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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

vεt+A(vε,εvεx)vεx = εB(vε)vε_{xx},

with data β(vε(t,0)) = g and vε(0,x) = v0. 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 v0 are constant.

When B is invertible, the boundary datum can be assigned by imposing vε(t,0) = vb, where vb is again a constant. The conservative case is included in the previous formulations.

It is assumed convergence of the vε, 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.

Available as PDF ( Kbytes).
Stefano Bianchini,
Laura V. Spinolo,
Publishing information:
Also at arXiv as math.AP/0605575
Updated 2009-10-01.
Submitted by:
; 2006-05-30.