Decay Rate for Travelling Waves of a Relaxation Model

Abstract

A relaxation model was proposed in [Shi Jin and Zhouping Xin, Comm. Pure Appl. Math. 48 (1995) preprint, 1995] when the original system is scalar. In this paper, we study the rate of the asymptotic convergence speed of thse travelling wave solutions. The analysis applies to the case of a nonconvex flux and when the shock speed coincides with characteristic speed of the state at infinity. The decay rate is obtained by applying the energy method and is shown to be the same as the one for the viscous conservation law [A. Matsumura and K. Nishihara, Comm. Math. Phys. 165 (1994), 83 96].

Asymptotic limit of initial boundary value problems for conservation laws with relaxational extensions

“…There are many recent studies concerning the asymptotic convergence of the relaxation systems to the corresponding equilibrium conservation laws as the rate of relaxation tends to zero; see, for example, [3,4,[9][10][11][12][13][14][15][16][17][18]. Most of these results deal with either large-time, nonlinear asymptotic stability or the zero relaxation limit for Cauchy problems for special systems.…”

Nonlinear Stability of Strong Planar Rarefaction Waves for the Relaxation Approximation of Conservation Laws in Several Space Dimensions

“…Since then, the stability of certain elementary waves was studied by H. Liu, C. Woo, and T. Yang [12], T. Luo [15], T. Luo and Z. P. Xin [16], C. Mascia and R. Natalini [18], M. Mei and T. Yang [22], R. H. Pan [26], C. J. Zhu [31], and P. Zingano [32], etc. The problem on the convergence to the diffusion waves was given by I.-L. Chern in [4].…”

Phase Transitions in a Relaxation Model of Mixed Type with Periodic Boundary Condition

“…With m 1 = f (m 0 ) from (1.6), we obtain from the second equation of (1.3) 19 which is equivalent to …”