Singularly perturbed ODEs and profiles for stationary symmetric Euler and Navier-Stokes shocks
Department of Mathematics, Penn State University, University Park, State College, PA 16802, United States, United States
Department of Mathematics, Univ. of North Carolina, Chapel Hill, NC 27599, United States
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.
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