Convergence to equilibrium rarefaction waves for discontinuous solutions of shallow water wave equations with relaxation
Authors:
Haitao Fan and Tao Luo
Journal:
Quart. Appl. Math. 63 (2005), 575-600
MSC (2000):
Primary 35L65, 35L67, 35L60
DOI:
https://doi.org/10.1090/S0033-569X-05-00980-4
Published electronically:
August 18, 2005
MathSciNet review:
2169035
Full-text PDF Free Access
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Abstract: The purpose of this paper is to study the discontinuous solutions to a shallow water wave equation with relaxation. The typical initial value problem of discontinuous solutions is the Riemann problem. Unlike the homogeneous hyperbolic conservation laws, due to the inhomogeneity of the system studied here, the solutions of the Riemann problem do not have a self-similar structure anymore. This problem can be formulated as a free boundary problem. We show that the Riemann solutions still have a piecewise smooth structure globally and converge to the rarefaction waves of the equilibrium equation as time tends to infinity.
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AG1 Amadori, D.; Guerra, G., Global BV solutions and relaxation limit for a system of conservation laws. Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), no. 1, 1β26.
bBianchini, S., A Glimm type functional for a special Jin-Xin relaxation model. Ann. Inst. H. PoincarΓ© Anal. Non LinΓ©aire 18 (2001), no. 1, 19β42.
chenil Chern, I. L., Long-time effect of relaxation for hyperbolic conservation laws. Comm. Math. Phys. 172 (1995), no. 1, 39β55.
CLL Chen, G.-Q.; Levermore, C.D.; Liu, P.-T., Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math., XLVII (1994), 1β45.
chen1 Chen, G. Q.; Liu, T. P., Zero relaxation and dissipation limits for hyperbolic conservation laws. Comm. Pure Appl. Math. 46 (1993), no. 5, 755β781.
CH Courant, R; Friedrichs K. O., Supersonic Flow and Shock Waves, Interscience Publishers Inc., New York, 1948.
fanFan H., Self-similar solutions for a modified Broadwell model and its hydrodynamic limits, SIAM J. Math. Anal. 28 (1997), 831-851.
fjm Fan, H.; Jin, S.; Miller, J., Wave patterns, stability, and slow motions in inviscid and viscous hyperbolic equations with stiff reaction terms. J. Differential Equations 189 (2003), no. 1, 267β291.
GH Greenberg, J; Hsiao L., The Riemann problem for $u_t+\sigma _x=0$ and $(\sigma -f(u))_t+(\sigma -\mu f(u))=0$. Arch. Ration. Mech. Anal. 82 (1983), 87-108.
Ha Ha, S.; Yu, S. H., Wave front tracing and asymptotic stability of planar traveling waves for a two-dimensional shallow river model. J. Differential Equations 186 (2002), no. 1, 230β258.
HL Hsiao L.; Luo, T., Nonlinear diffusive phenomena of entropy weak solutions for a system of quasilinear hyperbolic conservation laws with damping. Quart. Appl. Math. 56 (1998), no. 1, 173β189.
HT2 Hsiao, L.; Tang, S. Q., Construction and qualitative behavior of the solution of the perturbated Riemann problem for the system of one-dimensional isentropic flow with damping. J. Differential Equations 123 (1995), no. 2, 480β503.
hlm Hsiao, L.; Li, H.L.; Mei, M., Convergence rates to superposition of two travelling waves of the solutions to a relaxation hyperbolic system with boundary effects, Math. Models Methods Appl. Sci. 11 (2001), no. 7, 1143β1168.
hp Hsiao, L.; Pan, R., Nonlinear stability of rarefaction waves for a rate-type viscoelastic system. Chinese Ann. Math. Ser. B 20 (1999), no. 2, 223β232.
Jin Jin, S.; Katsoulakis, M., Hyperbolic systems with supercharacteristic relaxations and roll waves, SIAM J. Appl. Math. 61 (2000), no. 1, 273β292 (electronic).
JX Jin, S.; Xin, Z., The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Comm. Pure Appl. Math. 48 (1995), no. 3, 235β276.
lhlLiu, H. L., The $L^ p$ stability of relaxation rarefaction profiles. J. Differential Equations 171 (2001), no. 2, 397β411.
lh2 Liu, H. L., Asymptotic stability of relaxation shock profiles for hyperbolic conservation laws. J. Differential Equations 192 (2003), no. 2, 285β307.
lx Liu, J.-G.; Xin, Z., Boundary-layer behavior in the fluid-dynamic limit for a nonlinear model Boltzmann equation. Arch. Rational Mech. Anal. 135 (1996), no. 1, 61β105.
LY Li, T. T.; Yu W.C., Boundary value problems for quasilinear hyperbolic systems. Duke University Mathematics Series, V. Duke University, Mathematics Department, Durham, NC, 1985.
Liu1Liu, T.-P., Hyperbolic conservation laws with relaxation, Comm. Math. Phys. 108 (1987), 153-175.
Liu Liu, T. P., Linear and nonlinear large time behavior of general systems of hyperbolic conservation laws, Comm. Pure Appl. Math. 30 (1977), 767-796.
LYA Luo, T.; Yang, T., Interaction of elementary waves for the compressible Euler equations with frictional damping, J. Differential Equations 161 (2000), no. 1, 42β86.
