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Transactions of the American Mathematical Society

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Differentiability of solutions to hyperbolic initial-boundary value problems

Authors: Jeffrey B. Rauch and Frank J. Massey
Journal: Trans. Amer. Math. Soc. 189 (1974), 303-318
MSC: Primary 35L50
MathSciNet review: 0340832
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Abstract: This paper establishes conditions for the differentiability of solutions to mixed problems for first order hyperbolic systems of the form $ (\partial /\partial t - \sum {A_j}\partial /\partial {x_j} - B)u = F$ on $ [0,T] \times \Omega ,Mu = g$ on $ [0,T] \times \partial \Omega ,u(0,x) = f(x),x \in \Omega $. Assuming that $ {\mathcal{L}_2}$ a priori inequalities are known for this equation, it is shown that if $ F \in {H^s}([0,T] \times \Omega ),g \in {H^{s + 1/2}}([0,T] \times \partial \Omega ),f \in {H^s}(\Omega )$ satisfy the natural compatibility conditions associated with this equation, then the solution is of class $ {C^p}$ from [0, T] to $ {H^{s - p}}(\Omega ),0 \leq p \leq s$. These results are applied to mixed problems with distribution initial data and to quasi-linear mixed problems.

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  • [1] K. O. Friedrichs, Symmetric positive linear differential equations, Comm. Pure Appl. Math. 11 (1958), 333-418. MR 20 #7147. MR 0100718 (20:7147)
  • [2] L. Hörmander, Linear partial differential operators, Die Grundlehren der math. Wissenschaften, Band 116, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR 28 #4221.
  • [3] M. Ikawa, Mixed problem for a hyperbolic system of the first order, Publ. Res. Inst. Math. Sci. Kyoto Univ. 7 (1971/72), 427-454. MR 0330785 (48:9122)
  • [4] T. Kato, Linear evolution equations of ``hyperbolic'' type, J. Fac. Sci. Univ. Tokyo Sect. I 17 (1970), 241-258. MR 43 #5347. MR 0279626 (43:5347)
  • [5] H.-O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math. 23 (1970), 277-298. MR 0437941 (55:10862)
  • [6] P. D. Lax and R. S. Phillips, Local boundary conditions for dissipative symmetric linear differential operators, Comm. Pure Appl. Math. 13 (1960), 427-455. MR 22 #9718. MR 0118949 (22:9718)
  • [7] F. J. Massey III, Abstract evolution equations and the mixed problem for symmetric hyperbolic systems, Trans. Amer. Math. Soc. 168 (1972), 165-188. MR 0298231 (45:7283)
  • [8] J. B. Rauch, $ {\mathcal{L}_2}$ is a continuable initial condition for Kreiss' mixed problems, Comm. Pure Appl. Math. 25 (1972), 265-285. MR 0298232 (45:7284)
  • [9] J. B. Rauch and M. E. Taylor, Penetrations into shadow regions and unique continuation properties in hyperbolic mixed problems, Indiana Univ. Math. J. 22 (1972), 277-285. MR 0303098 (46:2240)
  • [10] R. Sakamoto, Mixed problems for hyperbolic equations. I, J. Math. Kyoto Univ. 10 (1970), 349-373; II, Existence theorems with zero initial datas and energy inequalities with initial datas, ibid., 403-417. MR 44 #632a,b.
  • [11] D. S. Tartakoff, Regularity of solutions to boundary value problems for first order systems, Indiana Univ. Math. J. 21 (1972), 1113-1129. MR 0440182 (55:13061)

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Keywords: A priori inequalities, compatibility conditions, differentiability of solutions, distribution solutions, generalized solutions, hyperbolic systems, mixed problems, quasilinear equations
Article copyright: © Copyright 1974 American Mathematical Society

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