Initial-boundary value problems for hyperbolic systems in regions with corners. II
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- by Stanley Osher PDF
- Trans. Amer. Math. Soc. 198 (1974), 155-175 Request permission
Abstract:
In the previous paper in this series we obtained conditions equivalent to the validity of certain energy estimates for a general class of hyperbolic systems in regions with corners. In this paper we examine closely the phenomena which occur near the corners if these conditions are violated. These phenomena include: the development of strong singularities (lack of existence), travelling waves which pass unnoticed through the corner (lack of uniqueness), existence and uniqueness if and only if additional conditions are imposed at the corner, and weak solutions which are not strong solutions. We also systematically analyze the conditions for certain important problems. We discuss the physical and computational significance of these results.References
-
T. Elvius and A. Sundström, Computationally efficient schemes and boundary conditions for afine mesh barotropic model based on shallow water equations, Tellus, 25 (1973), 132-156.
- V. A. Kondrat′ev, Boundary-value problems for elliptic equations in conical regions, Dokl. Akad. Nauk SSSR 153 (1963), 27–29 (Russian). MR 0158157
- Heinz-Otto Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math. 23 (1970), 277–298. MR 437941, DOI 10.1002/cpa.3160230304
- I. A. K. Kupka and S. J. Osher, On the wave equation in a multi-dimensional corner, Comm. Pure Appl. Math. 24 (1971), 381–393. MR 412616, DOI 10.1002/cpa.3160240304
- Stanley Osher, Initial-boundary value problems for hyperbolic systems in regions with corners. I, Trans. Amer. Math. Soc. 176 (1973), 141–164. MR 320539, DOI 10.1090/S0002-9947-1973-0320539-5 —, A symmetrizer for certain hyperbolic mixed problems with singular coefficients, Indiana J. Math. 22 (1973), 667-671.
- Stanley Osher, An ill posed problem for a hyperbolic equation near a corner, Bull. Amer. Math. Soc. 79 (1973), 1043–1044. MR 350211, DOI 10.1090/S0002-9904-1973-13324-5 —, On a generalized reflection principle and a transmission problem for a hyperbolic equation, Indiana J. Math. 79 (1973), 1043-1044.
- James V. Ralston, Note on a paper of Kreiss, Comm. Pure Appl. Math. 24 (1971), no. 6, 759–762. MR 606239, DOI 10.1002/cpa.3160240603
- Reiko Sakamoto, Mixed problems for hyperbolic equations. I. Energy inequalities, J. Math. Kyoto Univ. 10 (1970), 349–373. MR 283400, DOI 10.1215/kjm/1250523767
- Leonard Sarason, On weak and strong solutions of boundary value problems, Comm. Pure Appl. Math. 15 (1962), 237–288. MR 150462, DOI 10.1002/cpa.3160150301
Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 198 (1974), 155-175
- MSC: Primary 35L50
- DOI: https://doi.org/10.1090/S0002-9947-1974-0352715-0
- MathSciNet review: 0352715