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Long-time existence and growth of Sobolev norms for solutions of semi-linear Klein-Gordon equations and linear Schrödinger equations on some manifolds

Abstract : In recent years, several authors proved long time existence results for solutions of non-linear Klein-Gordon equation on some compact manifolds, like spheres, when the initial data are smooth and small enough, and when some mass parameter avoids a subset of zero measure of the real line. One of the fundamental assumptions in these works is a separation property of the eigenvalues of the laplacian on the manifolds under consideration. The goal of the first two papers of this thesis is to examine which results may be obtained when such a separation assumption does not hold. We study two cases: a Klein-Gordon operator associated to the harmonic oscillator on the Euclidean space, and the usual Klein-Gordon operator on the torus. We get, using normal forms methods, solutions existing over longer time intervals than the ones given by the local theory. The last paper of the thesis concerns long time estimates for Sobolev norms of solutions of a linear Schrödinger equation on the torus, with time dependent potential. We prove logarithmic bounds, when the potential is in the Gevrey class, extending results of Bourgain and Wang.
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Submitted on : Wednesday, February 16, 2011 - 3:19:10 PM
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Qidi Zhang. Long-time existence and growth of Sobolev norms for solutions of semi-linear Klein-Gordon equations and linear Schrödinger equations on some manifolds. Mathematics [math]. Université Paris-Nord - Paris XIII, 2010. English. ⟨tel-00566524⟩

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