Refined energy inequality with application to well-posedness for the fourth order nonlinear Schrodinger type equation on torus

We consider the time local and global well-posedness for the fourth order nonlinear Schrodinger type equation (4NLS) on the torus. The nonlinear term of (4NLS) contains the derivatives of unknown function and this prevents us to apply the classical energy method. To overcome this difficulty, we introduce the modified energy and derive an a priori estimate for the solution to (4NLS)

Nonlinear Schrödinger Equations with Rough Data

In this thesis we consider nonlinear Schrödinger equations with rough initial data. Roughness of the initial data in nonlinear Schrödinger equations can be understood as being of low regularity and as a lack of decay at infinity. Firstly we prove low regularity a priori estimates for the derivative nonlinear Schrödinger equation in Besov spaces with positive regularity index. These a priori estimates are sharp at the level of regularity but are conditional upon small mass. The proof uses the operator determinant characterization of the transmission coefficient introduced by Killip-Visan-Zhang. Secondly we show global wellposedness for the tooth problem of defocusing nonlinear Schrödinger equations, that is the Cauchy problem with initial data in the space $H^{s_1}(\mathbb{R}) + H^{s_2}(\mathbb{T})$. This result can be seen as an intermediate step between the wellposedness theory in the $L^2(\mathbb{T})$-based setting and more generic non-decaying behavior at infinity. In the case $s_1 = 1$ we obtain an at most exponentially growing energy, based on the Hamiltonian of the perturbed equation. For the cubic nonlinearity we may choose $s_2 > 3/2$ whereas for higher power nonlinearities our assumption is $s_2 > 5/2$. Finally we investigate the question of wellposedness of nonlinear Schrödinger equations with initial data in modulation spaces. Modulation spaces $M^s_{p,q}(\mathbb{R}^n)$ encode both regularity ($s$ and $q$) and decay ($p$) in their indices. By making use of multilinear interpolation we prove new local wellposedness results. The local wellposedness results we obtain are proven to be sharp with respect to the regularity index. Moreover we complement the local results by showing global wellposedness in several cases, including low regularity and very slow decay. This is done on the one hand by an extension of techniques developed by Oh-Wang to a broader range of modulation spaces, and on the other hand by applying calculations from Dodson-Soffer-Spencer to the modulation space setting

Short-time Fourier transform restriction phenomena and applications to nonlinear dispersive equations

Schippa R. Short-time Fourier transform restriction phenomena and applications to nonlinear dispersive equations. Bielefeld: Universität Bielefeld; 2019

