[Paper Review] The inviscid limit to a contact discontinuity for the compressible Navier-Stokes-Fourier system using the relative entropy method

Alexis Vasseur, Yi Wang|arXiv (Cornell University)|May 28, 2015

Navier-Stokes equation solutions26 references3 citations

TL;DR

This paper establishes the inviscid limit to a contact discontinuity for the compressible Navier-Stokes-Fourier system using the relative entropy method, proving convergence with a decay rate of $\kappa^{1/2}$ as heat conductivity $\kappa \to 0$, without smallness assumptions on the discontinuity or initial data. The result demonstrates that heat conductivity dominates dissipation in this regime, marking the first relative entropy-based convergence to a discontinuous solution in systems.

ABSTRACT

We consider the zero heat conductivity limit to a contact discontinuity for the mono-dimensional full compressible Navier-Stokes-Fourier system. The method is based on the relative entropy method, and do not assume any smallness conditions on the discontinuity, nor on the $BV$ norm of the initial data. It is proved that for any viscosity $ν\geq0$, the solution of the compressible Navier-Stokes-Fourier system (with well prepared initial value) converges, when the heat conductivity $κ$ tends to zero, to the contact discontinuity solution to the corresponding Euler system. We obtain the decay rate $κ^{\frac12}$. It implies that the heat conductivity dominates the dissipation in the regime of the limit to a contact discontinuity. This is the first result, based on the relative entropy, of an asymptotic limit to a discontinuous solutions for a system.

Motivation & Objective

• To establish the zero heat conductivity limit to a contact discontinuity in the compressible Navier-Stokes-Fourier system.
• To analyze the asymptotic behavior of solutions as $\kappa \to 0$ for arbitrary wave strength and initial data regularity.
• To extend the relative entropy method to convergence toward discontinuous solutions in hyperbolic systems.
• To demonstrate that heat conductivity dominates the dissipation in the limit to a contact discontinuity.

Proposed method

• Uses the relative entropy method to compare solutions of the Navier-Stokes-Fourier system with a contact discontinuity solution of the Euler system.
• Constructs a relative entropy functional measuring the distance between the viscous solution and the discontinuous limit state.
• Employs a weighted test function $\eta(x/\varepsilon)$ to localize the analysis near the discontinuity layer of width $\varepsilon$.
• Applies Gronwall's inequality to the resulting energy estimate after controlling boundary and bulk terms via state equations.
• Relies on thermodynamic assumptions: $p = p_e(\tau) + \vartheta p_\vartheta(\tau)$, $e = P_e(\tau) + Q(\vartheta)$, with $p_e' \leq 0$, $p_\vartheta' \leq 0$, and $C_v > 0$, to control singular terms.
• Sets $\varepsilon = \sqrt{\kappa}$ and initial relative entropy $\mathcal{E}(0) = O(\sqrt{\kappa})$ to achieve the $\kappa^{1/2}$ decay rate.

Experimental results

Research questions

• RQ1 Can the relative entropy method be applied to prove convergence to discontinuous solutions in systems, despite the lack of smoothness?
• RQ2 What is the optimal decay rate of the relative entropy as $\kappa \to 0$ in the absence of smallness assumptions on the discontinuity?
• RQ3 How does the dissipation from heat conductivity compare to viscosity in the limit to a contact discontinuity?
• RQ4 Does the contraction property of the contact discontinuity under a pseudo-distance enable convergence without shift or smallness?
• RQ5 Can the method handle large initial data and arbitrary wave strength in the contact discontinuity regime?

Key findings

• The solution of the compressible Navier-Stokes-Fourier system converges to the contact discontinuity solution of the Euler system as $\kappa \to 0$, for any $\nu \geq 0$, without smallness assumptions.
• The convergence rate is $\kappa^{1/2}$, established by choosing $\varepsilon = \sqrt{\kappa}$ and $\mathcal{E}(0) = O(\sqrt{\kappa})$.
• Heat conductivity dominates the dissipation in the limit, as the $\kappa^{1/2}$ rate is slower than any power of $\nu$, indicating $\kappa$ is the dominant dissipative mechanism.
• The relative entropy method successfully handles discontinuous limits in systems, extending its applicability beyond smooth solutions.
• The proof relies on thermodynamic state equations $p_e(\tau)$ and $p_\vartheta(\tau)$ satisfying $p_e' \leq 0$, $p_\vartheta' \leq 0$, and $P_e(\tau) \sim \tau^{-(\gamma-1)}$, $\gamma \geq 2$, to control singular terms.
• The contraction property of the contact discontinuity under a special inhomogeneous pseudo-distance enables the proof without shift, a key structural advantage.