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Charis Tsikkou, West Virginia University, Radial solutions to the Cauchy problem for the wave equation and compressible Euler system
March 20, 2019 | 3:00 pm - 4:00 pm EDT
In the first part of this work, we consider the strategy of realizing the solution of the three-dimensional linear wave equation with radial Cauchy data as a limit of radial exterior solutions satisfying vanishing Neumann and Dirichlet conditions, on the exterior of vanishing balls centered at the origin. We insist on robust arguments based on energy methods and strong convergence. Our findings show that while one can obtain existence of radial Cauchy solutions via exterior solutions, one should not expect such results to be optimal. In the second part, we review the construction of globally defined radial similarity shock and cavity flows, and give a detailed description of their behavior following collapse. We then prove that similarity shock solutions provide bona fide weak solutions, of unbounded amplitude, to the multi-dimensional Euler system. This is joint work with Helge Kristian Jenssen (PSU).