[Paper Review] The limit to rarefaction wave with vacuum for 1D compressible fluids with temperature-dependent viscosities

Mingjie Li, Teng Wang+1 more|arXiv (Cornell University)|Jul 12, 2013

Navier-Stokes equation solutions26 references3 citations

TL;DR

This paper establishes the zero dissipation limit of one-dimensional compressible Navier-Stokes equations with temperature-dependent viscosities and heat conduction, proving that solutions converge to a rarefaction wave with vacuum as viscosity and thermal conductivity vanish. The convergence is uniform, and the rate is quantified via sharp a priori estimates despite degeneracies at the vacuum region.

ABSTRACT

In this paper we study the zero dissipation limit of the one-dimensional full compressible Navier-Stokes(CNS) equations with temperature-dependent viscosity and heat-conduction coefficient. It is proved that given a rarefaction wave with one-side vacuum state to the full compressible Euler equations, we can construct a sequence of solutions to the full CNS equations which converge to the above rarefaction wave with vacuum as the viscosity and the heat conduction coefficient tend to zero. Moreover, the uniform convergence rate is obtained. The main difficulty in our proof lies in the degeneracies of the density, the temperature and the temperature-dependent viscosities at the vacuum region in the zero dissipation limit.

Motivation & Objective

• To justify the zero dissipation limit of the full compressible Navier-Stokes equations with temperature-dependent viscosities and heat conduction.
• To analyze the convergence of viscous solutions to a rarefaction wave solution of the Euler equations when one side of the wave is vacuum.
• To overcome the degeneracy of density, temperature, and viscosity at the vacuum region during the zero dissipation limit.
• To establish uniform convergence rates in the presence of singularities and degeneracies arising from vacuum states.
• To extend the validity of the zero dissipation limit to the full compressible fluid system with physical, temperature-dependent viscosity and heat conduction coefficients.

Proposed method

• Constructing a viscous approximation to the rarefaction wave with vacuum by solving the full compressible Navier-Stokes equations with $ε\mu(\theta) = \theta^\alpha$ and $\epsilon\kappa(\theta) = \theta^\alpha$.
• Using a Lagrangian transformation to reduce the problem to a moving boundary problem in the Lagrangian coordinate system.
• Applying a priori energy estimates in weighted Sobolev spaces to control the solution behavior near vacuum.
• Establishing uniform bounds on the solution and its derivatives through careful interpolation and weighted $L^2$ estimates.
• Employing matched asymptotic analysis and energy estimates to verify the a priori assumptions and extend local solutions globally in time.
• Deriving the convergence rate by comparing the viscous solution with the limiting rarefaction wave using $L^\infty$ norms and logarithmic corrections in $\epsilon$.

Experimental results

Research questions

• RQ1 Can the zero dissipation limit be justified for the full compressible Navier-Stokes equations with temperature-dependent viscosities and heat conduction?
• RQ2 How does the solution behave near the vacuum region during the zero dissipation limit, given the degeneracy of viscosity and temperature?
• RQ3 What is the uniform convergence rate of the viscous solutions to the rarefaction wave with vacuum as $\epsilon \to 0^+$?
• RQ4 Can global existence of solutions be established for the viscous system under the rarefaction wave with vacuum initial data?
• RQ5 How do the temperature-dependent viscosity and heat conduction affect the convergence and stability of the zero dissipation limit?

Key findings

• The viscous solutions to the full compressible Navier-Stokes equations converge uniformly to the rarefaction wave with vacuum as the viscosity and heat conduction coefficients tend to zero.
• A uniform convergence rate of order $\epsilon^a |\ln \epsilon|$ is established, where $a = \frac{24\gamma + 16\alpha(\gamma-1) - 3}{12(18\gamma + 12\alpha(\gamma-1))}$, with $\gamma > 1$ the adiabatic exponent and $\alpha > 0$ the temperature dependence exponent.
• The global existence of solutions to the viscous system is proven for small but fixed $\epsilon > 0$, extending local solutions to infinite time via a priori estimates.
• The degeneracy of density, temperature, and viscosity at the vacuum region is handled through weighted energy estimates and interpolation techniques.
• The convergence of momentum and total energy is also shown to hold with the same rate, confirming the full convergence of the state variables.
• The a priori estimates are stronger than the initial assumptions, allowing the extension of the local solution to a global one in time.