the 3-D full compressible Navier-Stokes (N-S) equations with temperature-dependent transport coefficients (including viscosity and heat-conductivity), and show that at arbitrary times {and arbitrary strength} this family of solutions converges to planar rarefaction waves connected to the vacuum as the viscosity vanishes in the sense of $L^\infty(\R^3)$. We consider the Cauchy problem in $\R^3$ with perturbations of the infinite global norm, particularly, periodic perturbations. To deal with the infinite oscillation, we construct a suitable ansatz carrying this periodic oscillation such that the difference between the solution and the ansatz belongs to some Sobolev space and thus the energy method is feasible. The novelty of this paper is that the viscosity and heat-conductivity are temperature-dependent and degeneracies caused by vacuum. Thus the a priori assumptions and two Gagliardo-Nirenberg type inequalities are essentially used. Next, more careful energy estimates are carried out in this paper, by studying the zero and non-zero modes of the solutions, we obtain not only the convergence rate concerning the viscosity and heat conductivity coefficients but also the exponential time decay rate for the non-zero mode.

Vanishing viscosity limit to the planar rarefaction wave with vacuum for 3-D full compressible Navier-Stokes equations with temperature-dependent transport coefficients

Abstract

In this paper, we construct a family of global-in-time solutions of the 3-D full compressible Navier-Stokes (N-S) equations with temperature-dependent transport coefficients (including viscosity and heat-conductivity), and show that at arbitrary times {and arbitrary strength} this family of solutions converges to planar rarefaction waves connected to the vacuum as the viscosity vanishes in the sense of . We consider the Cauchy problem in with perturbations of the infinite global norm, particularly, periodic perturbations. To deal with the infinite oscillation, we construct a suitable ansatz carrying this periodic oscillation such that the difference between the solution and the ansatz belongs to some Sobolev space and thus the energy method is feasible. The novelty of this paper is that the viscosity and heat-conductivity are temperature-dependent and degeneracies caused by vacuum. Thus the a priori assumptions and two Gagliardo-Nirenberg type inequalities are essentially used. Next, more careful energy estimates are carried out in this paper, by studying the zero and non-zero modes of the solutions, we obtain not only the convergence rate concerning the viscosity and heat conductivity coefficients but also the exponential time decay rate for the non-zero mode.
Paper Structure (12 sections, 15 theorems, 223 equations)

This paper contains 12 sections, 15 theorems, 223 equations.

Table of Contents

  1. Introduction and Main Result
  2. Preliminaries
  3. Rarefaction Wave
  4. Ansatz
  5. Decomposition
  6. Gagliardo-Nirenberg type inequality
  7. Vanishing Viscosity Limit
  8. The perturbation system
  9. A priori estimates
  10. Proof of Theorem \ref{['mt']}.
  11. Exponential decay rate
  12. Appendix: Proof of Lemma \ref{['lem-periodic-solution']}

Key Result

Theorem 1.1

Denote the planar rarefaction wave solution with one-side vacuum to the 3-d compressible Euler system EE as $\left(\rho^{r}, m^r=\rho^{r}u^{r}, n^r=\rho^{r}\theta^{r}\right)\left(\frac{x_{1}}{t}\right)$, which is defined in nrw2-ss1. Then there exists a constant $\varepsilon_{0}>0$ such that for an and for any small constant $h>0$, there exists a constant $C_{h}>0$ independent of $\varepsilon,\et

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 14 more