Random data Cauchy theory for supercritical wave equations I: local theory |
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Authors: | Nicolas Burq Nikolay Tzvetkov |
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Institution: | (1) Département de Mathématiques, Université Paris XI, 91 405 Orsay Cedex, France;(2) Institut Universitaire de France, Paris, France;(3) Département de Mathématiques, Université Lille I, 59 655 Villeneuve d’Ascq Cedex, France |
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Abstract: | We study the local existence of strong solutions for the cubic nonlinear wave equation with data in H
s
(M), s<1/2, where M is a three dimensional compact Riemannian manifold. This problem is supercritical and can be shown to be strongly ill-posed
(in the Hadamard sense). However, after a suitable randomization, we are able to construct local strong solution for a large
set of initial data in H
s
(M), where s≥1/4 in the case of a boundary less manifold and s≥8/21 in the case of a manifold with boundary.
Mathematics Subject Classification (2000) 35Q55, 35BXX, 37K05, 37L50, 81Q20 |
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Keywords: | |
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