# Preprint 2006-021

# The boundary Riemann solver coming from the real vanishing viscosity approximation

## Stefano Bianchini and Laura V. Spinolo

**Abstract:**
We study the limit of the hyperbolic-parabolic approximation

v^{ε}_{t}+A(v^{ε},εv^{ε}_{x})v^{ε}_{x} = εB(v^{ε})v^{ε}_{xx},

with data
β(v^{ε}(t,0)) = g and
v^{ε}(0,x) = v_{0}.
The function β is defined in such a way to guarantee
that the initial boundary value problem is well posed
even if B is not invertible.
The data g and v_{0}
are constant.

When B is invertible,
the boundary datum can be assigned by imposing
v^{ε}(t,0) = v_{b},
where v_{b} is again a constant.
The conservative case is included in the previous formulations.

It is assumed convergence of the
v^{ε},
smallness of the total variation
and other technical hypotheses
and it is provided a complete characterization of the limit.

The most interesting points are the following two.

First, the boundary characteristic case is considered, i.e. one eigenvalue of A can be 0.

Second, as pointed out before we take into account the possibility that B is not invertible. To deal with this case, we take as hypotheses conditions that were introduced by Kawashima and Shizuta relying on physically meaningful examples. We also introduce a new condition of block linear degeneracy. We prove that, if it is not satisfied, then pathological behaviours may occur.