Strichartz estimates and the nonlinear Schrödinger-type equations

Cette thèse est consacrée à l'étude des aspects linéaires et non-linéaires des équations de type Schrödinger [ i partial_t u + |nabla|^sigma u = F, quad |nabla| = sqrt {-Delta}, quad sigma in (0, infty).] Quand $sigma = 2$, il s'agit de l'équation de Schrödinger bien connue dans de nombreux contextes physiques tels que la mécanique quantique, l'optique non-linéaire, la théorie des champs quantiques et la théorie de Hartree-Fock. Quand $sigma in (0,2) backslash {1}$, c'est l'équation Schrödinger fractionnaire, qui a été découverte par Laskin (voir par exemple cite{Laskin2000} et cite{Laskin2002}) en lien avec l'extension de l'intégrale de Feynman, des chemins quantiques de type brownien à ceux de Lévy. Cette équation apparaît également dans des modèles de vagues (voir par exemple cite{IonescuPusateri} et cite{Nguyen}). Quand $sigma = 1$, c'est l'équation des demi-ondes qui apparaît dans des modèles de vagues (voir cite{IonescuPusateri}) et dans l'effondrement gravitationnel (voir cite{ElgartSchlein}, cite{FrohlichLenzmann}). Quand $sigma = 4$, c'est l'équation Schrödinger du quatrième ordre ou biharmonique introduite par Karpman cite{Karpman} et par Karpman-Shagalov cite{KarpmanShagalov} pour prendre en compte le rôle de la dispersion du quatrième ordre dans la propagation d'un faisceau laser intense dans un milieu massif avec non-linéarité de Kerr. Cette thèse est divisée en deux parties. La première partie étudie les estimations de Strichartz pour des équations de type Schrödinger sur des variétés comprenant l'espace plat euclidien, les variétés compactes sans bord et les variétés asymptotiquement euclidiennes. Ces estimations de Strichartz sont utiles pour l'étude de l'équations dispersives non-linéaire à régularité basse. La seconde partie concerne l'étude des aspects non-linéaires tels que les caractères localement puis globalement bien posés sous l'espace d'énergie, ainsi que l'explosion de solutions peu régulières pour des équations non-linéaires de type Schrödinger. Dans le Chapitre 1, nous discutons des estimations de Strichartz pour les équations de type Schrödinger avec $sigma in (0, infty)$ sur l'espace euclidien $mathbb{R}^d$. Dans le Chapitre 2, nous prouvons des estimations de Strichartz pour les équations de type Schrödinger avec $sigma in (0, infty) backslash {1}$ sur $mathhbb{R}^d$ équipé d'une métrique lisse bornée $g$. Au Chapitre 3, nous utilisons les estimations de Strichartz prouvées au Chapitre 2 pour montrer les estimations de Strichartz pour les équations de type Schrödinger avec $sigma in (0, infty) backslash {1 }$ sur les variétés compactes sans bord. Au Chapitre 4, nous montrons des estimations de Strichartz globales pour les équations de type Schrödinger avec $sigma in (0, infty) backslash {1}$ sur les variétés asymptotiquement euclidiennes sous la condition de non-capture. Dans le Chapitre 5, nous utilisons les estimations de Strichartz données au Chapitre 1 (entre autres) pour étudier le caractère localement bien posé des équations non-linéaires de type Schrödinger avec la non-linéarité de type puissance et $sigma in (0, infty)$ posées sur $mathbb{R}^d$. Dans le Chapitre 6, nous étudions le le caractère globalement bien posé de l'équation de Schrödinger non-linéaire du quatrième ordre $sigma = 4$ défocalisante et $L^2$ critique, en considérant séparément deux cas $d = 4$ et $d geq 5$ qui correspondent respectivement à la non-linéarité algébrique et non-algébrique. Dans le Chapitre 7, nous étudions l'explosion des solutions peu régulières de l'équation de Schrödinger non-linéaire du quatrième ordre focalisante $L^2$ critique. Comme au Chapitre 6, nous considérons aussi séparément deux cas $d = 4$ et $d geq 5$.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 $sigma = 2$, 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 $sigma in (0,2) backslash {1}$, 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 $sigma = 1$, 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 $sigma =4$, 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. In Chapter 1, we discuss Strichartz estimates for Schrödinger-type equations with $sigma in (0, infty)$ on the Euclidean space $R^d$. In Chapter 2, we derive Strichartz estimates for Schrödinger-type equations with $sigma in (0, infty) backslash {1}$ on $R^d$ equipped with a smooth bounded metric $g$.In Chapter 3, we make use of Strichartz estimates proved in Chapter 2 to show Strichartz estimates for Schrödinger-type equations with $sigma in (0, infty) backslash {1}$ on compact manifolds without boundary. In Chapter 4, we prove global in time Strichartz estimates for Schrödinger-type equations with $sigma in (0, infty) backslash {1}$ on asymptotically Euclidean manifolds under the non-trapping condition. In Chapter 5, we use Strichartz estimates given in Chapter 1 (among other things) to study the local well-posedness of the power-type nonlinear Schrödinger-type equations with $sigma in (0, infty)$ posed on $R^d$. In Chapter 6, we study the global well-posedness for the defocusing mass-critical nonlinear fourth-order Schrödinger equation $sigma =4$ below the energy space. We will consider separately two cases $d=4$ and $d geq 5$ which respectively correspond to the algebraic and non-algebraic nonlinearity. In Chapter 7, we study the blowup of rough solutions to the focusing mass-critical nonlinear fourth-order Schrödinger equation. As in Chapter 6, we also consider separately two cases $d=4$ and $d geq 5$

Rare Events and Exit Problems for Stochastic Equations: Theory and Numerics

This dissertation is concerned with the small-noise asymptotics of stochastic differential equations and stochastic partial differential equations. In the first part of the manuscript, we present an overview of large deviations theory in the context of stochastic differential equations, with a particular focus on describing the long time behavior of the system in the presence of point attractors. We then present results describing a novel algorithm for computing the \textit{quasi-potential}, a key quantity in large deviations theory, for two-dimensional stochastic differential equations. Our solver, the Efficient Jet Marcher, computes the quasi-potential $U(x)$ on a mesh by propagating $U$ and $\nabla U$ outward away from the attractor. By using higher-order interpolation schemes and approximations for the minimum action paths, we are able to achieve 2nd order accuracy in the mesh spacing $h$. In the second part of the manuscript, we consider two important problems in large deviations theory for stochastic partial differentiable equations. First, we consider stochastic reaction-diffusion equations posed on a bounded domain, which contain both a large diffusion term and a small noise term. We prove that in the joint small noise and large diffusion limits, the system satisfies a large deviations principle with respect to an action functional that is finite only on paths that are constant in the spatial variable. We then use this result to compute asymptotics of the first exit time of the solution from bounded domains in function spaces. Second, we consider the two-dimensional stochastic Navier-Stokes equations posed on the torus. In the simultaneous limit as the noise magnitude and noise regularization are both sent to $0$, the solutions converge to the deterministic Navier-Stokes equations. We prove that the invariant measures, which converge to a Dirac mass at $0$, also satisfy a large deviation principle with action functional given by the enstrophy