Gui-Qiang G. Chen, Vaibhav Kukreja
We are concerned with the stability of compressible vortex sheets and entropy waves in two-dimensional steady supersonic Euler flows over Lipschitz walls under a $BV$ boundary perturbation. In this paper, we establish the $L^{1}$ well-posedness for two-dimensional steady supersonic Euler flows containing a strong vortex sheet/entropy wave past a Lipschitz wall whose boundary slope function has small total variation, when the total variation of the incoming flow perturbation around the background strong vortex sheet/entropy wave is small. In this case, both the Lipschitz wall and incoming flow perturb the background strong vortex sheet/entropy wave, and the waves reflect after interacting with the strong vortex sheet/entropy wave and the wall boundary. We first establish the existence of solutions in $BV$, when the incoming flow perturbation of the background strong vortex sheet/entropy wave has small total variation by the wave-front tracking method and then establish the $L^{1}$-stability of the solutions with respect to the incoming flows. In particular, we incorporate the nonlinear waves generated both by the wall boundary and from the incoming flow to develop a Lyapunov functional between two solutions containing strong vortex sheets/entropy waves, which is equivalent to the $L^{1}$-norm, and prove that the functional decreases in the flow direction. Then the $L^{1}$-stability is established, so is the uniqueness of the solutions by the wave-front tracking method. Finally, the uniqueness of solutions in a broader class, the class of viscosity solutions, is also obtained.
View original:
http://arxiv.org/abs/1205.4429
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