We study the limit of the hyperbolic-parabolic approximation 8 < : v ε t + A˜ ` v ε , εv ε x ´ v ε x = εB˜(v ε )v ε xx v ε ∈ R N ˜ ß(v ε (t, 0)) ≡ g¯ v ε (0, x) ≡ v¯0. The function ß is deﬁned in such a way to guarantee that the initial boundar ˜ y value problem is well posed even if B˜ is not invertible. The data ¯g and ¯v0 are constant. When B˜ is invertible, the previous problem takes the simpler form 8 < : v ε t + A˜ ` v ε , εv ε x ´ v ε x = εB˜(v ε )v ε xx v ε ∈ R N v ε (t, 0) ≡ v¯b v ε (0, x) ≡ v¯0. Again, the data ¯vb and ¯v0 are 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.
|Titolo:||THE BOUNDARY RIEMANN SOLVER COMING FROM THE REAL VANISHING VISCOSITY APPROXIMATION|
|Autori:||BIANCHINI S; SPINOLO L. V|
|Rivista:||ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS|
|Data di pubblicazione:||2009|
|Appare nelle tipologie:||1.1 Journal article|