Evolution systems for paraxial wave equations of Schrödinger-type with non-smooth coefficients |
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Authors: | Maarten de Hoop Michael Oberguggenberger |
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Affiliation: | a Center for Computational and Applied Mathematics, Purdue University, West Lafayette, USA b Fakultät für Mathematik, Universität Wien, Austria c Institut für Grundlagen der Bauingenieurwissenschaften, Universität Innsbruck, Austria |
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Abstract: | We prove existence of strongly continuous evolution systems in L2 for Schrödinger-type equations with non-Lipschitz coefficients in the principal part. The underlying operator structure is motivated from models of paraxial approximations of wave propagation in geophysics. Thus, the evolution direction is a spatial coordinate (depth) with additional pseudodifferential terms in time and low regularity in the lateral space variables. We formulate and analyze the Cauchy problem in distribution spaces with mixed regularity. The key point in the evolution system construction is an elliptic regularity result, which enables us to precisely determine the common domain of the generators. The construction of a solution with low regularity in the coefficients is the basis for an inverse analysis which allows to infer the lack of lateral regularity in the medium from measured data. |
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