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Transactions of the American Mathematical Society

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Initial-boundary value problems for hyperbolic systems in regions with corners. II

Author: Stanley Osher
Journal: Trans. Amer. Math. Soc. 198 (1974), 155-175
MSC: Primary 35L50
MathSciNet review: 0352715
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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.

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  • [1] 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.
  • [2] V. A. Kondrat'ev, Boundary value problems for elliptic equations in conical regions, Dokl. Akad. Nauk SSSR 153 (1964), 27-29 = Soviet Math. Dokl. 4 (1964), 1600-1602. MR 28 #1383. MR 0158157 (28:1383)
  • [3] H. -O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure. Appl. Math. 23 (1970), 277-298. MR 0437941 (55:10862)
  • [4] I. A. K. Kupka and S. Osher, On the wave equation in a multidimensional corner, Comm. Pure. Appl. Math. 24 (1971), 381-393. MR 0412616 (54:738)
  • [5] S. Osher, Initial-boundary value problems for hyperbolic systems in regions with corners. I, Trans. Amer. Math. Soc. 176 (1973), 141-165. MR 0320539 (47:9076)
  • [6] -, A symmetrizer for certain hyperbolic mixed problems with singular coefficients, Indiana J. Math. 22 (1973), 667-671.
  • [7] -, An ill posed problem for a hyperbolic equation near a corner, Bull. Amer. Math. Soc. (to appear). MR 0350211 (50:2704)
  • [8] -, On a generalized reflection principle and a transmission problem for a hyperbolic equation, Indiana J. Math. 79 (1973), 1043-1044.
  • [9] J. Ralston, Note on a paper of Kreiss, Comm. Pure. Appl. Math. 24 (1971), 759-762. MR 0606239 (58:29326)
  • [10] R. Sakomoto, Mixed problem for hyperbolic equations. I, II, J. Math. Kyoto Univ. 10 (1970), 349-373, 403-417. MR 44 #632a,b. MR 0283400 (44:632a)
  • [11] L. Sarason, On weak and strong solutions of boundary value problems, Comm. Pure Appl. Math. 15 (1962), 237-288. MR 27 #460. MR 0150462 (27:460)

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Keywords: Hyperbolic equations, initial boundary conditions, energy estimates, existence and uniqueness
Article copyright: © Copyright 1974 American Mathematical Society

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