Hyperbolic Systems with Analytic Coefficients [electronic resource] :Well-posedness of the Cauchy Problem /Material type: BookSeries: Lecture Notes in Mathematics: 2097Publisher: Cham : Springer International Publishing : 2014.Description: VIII, 237 p. online resource.ISBN: 9783319022734.Subject(s): Mathematics | Differential equations, partial | Mathematical physics | Mathematics | Partial Differential Equations | Mathematical Methods in PhysicsDDC classification: 515.353 Online resources: Click here to access online
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Introduction -- Necessary conditions for strong hyperbolicity -- Two by two systems with two independent variables -- Systems with nondegenerate characteristics -- Index.
This monograph focuses on the well-posedness of the Cauchy problem for linear hyperbolic systems with matrix coefficients. Mainly two questions are discussed: (A) Under which conditions on lower order terms is the Cauchy problem well posed? (B) When is the Cauchy problem well posed for any lower order term? For first order two by two systems with two independent variables with real analytic coefficients, we present complete answers for both (A) and (B). For first order systems with real analytic coefficients we prove general necessary conditions for question (B) in terms of minors of the principal symbols. With regard to sufficient conditions for (B), we introduce hyperbolic systems with nondegenerate characteristics, which contains strictly hyperbolic systems, and prove that the Cauchy problem for hyperbolic systems with nondegenerate characteristics is well posed for any lower order term. We also prove that any hyperbolic system which is close to a hyperbolic system with a nondegenerate characteristic of multiple order has a nondegenerate characteristic of the same order nearby.