# Preprint 2009-018

# 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
*w*^{ε}
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 *w*^{ε},
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.