Asymptotic limit of initial boundary value problems for conservation laws with relaxational extensions
1998; Wiley; Volume: 51; Issue: 5 Linguagem: Inglês
10.1002/(sici)1097-0312(199805)51
ISSN1097-0312
Autores Tópico(s)Cosmology and Gravitation Theories
ResumoCommunications on Pure and Applied MathematicsVolume 51, Issue 5 p. 505-535 Asymptotic limit of initial boundary value problems for conservation laws with relaxational extensions Wei-Cheng Wang, Corresponding Author Wei-Cheng Wang Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720Search for more papers by this authorZhouping Xin, Zhouping Xin Courant Institute, 251 Mercer Street, New York, NY 10012Search for more papers by this author Wei-Cheng Wang, Corresponding Author Wei-Cheng Wang Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720Search for more papers by this authorZhouping Xin, Zhouping Xin Courant Institute, 251 Mercer Street, New York, NY 10012Search for more papers by this author First published: 06 December 1998 https://doi.org/10.1002/(SICI)1097-0312(199805)51:5 3.0.CO;2-CCitations: 38AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onEmailFacebookTwitterLinkedInRedditWechat Abstract We study the boundary layer effect in the small relaxation limit to the equilibrium scalar conservation laws in one space dimension for the relaxation system proposed in [6]. First, it is shown that for initial and boundary data satisfying a strict version of the subcharacteristic condition, there exists a unique global (in time) solution, (uε, vε), to the relaxation system (1.4) for each ε > 0. The spatial total variation of (uε, vε) is shown to be bounded independently of ε, and consequently, a subsequence of (uε, vε) converges to a limit (u, v) as ε → 0+. Furthermore, u(x, t) is a weak solution to the scalar conservation law (1.5) and v = f(u). Next, we prove that for data that are suitably small perturbations of a nontransonic state, the relaxation limit function satisfies the boundary-entropy condition (2.11). Finally, the weak solutions to (1.5) with the boundary-entropy condition (2.11) is shown to be unique. Consequently, the relaxation limit of solutions to (1.4) is unique, and the whole sequence converges to the unique limit. One consequence of our analysis shows that the boundary layer occurs only in the u-component in the sense that vε(0, ·) converges strongly to γ ○ v = f(γ ○ u), the trace of f(u) on the t-axis. © 1998 John Wiley & Sons, Inc. Bibliography 1 M. Abramowitz, and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1992. Google Scholar 2 Bardos, C., le Roux, A. Y., and Nédélec, J.-C., First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations 4 (9), 1979, pp. 1017–1034. 10.1080/03605307908820117 Google Scholar 3 Chen, G. Q., Levermore, C. S., and Liu, T.-P., Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. 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