Convergence to nonlinear diffusion waves for solutions of the initial boundary problem to the hyperbolic conservation laws with damping

Marcati, Pierangelo; Mei, Ming

doi:10.1090/qam/1788427

Authors: Pierangelo Marcati and Ming Mei
Journal: Quart. Appl. Math. 58 (2000), 763-784
MSC: Primary 35L65; Secondary 35A05, 35B40
DOI: https://doi.org/10.1090/qam/1788427
MathSciNet review: MR1788427
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider a model of hyperbolic balance laws with damping on the quarter plane $(x, t) \in {\mathbb {R}_ + } \times {\mathbb {R}_ + }$. By means of a suitable shift function, which will play a key role to overcome the difficulty of large boundary perturbations, we show that the IBVP solutions converge time-asymptotically to the shifted nonlinear diffusion wave solutions of the Cauchy problem to the nonlinear parabolic equation given by the related Darcy’s law. We obtain also the time decay rates, which are the optimal ones in the ${L^{2}}$-sense. Our proof is based on the use of the classical energy method.

I-Liang Chern, Multiple-mode diffusion waves for viscous non-strictly hyperbolic conservation laws, Comm. Math. Phys. 138 (1991), no. 1, 51–61. MR 1108036
I-Liang Chern, Long-time effect of relaxation for hyperbolic conservation laws, Comm. Math. Phys. 172 (1995), no. 1, 39–55. MR 1346371
I-Liang Chern and Tai-Ping Liu, Convergence to diffusion waves of solutions for viscous conservation laws, Comm. Math. Phys. 110 (1987), no. 3, 503–517. MR 891950
S. Claudi and F. R. Guarguaglini, Large time behaviour for the heat equation with absorption and convection, Adv. in Appl. Math. 16 (1995), no. 4, 377–401. MR 1362847, DOI https://doi.org/10.1006/aama.1995.1018
S. Claudi, R. Natalini, and A. Tesei, Large time behaviour of a diffusion equation with strong convection, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 21 (1994), no. 3, 445–474. MR 1310636

C. T. Van Duyn and L. A. Peletier, A class of similarity solutions of the nonlinear diffusion equation, Nonlinear Analysis T.M.A. 1, 223–233 (1977)

Ling Hsiao and Tai-Ping Liu, Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Comm. Math. Phys. 143 (1992), no. 3, 599–605. MR 1145602
Ling Hsiao and Tai-Ping Liu, Nonlinear diffusive phenomena of nonlinear hyperbolic systems, Chinese Ann. Math. Ser. B 14 (1993), no. 4, 465–480. A Chinese summary appears in Chinese Ann. Math. Ser. A 14 (1993), no. 6, 740. MR 1254543
L. Hsiao and Tao Luo, Nonlinear diffusive phenomena of solutions for the system of compressible adiabatic flow through porous media, J. Differential Equations 125 (1996), no. 2, 329–365. MR 1378760, DOI https://doi.org/10.1006/jdeq.1996.0034
Ta-tsien Li, Nonlinear heat conduction with finite speed of propagation, China-Japan Symposium on Reaction-Diffusion Equations and their Applications and Computational Aspects (Shanghai, 1994) World Sci. Publ., River Edge, NJ, 1997, pp. 81–91. MR 1654353
Tai-Ping Liu and Kenji Nishihara, Asymptotic behavior for scalar viscous conservation laws with boundary effect, J. Differential Equations 133 (1997), no. 2, 296–320. MR 1427855, DOI https://doi.org/10.1006/jdeq.1996.3217
Tai-Ping Liu and Shih-Hsien Yu, Propagation of a stationary shock layer in the presence of a boundary, Arch. Rational Mech. Anal. 139 (1997), no. 1, 57–82. MR 1475778, DOI https://doi.org/10.1007/s002050050047
Pierangelo Marcati and Albert Milani, The one-dimensional Darcy’s law as the limit of a compressible Euler flow, J. Differential Equations 84 (1990), no. 1, 129–147. MR 1042662, DOI https://doi.org/10.1016/0022-0396%2890%2990130-H
Pierangelo Marcati, Albert J. Milani, and Paolo Secchi, Singular convergence of weak solutions for a quasilinear nonhomogeneous hyperbolic system, Manuscripta Math. 60 (1988), no. 1, 49–69. MR 920759, DOI https://doi.org/10.1007/BF01168147
Pierangelo Marcati and Bruno Rubino, Hyperbolic to parabolic relaxation theory for quasilinear first order systems, J. Differential Equations 162 (2000), no. 2, 359–399. MR 1751710, DOI https://doi.org/10.1006/jdeq.1999.3676

A. Matsumura, Nonlinear hyperbolic equations and related topics in fluid dynamics, T. Nishida (ed.), Pub. Math. D’Orsay, 53–57 (1978)

Akitaka Matsumura and Ming Mei, Convergence to travelling fronts of solutions of the $p$-system with viscosity in the presence of a boundary, Arch. Ration. Mech. Anal. 146 (1999), no. 1, 1–22. MR 1682659, DOI https://doi.org/10.1007/s002050050134
Ming Mei and Bruno Rubino, Convergence to traveling waves with decay rates for solutions of the initial-boundary problem to a relaxation model, J. Differential Equations 159 (1999), no. 1, 138–185. MR 1726921, DOI https://doi.org/10.1006/jdeq.1999.3640
Kenji Nishihara, Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping, J. Differential Equations 131 (1996), no. 2, 171–188. MR 1419010, DOI https://doi.org/10.1006/jdeq.1996.0159
Takaaki Nishida, Nonlinear hyperbolic equations and related topics in fluid dynamics, Département de Mathématique, Université de Paris-Sud, Orsay, 1978. Publications Mathématiques d’Orsay, No. 78-02. MR 0481578
Songmu Zheng, Nonlinear parabolic equations and hyperbolic-parabolic coupled systems, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 76, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1995. MR 1375458