[Paper Review] Contraction property for large perturbations of shocks of the barotropic Navier-Stokes system

Moon-Jin Kang, Alexis Vasseur|arXiv (Cornell University)|Dec 20, 2017

Navier-Stokes equation solutions31 references3 citations

TL;DR

This paper establishes a contraction property for viscous shock profiles in the barotropic Navier-Stokes equations under large perturbations, using a novel pseudo-norm constructed via the relative entropy method. The key result is that shock stability holds independently of viscosity strength, extending a-contraction theory to the viscous, compressible case with degenerate viscosity ($\alpha = \gamma$).

ABSTRACT

This paper is dedicated to the construction of a pseudo-norm, for which small shock profiles of the barotropic Navier-Stokes equation have a contraction property. This contraction property holds in the class of any large 1D weak solutions to the barotropic Navier-Stokes equation. It implies a stability condition which is independent of the strength of the viscosity. The proof is based on the relative entropy method, and is reminiscent to the notion of a-contraction first introduced by the authors in the hyperbolic case.

Motivation & Objective

• To establish a contraction property for viscous shock profiles in the barotropic Navier-Stokes system under arbitrary large perturbations.
• To develop a pseudo-norm that ensures contraction of solutions toward a shifted shock profile, regardless of the initial perturbation size.
• To extend the a-contraction framework—previously used in hyperbolic conservation laws—to the viscous, compressible Navier-Stokes system with degenerate viscosity.
• To prove stability of shocks without requiring smallness assumptions on perturbations or integral zero conditions.
• To construct a weight function and use relative entropy to derive nonlinear Poincaré-type inequalities for large deviations.

Proposed method

• Construct a pseudo-norm based on relative entropy between the solution and a shifted shock profile, tailored to the barotropic Navier-Stokes system with $\alpha = \gamma$.
• Apply the relative entropy method to derive evolution estimates for the difference between the solution and the viscous shock profile.
• Introduce a weight function that adapts to the non-homogeneous structure of the system, enabling control of large perturbations.
• Use a nonlinear Poincaré-type inequality to bound the relative entropy in terms of spatial derivatives, leveraging the structure of the pressure law $p(v) = v^{-\gamma}$.
• Perform a truncation of large values of $|p(v) - p(\tilde{v}_\varepsilon)|$ to control nonlinearity in the entropy dissipation.
• Analyze the asymptotic behavior of the system through expansion in the shock size and prove the existence of a global contraction via functional inequalities.

Experimental results

Research questions

• RQ1 Can a contraction property be established for viscous shocks in the barotropic Navier-Stokes system under arbitrarily large perturbations?
• RQ2 Is it possible to construct a pseudo-norm that ensures contraction independent of the viscosity strength?
• RQ3 How can the relative entropy method be adapted to handle large deviations in the presence of degenerate viscosity ($\mu(v) = b v^{-\gamma}$)?
• RQ4 Does the a-contraction framework extend to viscous, compressible systems with non-constant viscosity?
• RQ5 What functional inequalities are necessary to control the relative entropy evolution in the large perturbation regime?

Key findings

• A contraction property is established for viscous shock profiles in the barotropic Navier-Stokes system with $\alpha = \gamma$, valid for any large perturbation.
• The contraction is proven via a specially constructed pseudo-norm that depends on the relative entropy and a weight function adapted to the shock structure.
• The stability result holds independently of the viscosity strength, removing the need for smallness assumptions on perturbations or integral-zero conditions.
• The proof relies on a nonlinear Poincaré-type inequality derived from the structure of the pressure law and the viscosity coefficient.
• The authors construct a shift $X(t)$ such that the solution remains close to the shifted shock profile in the pseudo-norm, ensuring long-time stability.
• A contradiction argument is used to show that the maximum of a key auxiliary function $g(x)$ occurs only at endpoints, proving non-positivity and enabling the contraction estimate.