2012/05/20 by Gui‐Qiang Chen, Chen, Gui-Qiang G., Vaibhav Kukreja +1
Engineering · Mathematics · #35A05 #35B35 #35B40 #35L65 #76J20 #85A05 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.1205.4429
openalex publication_date 2012/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the well-posedness of compressible vortex sheets and entropy waves\nin two-dimensional steady supersonic Euler flows over Lipschitz walls with BV\nincoming flows. Both the Lipschitz wall of BV tangential angle function and\nthe BV incoming flow perturb a background strong vortex sheet/entropy wave.\nIn particular, when the total variation of the incoming flow perturbation\naround the background strong vortex sheet/entropy wave is small, we prove that\nthe two-dimensional steady supersonic Euler flows containing a strong vortex\nsheet/entropy wave past the Lipschitz wall are L1--stable. The weak waves\nare reflected after the nonlinear waves interact with the strong vortex\nsheet/entropy wave and the wall boundary. Using the wave-front tracking method,\nthe existence of solutions in BV over the Lipschitz walls is first shown,\nwhen the total variation of the incoming flow perturbation around the\nbackground strong vortex sheet/entropy wave is suitably small. Then we\nestablish the L1--stability of the solutions with respect to the incoming\nflows. To achieve this, a Lyapunov functional, equivalent to the\nL1--distance between two solutions containing the strong vortex\nsheets/entropy waves, is carefully constructed to include the nonlinear waves\ngenerated by both the wall boundary and the incoming flow. This Lyapunov\nfunctional is then proved to decrease in the flow direction, leading to the\nL1--stability of the solutions. Furthermore, the uniqueness of these\nsolutions extends to a larger class of viscosity solutions.\n