[Paper Review] $L^1$-Stability of Vortex Sheets and Entropy Waves in Steady Compressible Supersonic Euler Flows over Lipschitz Walls

Gui‐Qiang Chen, Vaibhav Kukreja|arXiv (Cornell University)|May 20, 2012

Navier-Stokes equation solutions16 references3 citations

TL;DR

This paper establishes the $L^1$-stability of compressible vortex sheets and entropy waves in two-dimensional steady supersonic Euler flows over Lipschitz walls with $BV$-regular incoming flows. Using wave-front tracking and a tailored Lyapunov functional, it proves global existence, uniqueness, and $L^1$-stability of solutions when the total variation of incoming perturbations is sufficiently small.

ABSTRACT

We study the well-posedness of compressible vortex sheets and entropy waves in two-dimensional steady supersonic Euler flows over Lipschitz walls with $BV$ incoming flows. Both the Lipschitz wall of $BV$ tangential angle function and the $BV$ incoming flow perturb a background strong vortex sheet/entropy wave. In particular, when the total variation of the incoming flow perturbation around the background strong vortex sheet/entropy wave is small, we prove that the two-dimensional steady supersonic Euler flows containing a strong vortex sheet/entropy wave past the Lipschitz wall are $L^{1}$--stable. The weak waves are reflected after the nonlinear waves interact with the strong vortex sheet/entropy wave and the wall boundary. Using the wave-front tracking method, the existence of solutions in $BV$ over the Lipschitz walls is first shown, when the total variation of the incoming flow perturbation around the background strong vortex sheet/entropy wave is suitably small. Then we establish the $L^{1}$--stability of the solutions with respect to the incoming flows. To achieve this, a Lyapunov functional, equivalent to the $L^{1}$--distance between two solutions containing the strong vortex sheets/entropy waves, is carefully constructed to include the nonlinear waves generated by both the wall boundary and the incoming flow. This Lyapunov functional is then proved to decrease in the flow direction, leading to the $L^{1}$--stability of the solutions. Furthermore, the uniqueness of these solutions extends to a larger class of viscosity solutions.

Motivation & Objective

• To establish the well-posedness of compressible vortex sheets and entropy waves in steady supersonic Euler flows over Lipschitz walls with $BV$ incoming flows.
• To prove $L^1$-stability of solutions containing strong vortex sheets or entropy waves under small $BV$ perturbations of the incoming flow.
• To extend the uniqueness of entropy solutions to a broader class of viscosity solutions using the wave-front tracking semigroup.
• To construct a Lyapunov functional equivalent to the $L^1$-distance that decreases along the flow direction, ensuring stability.
• To show that the wave-front tracking semigroup coincides with the standard Riemann semigroup, guaranteeing uniqueness in the viscosity solution class.

Proposed method

• Formulates the 2D steady Euler system as a hyperbolic system of conservation laws with full thermodynamic relations.
• Applies the wave-front tracking method to construct approximate solutions in $BV$ space, ensuring convergence under small total variation of incoming perturbations.
• Constructs a Lyapunov functional that captures both the $L^1$-distance between solutions and the nonlinear waves generated by wall reflections and incoming perturbations.
• Proves that the Lyapunov functional decreases monotonically in the flow direction, implying $L^1$-stability of the solution semigroup.
• Uses the convergence of the wave-front tracking scheme to define a uniformly Lipschitz semigroup $\mathscr{S}$ in $L^1$.
• Establishes equivalence between the wave-front tracking semigroup and the standard Riemann semigroup, proving uniqueness in the viscosity solution class.

Experimental results

Research questions

• RQ1 Under what conditions does a steady supersonic Euler flow with a strong vortex sheet or entropy wave remain $L^1$-stable over a Lipschitz wall with $BV$-regular tangential angle?
• RQ2 Can the wave-front tracking method be adapted to handle nonlinear wave interactions with both a strong vortex sheet/entropy wave and a Lipschitz wall boundary?
• RQ3 Is the solution semigroup generated by wave-front tracking equivalent to the standard Riemann semigroup in the context of $L^1$-stability?
• RQ4 How can a Lyapunov functional be constructed to account for both incoming flow perturbations and wall-induced nonlinear waves?
• RQ5 Does the uniqueness of entropy solutions extend to the broader class of viscosity solutions under the given conditions?

Key findings

• The solution to the initial-boundary value problem for 2D steady supersonic Euler flows with a strong vortex sheet or entropy wave is $L^1$-stable when the total variation of the incoming $BV$ perturbation is sufficiently small.
• The wave-front tracking algorithm produces a Cauchy sequence in $L^1$, converging to a unique limit solution that defines a uniformly Lipschitz semigroup $\mathscr{S}$ in $L^1$.
• A Lyapunov functional equivalent to the $L^1$-distance between solutions is constructed, which decreases monotonically in the flow direction, proving $L^1$-stability.
• The semigroup $\mathscr{S}$ generated by wave-front tracking coincides with the standard Riemann semigroup, ensuring uniqueness of entropy solutions in the viscosity solution class.
• The solution is unique not only among entropy solutions but also within the broader class of viscosity solutions as defined by Bressan.
• The results extend to isentropic and isothermal flows, indicating the robustness of the $L^1$-stability framework for hyperbolic systems with free boundaries.