We consider the special Jin-Xin relaxation model (0.1) ut + A(u)ux = (uxx − utt ), We assume that the initial data (u0 , u0,t ) are sufficiently smooth and close to ( ̄, 0) in L∞ and have u small total variation. Then we prove that there exists a solution (u (t), ut (t)) with uniformly small total variation for all t ≥ 0, and this solution depends Lipschitz continuously in the L1 norm w.r.t. the initial data and time. We then take the limit → 0, and show that u (t) tends to a unique Lipschitz continuous semigroup S on a domain D containing the functions with small total variation and close to u. The semigroup S ̄ defines a semigroup of relaxation limiting solutions to the quasilinear non conservative system (0.2) ut + A(u)ux = 0. Moreover this semigroup coincides with the trajectory of a Riemann Semigroup, which is determined by the unique Riemann solver compatible with (0.1).
|Titolo:||Relaxation Limit of the Jin-Xin relaxation model|
|Rivista:||COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS|
|Data di pubblicazione:||2006|
|Appare nelle tipologie:||1.1 Journal article|