Initial-boundary value problem to systems of conservation laws with relaxation
Authors:
Zhouping Xin and Wen-Qing Xu
Journal:
Quart. Appl. Math. 60 (2002), 251-281
MSC:
Primary 35L65; Secondary 35B30
DOI:
https://doi.org/10.1090/qam/1900493
MathSciNet review:
MR1900493
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Abstract: In this paper we consider the initial-boundary value problem (IBVP) for the one-dimensional Jin-Xin relaxation model. The main interest is to study the boundary layer behaviors in the solutions to the IBVP of the relaxation system and their asymptotic convergence to solutions of the corresponding hyperbolic conservation laws in the limit of small relaxation rate. First we develop a general expansion theory for the relaxation IBVP using a matched asymptotic analysis. This formal procedure determines a unique equilibrium limit, and also reveals rich initial and boundary layer structures in the solutions of the relaxation system. Arbitrarily accurate solutions to the IBVP of the relaxation system are then constructed by combining the various orders of the equilibrium solutions, the initial and boundary layer solutions. The validity of the initial and boundary layers and the asymptotic convergence results are rigorously justified through a stability analysis for a broad class of boundary conditions in the case when the relaxation system is $2 \times 2$.
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Gui-Qiang Chen, David Levermore, and Tai-Ping Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math. 47, 787β830 (1994)
Earl. A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955
Shi Jin and Zhouping Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48, 235β277 (1995)
Heinz-Otto Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math. 23, 277β298 (1970)
Jian-Guo Liu and Zhouping Xin, Boundary layer behavior in the fluid-dynamic limit for a nonlinear model Boltzmann equation, Arch. Rational Mech. Anal. 135, 61β105 (1996)
Tai-Ping Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys. 108, 153β175 (1987)
Jens Lorenz and H. Joachim Schroll, Stiff well-posedness for hyperbolic systems with large relaxation terms, Adv. in Differential Equations 2, 643β666 (1997)
Roberto Natalini, Convergence to equilibrium for the relaxation approximations of conservation laws, Comm. Pure Appl. Math. 49, 795β823 (1996)
Roberto Natalini, Recent results on hyperbolic relaxation problems, Analysis of Systems of Conservation Laws (Aachen, 1997), 128β198, Chapman and Hall/CRC Monographs, Surv. Pure Appl. Math., 99, Chapman and Hall/CRC, Boca Raton, FL, 1999
Jeffrey Rauch, $L_{2}$ is a continuable initial condition for Kreissβ mixed problems, Comm. Pure Appl. Math. 25, 265β285 (1972)
Eitan Tadmor and Tao Tang, Pointwise error estimates for relaxation approximations to conservation laws, SIAM J. Math. Anal. 32, 870β886 (2000)
Wei-Cheng Wang and Zhouping Xin, Asymptotic limit of initial boundary value problems for conservation laws with relaxational extensions, Comm. Pure Appl. Math. 51, 505β535 (1998)
G. B. Whitham, Linear and Nonlinear Waves, John Wiley and Sons, New York, NY, 1974
Zhouping Xin and Wen-Qing Xu, Still well-posedness and asymptotic convergence for a class of linear relaxation systems in a quarter plane, J. Differential Equations 167, 388β437 (2000)
Wen-Qing Xu, Relaxation limit for pointwise smooth solutions to systems of conservation laws, J. Differential Equations 162, 140β173 (2000)
Wen-An Yong, Boundary conditions for hyperbolic systems with stiff source terms, Indiana Univ. Math. J. 48, 115β137 (1999)
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© Copyright 2002
American Mathematical Society