Symmetric Euler and Navier–Stokes shocks in stationary barotropic flow on a bounded domain
Erik Endres, Helge Kristian Jenssen and Mark Williams
Abstract: We construct stationary solutions to the barotropic, compressible Euler and Navier–Stokes equations in several space dimensions with spherical or cylindrical symmetry. For given Dirichlet data on a sphere or a cylinder we first construct smooth and radially symmetric solutions to the Euler equations in the exterior domain. On the other hand, stationary smooth solutions in the interior domain necessarily become sonic and can not be continued beyond a critical inner radius. We then use these solutions to construct entropy-satisfying shocks for the Euler equations in the region between two concentric spheres or cylinders. Next we construct smooth Navier-Stokes solutions converging to the previously constructed Euler shocks in the small viscosity limit. In the process we introduce a new technique for constructing smooth solutions, which exhibit a fast transition in the interior, to a class of two-point boundary problems.