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Strichartz estimates and the nonlinear Schrödinger-type equations

Abstract : This dissertation is devoted to the study of linear and nonlinear aspects of the Schrödinger-type equations [ i partial_t u + |nabla|^sigma u = F, quad |nabla| = sqrt {-Delta}, quad sigma in (0, infty).] When dollar sigma = 2 dollar, it is the well-known Schrödinger equation arising in many physical contexts such as quantum mechanics, nonlinear optics, quantum field theory and Hartree-Fock theory. When dollar sigma in (0,2) backslash {1} dollar, it is the fractional Schrödinger equation, which was discovered by Laskin (see e.g. cite{Laskin2000} and cite{Laskin2002}) owing to the extension of the Feynman path integral, from the Brownian-like to Lévy-like quantum mechanical paths. This equation also appears in the water waves model (see e.g. cite{IonescuPusateri} and cite{Nguyen}). When dollar sigma = 1 dollar, it is the half-wave equation which arises in water waves model (see cite{IonescuPusateri}) and in gravitational collapse (see cite{ElgartSchlein}, cite{FrohlichLenzmann}). When dollar sigma =4 dollar, it is the fourth-order or biharmonic Schrödinger equation introduced by Karpman cite {Karpman} and by Karpman-Shagalov cite{KarpmanShagalov} taking into account the role of small fourth-order dispersion term in the propagation of intense laser beam in a bulk medium with Kerr nonlinearity. This thesis is divided into two parts. The first part studies Strichartz estimates for Schrödinger-type equations on manifolds including the flat Euclidean space, compact manifolds without boundary and asymptotically Euclidean manifolds. These Strichartz estimates are known to be useful in the study of nonlinear dispersive equation at low regularity. The second part concerns the study of nonlinear aspects such as local well-posedness, global well-posedness below the energy space and blowup of rough solutions for nonlinear Schrödinger-type equations.[...]
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van Duong Dinh. Strichartz estimates and the nonlinear Schrödinger-type equations. Analysis of PDEs [math.AP]. Université Paul Sabatier - Toulouse III, 2018. English. ⟨NNT : 2018TOU30247⟩. ⟨tel-02379637⟩

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