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Theses

Invariant measures for Hamiltonian PDE

Abstract : In this thesis, we are concerned with the qualitative study of solutions of Hamiltonian partial differential equations by the way of the invariant measures theory. Indeed, existence of such a measure provides some informations concerning the large time dynamics of the PDE in question. In this thesis we treat two "extremal" situations. In the first part, we consider equations with infinitely many conservation laws, and in the second, we study equations for which we know only one non-trivial conservation law. We study the first equations by considering the Benjamin-Ono equation. The latter is a model describing internal waves in a fluide of great depth. We are concerned with the dynamics of that equation on the space C^∞(T) by constructing for it an invariant measure on that space. Accordingly, an almost sure (w.r.t. this measure) recurrence property is established for infinitely smooth solutions of that equation. Then, we prove qualitative properties for the constructed measure by showing that there are infinitely many independent observables whose distributions via this measure are absolutely continuous w.r.t. the Lebesgue measure on R. Moreover, we establish that the measure is of at least 2-dimensional nature. In this work, we used the Fluctuation-Dissipation-Limit (FDL) approach introduced by Kuksin and Shirikyan. Notice that an almost sure recurrence property for the Benjamin-Ono equation was established on Sobolev spaces by Deng, Tzvetkov and Visciglia.In the second part of the thesis, we consider the cubic Klein-Gordon equation, which is an example of Hamiltonian PDEs for which we know only one conservation law. This equation models the evolution of a massive relativistic particle. Here, we consider both the case of the tri-dimensional periodic solutions and those defined on a bounded domain of R³. In both settings, we construct an invariant measure concentrated on the Sobolev space H²xH¹, again with use of the FDL approach. Another aspect of this work is to extend the FDL approach to the context of PDEs having only one conservation law; indeed, in previous works, this approach required two conservation laws. Qualitative properties for the measure and almost sure (w.r.t. this measure) recurrence for H²-solutions are proven. Notice that previous works by Burq-Tzvetkov, de Suzzoni, Bourgain-Bulut and Xu have treated the invariant Gibbs measure problem in the radial symmetry context for waves equations.
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Mouhamadou Sy. Invariant measures for Hamiltonian PDE. Analysis of PDEs [math.AP]. Université de Cergy Pontoise, 2017. English. ⟨NNT : 2017CERG0949⟩. ⟨tel-02115899⟩

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