[Paper Review] Nonlinear asymptotic stability of compressible vortex sheets with viscosity effects

Feimin Huang, Zhouping Xin+2 more|arXiv (Cornell University)|Aug 11, 2023

Navier-Stokes equation solutionsMathematics3 citations

TL;DR

This paper establishes the nonlinear asymptotic stability of compressible vortex sheets in the presence of viscosity by constructing a viscous wave approximation that remains stable under small perturbations in $L^rown ext{infty}$-norm, regardless of vortex sheet amplitude. The analysis uses an $L^2$-energy method with a tailored ansatz to cancel spatial oscillations and a Galilean transformation to enable anti-derivative estimates, achieving optimal decay rates.

ABSTRACT

This paper concerns the stabilizing effect of viscosity on the vortex sheets. It is found that although a vortex sheet is not a time-asymptotic attractor for the compressible Navier-Stokes equations, a viscous wave that approximates the vortex sheet on any finite time interval can be constructed explicitly, which is shown to be time-asymptotically stable in the $ L^\infty $-space with small perturbations, regardless of the amplitude of the vortex sheet. The result shows that the viscosity has a strong stabilizing effect on the vortex sheets, which are generally unstable for the ideal compressible Euler equations even for short time [26,8,1]. The proof is based on the $ L^2 $-energy method.In particular, the asymptotic stability of the vortex sheet under small spatially periodic perturbations is proved by studying the dynamics of these spatial oscillations. The first key point in our analysis is to construct an ansatz to cancel these oscillations. Then using the Galilean transformation, we are able to find a shift function of the vortex sheet such that an anti-derivative technique works, which plays an important role in the energy estimates. Moreover, by introducing a new variable and using the intrinsic properties of the vortex sheet, we can achieve the optimal decay rates to the viscous wave.

Motivation & Objective

• To investigate the stabilizing effect of viscosity on compressible vortex sheets, which are unstable under the ideal Euler equations.
• To construct a viscous wave that approximates a vortex sheet over any finite time interval.
• To prove the time-asymptotic stability of this viscous wave in $L^rown ext{infty}$-space under small perturbations.
• To achieve optimal decay rates for the viscous wave using intrinsic properties of the vortex sheet and a novel variable transformation.
• To resolve the challenge of spatial oscillations in perturbations through a carefully designed ansatz and shift function.

Proposed method

• Construct an ansatz to cancel spatial oscillations arising from small perturbations around the vortex sheet.
• Apply a Galilean transformation to shift the vortex sheet and enable the use of an anti-derivative technique in energy estimates.
• Introduce a new variable to exploit intrinsic properties of the vortex sheet and improve energy estimates.
• Use the $L^2$-energy method to control higher-order derivatives and derive decay estimates.
• Employ weighted norms and time-decay estimates to handle the singular behavior near the interface.
• Combine pointwise and integral estimates to achieve optimal decay rates in $L^rown ext{infty}$-norm.

Experimental results

Research questions

• RQ1 Can viscosity stabilize compressible vortex sheets that are nonlinearly unstable under the Euler equations?
• RQ2 Is there a viscous wave approximation to a vortex sheet that remains stable under small perturbations in $L^rown ext{infty}$-norm?
• RQ3 How can spatial oscillations in perturbations be effectively canceled to enable energy estimates?
• RQ4 What is the optimal decay rate of the viscous wave toward the vortex sheet in the presence of viscosity?
• RQ5 Can the $L^2$-energy method be adapted to handle the nonlinear and degenerate structure of vortex sheet problems?

Key findings

• Viscosity provides a strong stabilizing effect on compressible vortex sheets, rendering them nonlinearly asymptotically stable in $L^rown ext{infty}$-norm despite instability in the Euler system.
• A viscous wave can be explicitly constructed to approximate the vortex sheet on any finite time interval.
• The viscous wave is time-asymptotically stable under small perturbations, regardless of the amplitude of the original vortex sheet.
• The $L^2$-energy method, combined with a Galilean transformation and anti-derivative technique, enables control of higher-order derivatives and optimal decay rates.
• The use of a shift function and a new variable allows the cancellation of oscillatory terms and improves the regularity of the energy estimates.
• Optimal decay rates are achieved for the viscous wave, with decay rates matching those expected from the underlying parabolic structure of the Navier-Stokes equations.