Singularly perturbed ODEs and profiles for stationary symmetric Euler and Navier–Stokes shocks
Erik Endres, Helge Kristian Jenssen and Mark Williams
Abstract: We construct stationary solutions to the non-barotropic, compressible Euler and Navier–Stokes equations in several space dimensions with spherical or cylindrical symmetry. The equation of state is assumed to satisfy standard monotonicity and convexity assumptions. For given Dirichlet data on a sphere or a cylinder we first construct smooth and radially symmetric solutions to the Euler equations in an exterior domain. On the other hand, stationary smooth solutions in an interior domain necessarily become sonic and cannot 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 solutions wε to the Navier–Stokes system converging to the previously constructed Euler shocks in the small viscosity limit ε→0. The viscous solutions are obtained by a new technique for constructing solutions to a class of two-point boundary problems with a fast transition region. The construction is explicit in the sense that it produces high order expansions in powers of ε for wε, and the coefficients in the expansion satisfy simple, explicit ODEs, which are linear except in the case of the leading term. The solutions to the Euler equations described above provide the slowly varying contribution to the leading term in the expansion.
The approach developed here is applicable to a variety of singular perturbation problems, including the construction of heteroclinic orbits with fast transitions. For example, a variant of our method is used in [W] to give a new construction of detonation profiles for the reactive Navier–Stokes equations.