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Instabilité et croissance des normes de Sobolev pour certaines EDP hamiltoniennes

Abstract : In this thesis we study global smooth solutions of certain Hamiltonian PDEs, in order to capture the possible growth of their Sobolev norms. Such a phenomenon is typical for what is sometimes called "weak turbulence" : a change in the distribution of energy between Fourier modes. We first study a nonlinear evolution equation involving a fractional Laplacian, and we prove a priori estimates on the growth of Sobolev norms. We then introduce an equation where these estimates turn out to be optimal : an integrable Szegő equation with a quadratic nonlinearity, which admits exponentially growing smooth solutions that remain bounded in the energy space. We classify the traveling wave solutions of this quadratic Szegő equation, and show that some of them are unstable. Eventually we find a hierarchy of conservation laws for this equation, which leads us into a deeper study of rational turbulent solutions.
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Submitted on : Monday, September 10, 2018 - 11:35:06 AM
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Joseph Thirouin. Instabilité et croissance des normes de Sobolev pour certaines EDP hamiltoniennes. Equations aux dérivées partielles [math.AP]. Université Paris-Saclay, 2018. Français. ⟨NNT : 2018SACLS195⟩. ⟨tel-01871043⟩

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