$ L^1 $–stability of vortex sheets and entropy waves in steady supersonic Euler flows over Lipschitz walls

Gui-Qiang G. Chen; Vaibhav Kukreja

doi:10.3934/dcds.2022124

Article Contents

2023, Volume 43, Issue 3&4: 1239-1268. Doi: 10.3934/dcds.2022124

This issue Previous Article Controlled boundary explosions: Dynamics after blow-up for some semilinear problems with global controls Next Article On some refraction billiards

$ L^1 $–stability of vortex sheets and entropy waves in steady supersonic Euler flows over Lipschitz walls

Gui-Qiang G. Chen^1, , and
Vaibhav Kukreja^2,3,

1.
Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK
2.
Moshman Research, Portland, OR 97219, USA
3.
Department of Mathematics, Northwestern University, Evanston, IL 60208, USA

^*Corresponding author
^*Corresponding author

To Juan Luis Vázquez with Admiration and Friendship

Received: April 2022

Revised: August 2022

Early access: September 7, 2022

Published online: September 7, 2022

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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. 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, by using the wave-front tracking method. 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.

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Mathematics Subject Classification: Primary: 35B35, 35B40, 76J20; Secondary: 35L65, 35A05, 85A05.

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Figure 1.1. Stability of the compressible vortex sheet/entropy wave in supersonic flow

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