Zero relaxation limit to centered rarefaction waves for Jin–Xin relaxation system

• Published in: Nonlinear analysis 2011-03, Vol.74 (6), p.2249-2261
• Main Authors: Zhang, Yinghui, Tan, Zhong, Sun, Ming-Bao
• Format: Article
• Language: English
• Subjects: Centered rarefaction waves; Conservation; Discontinuity; Exact sciences and technology; Jin–Xin relaxation system; Law; Mathematical analysis; Mathematical models; Mathematics; Partial differential equations; Positioning; Rarefaction; Relaxation time; Sciences and techniques of general use; Zero relaxation limit
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2010.11.030

In this paper, we study the zero relaxation limit problem for the following Jin–Xin relaxation system (E) { u t + v x = 0 v t + a 2 u x = 1 ϵ ( f ( u ) − v ) with initial data (I) ( u , v ) ( x , 0 ) = ( u 0 ( x ) , v 0 ( x ) ) → ( u ± , v ± ) , u ± > 0 , as  x → ± ∞ . This system was proposed by Jin and Xin (1995) [1] with an interesting numerical origin. As the relaxation time tends to zero, this system converges to the equilibrium conservation law formally. Our interest is to study the case where the initial data are allowed to have jump discontinuities such that the corresponding solutions to the equilibrium conservation law contain centered rarefaction waves and the limits ( u ± , v ± ) of the initial data at x = ± ∞ do not satisfy the equilibrium equation, i.e., v ± ≠ f ( u ± ) . In particular, Riemann data connected by rarefaction curves are included. We show that if the wave strength is sufficiently small, then the solution for the relaxation system exists globally in time and converges to the solution of the corresponding rarefaction waves uniformly as the relaxation time goes to zero except for an initial layer.