This thesis contributes towards the maximal-in-time well-posedness theory of three nonlinear dispersive partial differential equations (PDEs). We are interested in questions that extend beyond the usual well-posedness theory: what is the ultimate fate of solutions? How does Hamiltonian structure influence PDE dynamics? How does randomness, within the PDE or the initial data, interact with well-posedness of the Cauchy problem? The first topic of this thesis is the analysis of blow-up solutions to the elliptic-elliptic Davey-Stewartson system, which appears in the description of surface water waves. We prove a mass concentration property for H^1-solutions, analogous to the one known for the L^2-critical nonlinear Schrodinger equation. We also prove a mass concentration result for L^2-solutions. The second topic of this thesis is the invariance of the Gibbs measure for the (gauge transformed) periodic quartic KdV equation. The Gibbs measure is a probability measure supported on H^s for s<1/2, and local solutions to the quartic KdV cannot be obtained below H^{1/2} by using the standard fixed point method. We exhibit nonlinear smoothing when the initial data are randomized, and establish almost sure local well-posedness for the (gauge transformed) quartic KdV below H^{1/2}. Then, using the invariance of the Gibbs measure for the finite-dimensional system of ODEs given by projection onto the first N>0 modes of the trigonometric basis, we extend the local solutions of the (gauge transformed) quartic KdV to global solutions, and prove the invariance of the Gibbs measure under the flow. Inverting the gauge, we establish almost sure global well-posedness of the (ungauged) periodic quartic KdV below H^{1/2}. The third topic of this thesis is well-posedness of the stochastic KdV-Burgers equation. This equation is studied as a toy model for the stochastic Burgers equation, which appears in the description of a randomly growing interface. We are interested in rigorously proving the invariance of white noise for the stochastic KdV-Burgers equation. This thesis provides a result in this direction: after smoothing the additive noise (by a fractional derivative), we establish (almost sure) local well-posedness of the stochastic KdV-Burgers equation with white noise as initial data. We also prove a global well-posedness result under an additional smoothing of the noise.

This thesis contributes towards the maximal-in-time well-posedness theory of three nonlinear dispersive partial differential equations (PDEs). We are interested in questions that extend beyond the usual well-posedness theory: what is the ultimate fate of solutions? How does Hamiltonian structure influence PDE dynamics? How does randomness, within the PDE or the initial data, interact with well-posedness of the Cauchy problem? The first topic of this thesis is the analysis of blow-up solutions to the elliptic-elliptic Davey-Stewartson system, which appears in the description of surface water waves. We prove a mass concentration property for H^1-solutions, analogous to the one known for the L^2-critical nonlinear Schrodinger equation. We also prove a mass concentration result for L^2-solutions. The second topic of this thesis is the invariance of the Gibbs measure for the (gauge transformed) periodic quartic KdV equation. The Gibbs measure is a probability measure supported on H^s for s<1/2, and local solutions to the quartic KdV cannot be obtained below H^{1/2} by using the standard fixed point method. We exhibit nonlinear smoothing when the initial data are randomized, and establish almost sure local well-posedness for the (gauge transformed) quartic KdV below H^{1/2}. Then, using the invariance of the Gibbs measure for the finite-dimensional system of ODEs given by projection onto the first N>0 modes of the trigonometric basis, we extend the local solutions of the (gauge transformed) quartic KdV to global solutions, and prove the invariance of the Gibbs measure under the flow. Inverting the gauge, we establish almost sure global well-posedness of the (ungauged) periodic quartic KdV below H^{1/2}. The third topic of this thesis is well-posedness of the stochastic KdV-Burgers equation. This equation is studied as a toy model for the stochastic Burgers equation, which appears in the description of a randomly growing interface. We are interested in rigorously proving the invariance of white noise for the stochastic KdV-Burgers equation. This thesis provides a result in this direction: after smoothing the additive noise (by a fractional derivative), we establish (almost sure) local well-posedness of the stochastic KdV-Burgers equation with white noise as initial data. We also prove a global well-posedness result under an additional smoothing of the noise.

Ce travail cherche a comprendre comment l'ajout d'une géométrie non euclidienne dans un problème de Schrödinger non linéaire influe sur l'existence et l'unicité des solutions explosives de masse critique. On s'inspire pour beaucoup des travaux de Merle et Raphaël sur la méthode de modulation des paramètres d'invariance géométrique pour une EDP qui possède de bonnes lois de conservations. On s'appuie ici plus particulièrement sur un article de Raphaël et Szeftel qui prouve l'existence et l'unicité d'une solution de masse critique en dimension 2 pour l'équation de Schrödinger non linéaire avec potentiel d'inhomogénéité devant la non-linéarité, et qui explose par ailleurs au maximum de l'inhomogénéité. Dans un premier temps, il s'agit de reprendre la méthode dans son ensemble afin de l'adapter à des cas où le Laplacien n'est plus plat, et est remplacé par un opérateur de type Laplace-Beltrami ou Laplacien généralisé. Ayant mis en avant le rôle de la courbure au point d'explosion, en termes de conditions sur les dérivées de termes métriques, on reprend dans un deuxième temps l'étude dans le cas plus général d'une variété riemannienne. Grâce à un ansatz sur la solution qui intègre maintenant la transformation induite par la métrique, on est capable d'énoncer un résultat d'existence et d'unicité en termes de conditions géométriques sur la variété elle même. Par soucis de simplicité, on se limite néanmoins au rôle local de la métrique, en la supposant globalement définie dans une certaine carte, et asymptotiquement équivalente a la métrique euclidienne. / The present work aims at investigating the effects of a non-euclidean geometry on existence and uniqueness results for critical blow up NLS solutions. We will use many ideas from the works of Merle and Raphaël, particularly ideas from modulation theory which describes a solution in terms of geometric invariants parameters. We will rely more specically on a paper from Raphaël and Szeftel for existence and uniqueness of a critical mass blow up solution in dimension two tothe nonlinear Schrödinger equation with inhomogeneous potential acting on the nonlinearity, and which blows up where the inhomogeneity reaches its maximum. At first, we consider a generalized Laplacian operator and deploy the classical ansatz method to point out difficulties inherited from the non-flat metric terms, and in particular the key role played by the curvature at the blow-up point. In a second part, we reproduce the method when modifying the geometrical ansatz on which the parametrix is constructed, and investigate more precisely what is needed for existence and then uniqueness when dealing with a Laplace-Beltrami operator associated to a riemannian manifold. For simplicity, we shall only consider the role of g locally around the blow up point we are constructing, by assuming g is globally defined in some map, and asymptotically equals the usual euclidean metric.