[Paper Review] Uniform regularity and vanishing viscosity limit for the compressible Navier-Stokes with general Navier-slip boundary conditions in 3-dimensional domains

Yong Wang, Xin Zhouping+1 more|arXiv (Cornell University)|Jan 8, 2015

Navier-Stokes equation solutions21 references3 citations

TL;DR

This paper establishes uniform regularity and vanishing viscosity limits for the 3D compressible Navier-Stokes equations with general Navier-slip boundary conditions. It proves the existence of a unique strong solution uniformly bounded in conormal Sobolev and $L^∞$ spaces over a time interval independent of viscosity, and demonstrates convergence to the Euler equations with a quantified rate, showing weaker boundary layers for density than for velocity.

ABSTRACT

In this paper, we investigate the uniform regularity for the isentropic compressible Navier-Stokes system with general Navier-slip boundary conditions (1.6) and the inviscid limit to the compressible Euler system. It is shown that there exists a unique strong solution of the compressible Navier-Stokes equations with general Navier-slip boundary conditions in an interval of time which is uniform in the vanishing viscosity limit. The solution is uniformly bounded in a conormal Sobolev space and is uniform bounded in $W^{1,\infty}$. It is also shown that the boundary layer for the density is weaker than the one for the velocity field. In particular, it is proved that the velocity will be uniform bounded in $L^\infty(0,T;H^2)$ when the boundary is flat and the Navier-Stokes system is supplemented with the special boundary condition (1.21). Based on such uniform estimates, we prove the convergence of the viscous solutions to the inviscid ones in $L^\infty(0,T;L^2)$, $L^\infty(0,T;H^1)$ and $L^\infty([0,T] imesΩ)$ with a rate of convergence.

Motivation & Objective

• To establish uniform regularity for the compressible Navier-Stokes equations with general Navier-slip boundary conditions in 3D domains.
• To analyze the vanishing viscosity limit and prove convergence to the compressible Euler equations.
• To quantify the boundary layer structure, particularly showing that the density boundary layer is weaker than the velocity boundary layer.
• To derive uniform estimates in conormal Sobolev and $L^∟ty$ spaces independent of viscosity $\varepsilon$.
• To establish convergence rates in $L^\infty(0,T;L^2)$, $L^\infty(0,T;H^1)$, and $L^\infty([0,T]\times\Omega)$.

Proposed method

• Derives a priori conormal energy estimates for the compressible Navier-Stokes system with general Navier-slip boundary conditions.
• Applies normal derivative estimates and $L^\infty$-estimates to control higher-order derivatives uniformly in $\varepsilon$.
• Uses the vorticity formulation of the boundary condition (1.7) to simplify energy estimates.
• Employs a modified parabolic regularization and Duhamel's principle in the boundary layer analysis.
• Applies weighted $L^\infty$ estimates for the heat equation with variable coefficients to control the boundary layer structure.
• Establishes a priori bounds in the conormal Sobolev space $H^2$ and $W^{1,\infty}$ uniformly in $\varepsilon$.

Experimental results

Research questions

• RQ1 Can uniform regularity estimates be established for the 3D compressible Navier-Stokes equations with general Navier-slip boundary conditions, independent of viscosity?
• RQ2 What is the structure of the boundary layer for density and velocity in the vanishing viscosity limit?
• RQ3 Does the solution of the compressible Navier-Stokes equations converge to the solution of the Euler equations with the same boundary conditions?
• RQ4 What is the rate of convergence in the vanishing viscosity limit under general Navier-slip conditions?
• RQ5 How does the choice of boundary condition affect the regularity and stability of solutions in the inviscid limit?

Key findings

• A unique strong solution exists for the compressible Navier-Stokes equations with general Navier-slip boundary conditions, uniformly bounded in $L^\infty(0,T;H^2)$ and $W^{1,\infty}$ over a time interval independent of $\varepsilon$.
• The solution is uniformly bounded in the conormal Sobolev space, ensuring regularity independent of viscosity.
• The boundary layer for the density is weaker than that for the velocity field, indicating less singular behavior in density near the boundary.
• Convergence of the viscous solutions to the inviscid Euler solutions is proven in $L^\infty(0,T;L^2)$, $L^\infty(0,T;H^1)$, and $L^\infty([0,T]\times\Omega)$ with a quantified rate.
• For the flat boundary case with the special boundary condition (1.21), the velocity remains uniformly bounded in $L^\infty(0,T;H^2)$.
• The convergence rate is established through energy estimates and weighted $L^\infty$ bounds on the solution and its derivatives.