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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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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DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0340832-0
PII: S 0002-9947(1974)0340832-0
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