LYAN Luo, T.; Yang, T., Global Structure and Asymptotic Behavior of Weak Solutions to Flood Wave Equations, J. Differential Equations 207 (2004), no. 1, 117β160.
lny Luo, T.; Natalini, R.; Yang, T., Global BV solutions to a $p$-system with relaxation. J. Differential Equations 162 (2000), no. 1, 174β198.
lux Luo, T.; Xin, Z., Nonlinear stability of shock fronts for a relaxation system in several space dimensions. J. Differential Equations 139 (1997), no. 2, 365β408.
luo Luo, T., Asymptotic stability of planar rarefaction waves for the relaxation approximation of conservation laws in several dimensions. J. Differential Equations 133 (1997), no. 2, 255β279.
macartiMarcati, P.; Rubino, B., Hyperbolic to parabolic relaxation theory for quasilinear first order systems. J. Differential Equations 162 (2000), no. 2, 359β399.
mn Mascia, C.; Natalini, R., $L^ 1$ nonlinear stability of traveling waves for a hyperbolic system with relaxation. J. Differential Equations 132 (1996), no. 2, 275β292.
msMascia, C.; Sinestrari, C.; The perturbed Riemann problem for a balance law. Adv. Differential Equations 2 (1997), no. 5, 779β810.
mz Mascia, C.; Zumbrun, K., Pointwise Greenβs function bounds and stability of relaxation shocks. Indiana Univ. Math. J. 51 (2002), no. 4, 773β904.
na Natalini, R., Recent results on hyperbolic relaxation problems. Analysis of systems of conservation laws (Aachen, 1997), 128β198, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., 99, Chapman & Hall/CRC, Boca Raton, FL, 1999.
na1 Natalini, R., Convergence to equilibrium for the relaxation approximations of conservation laws. Comm. Pure Appl. Math. 49 (1996), no. 8, 795β823.
yu Nishibata, S; Yu, S. H., The asymptotic behavior of the hyperbolic conservation laws with relaxation on the quarter-plane. SIAM J. Math. Anal. 28 (1997), no. 2, 304β321.
Sm Smoller, J., Shock waves and reaction-diffusion equations, Springer-Verlag, New York, 1996.
shen Shen, W.; Tveito, A.; Winther, R., On the zero relaxation limit for a system modeling the motions of a viscoelastic solid. SIAM J. Math. Anal. 30 (1999), no. 5, 1115β1135.
serreSerre, D., The stability of constant equilibrium states in relaxation models. Ann. Univ. Ferrara Sez. VII (N.S.) 48 (2002), 253β274.
slemerodSlemrod, M.; Tzavaras, A. E., Self-similar fluid-dynamic limits for the Broadwell system. Arch. Rational Mech. Anal. 122 (1993), no. 4, 353β392.
tt Tadmor, E.; Tang, T., Pointwise error estimates for relaxation approximations to conservation laws. SIAM J. Math. Anal. 32 (2000), no. 4, 870β886.
teng Teng, Z.-H., First-order $L^ 1$-convergence for relaxation approximations to conservation laws. Comm. Pure Appl. Math. 51 (1998), no. 8, 857β895.
xw Wang, W.C.; Xin, Z., Asymptotic limit of initial-boundary value problems for conservation laws with relaxational extensions. Comm. Pure Appl. Math. 51 (1998), no. 5, 505β535.
WH Whitham, G. B., Linear and nonlinear waves, John Wiley & Sons, New York, 1974.
wang Wang, W.-C., Nonlinear stability of centered rarefaction waves of the Jin-Xin relaxation model for $2\times 2$ conservation laws. Electron. J. Differential Equations 2002, No. 57, 20 pp. (electronic).
wangxin Wang, W.-C.; Xin, Zhouping, Fluid-dynamic limit for the centered rarefaction wave of the Broadwell equation. J. Differential Equations 150 (1998), no. 2, 438β461.
xin Xin, Z., The fluid-dynamic limit of the Broadwell model of the nonlinear Boltzmann equation in the presence of shocks. Comm. Pure Appl. Math. 44 (1991), no. 6, 679β713.
x Xin, Z.; Xu, W. Q., Stiff well-posedness and asymptotic convergence for a class of linear relaxation systems in a quarter plane. J. Differential Equations 167 (2000), no. 2, 388β437.
xinxu Xin, Z.; Xu, W.-Q., Initial-boundary value problem to systems of conservation laws with relaxation. Quart. Appl. Math. 60 (2002), no. 2, 251β281
xu Xu, W.-Q., Boundary conditions and boundary layers for a multi-dimensional relaxation model. J. Differential Equations 197 (2004), no. 1, 85β117.
xu1 Xu, W.-Q., Relaxation limit for piecewise smooth solutions to systems of conservation laws. J. Differential Equations 162 (2000), no. 1, 140β173.
ZY Yang, T.; Zhu, C., Existence and nonexistence of smooth solutions to $p$-system with relaxation, J. Differential Equations, 161 (2000), 321-336.
yong Yong, W.-A., Singular perturbations of first-order hyperbolic systems with stiff source terms. J. Differential Equations 155 (1999), no. 1, 89β132.
yz Yong, W.-A.; Zumbrun, K., Existence of relaxation shock profiles for hyperbolic conservation laws. SIAM J. Appl. Math. 60 (2000), no. 5, 1565β1575.
zeng Zeng, Y., Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation. Arch. Ration. Mech. Anal. 150 (1999), no. 3, 225β279.
zhu Zhu, C. J., Asymptotic behavior of solutions for $p$-system with relaxation. J. Differential Equations 180 (2002), no. 2, 273β306.
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Additional Information
Haitao Fan
Affiliation:
Department of Mathematics, Georgetown University, Washington, DC 20057-1233
Email:
fanh@georgetown.edu
Tao Luo
Affiliation:
Department of Mathematics, Georgetown University, Washington, DC 20057-1233
Email:
tl48@georgetown.edu
Keywords:
Shallow water wave equations,
relaxation,
shock waves,
rarefaction waves,
free boundary problem
Received by editor(s):
February 10, 2005
Published electronically:
August 18, 2005
Article copyright:
© Copyright 2005
Brown University
The copyright for this article reverts to public domain 28 years after publication.