We consider the special Jin-Xin relaxation model (0.1) u 1 + A(u)u x = ε (u xx - u tt). We assume that the initial data (u 0, εu 0,t) are sufficiently smooth and close to (ū, 0) in L ∞ and have small total variation. Then we prove that there exists a solution (u ε (t), εu t ε(t)) with uniformly small total variation for all t ≥ 0, and this solution depends Lipschitz-continuously in the L 1 norm with respect to time and the initial data. Letting ε → 0, the solution u ε converges to a unique limit, providing a relaxation limit solution to the quasi-linear, nonconservative system (0.2) u t + A(u)u x= 0. These limit solutions generate a Lipschitz semigroup S on a domain D containing the functions with small total variation and close to ū. This is precisely the Riemann semigroup determined by the unique Riemann solver compatible with (0.1).

Hyperbolic limit of the Jin-Xin relaxation model / Bianchini, Stefano. - In: COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS. - ISSN 0010-3640. - 59:5(2006), pp. 688-753. [10.1002/cpa.20114]

Hyperbolic limit of the Jin-Xin relaxation model

Bianchini, Stefano
2006-01-01

Abstract

We consider the special Jin-Xin relaxation model (0.1) u 1 + A(u)u x = ε (u xx - u tt). We assume that the initial data (u 0, εu 0,t) are sufficiently smooth and close to (ū, 0) in L ∞ and have small total variation. Then we prove that there exists a solution (u ε (t), εu t ε(t)) with uniformly small total variation for all t ≥ 0, and this solution depends Lipschitz-continuously in the L 1 norm with respect to time and the initial data. Letting ε → 0, the solution u ε converges to a unique limit, providing a relaxation limit solution to the quasi-linear, nonconservative system (0.2) u t + A(u)u x= 0. These limit solutions generate a Lipschitz semigroup S on a domain D containing the functions with small total variation and close to ū. This is precisely the Riemann semigroup determined by the unique Riemann solver compatible with (0.1).
2006
59
5
688
753
https://doi.org/10.1002/cpa.20114
Bianchini, Stefano
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/12942
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