On the stochastic nonlinear Schrödinger equations at critical regularities.

We consider the Cauchy problem for the defocusing stochastic nonlinear Schr\"odinger equations (SNLS) with an additive noise in the mass-critical and energy-critical settings. By adapting the probabilistic perturbation argument employed in the context of the random data Cauchy theory by the first author with B\'enyi and Pocovnicu (2015) to the current stochastic PDE setting, we present a concise argument to establish global well-posedness of the mass-critical and energy-critical SNLS.Comment: 23 pages. Appendix A added. Published in Stoch. Partial Differ. Equ. Anal. Compu

Long time dynamics for the focusing inhomogeneous fractional Schr\"odinger equation

We consider the following fractional NLS with focusing inhomogeneous power-type nonlinearity $$i\partial_t u -(-\Delta)^s u + |x|^{-b}|u|^{p-1}u=0,\quad (t,x)\in \mathbb{R}\times \mathbb{R}^N,$$ where $N\geq 2$, $1/2<s<1$, $0<b<2s$ and $1+\frac{2(2s-b)}{N}<p<1+\frac{2(2s-b)}{N-2s}$. We prove the ground state threshold of global existence and scattering versus finite time blow-up of energy solutions in the inter-critical regime with spherically symmetric initial data. The scattering is proved by the new approach of Dodson-Murphy ({Proc. Am. Math. Soc.} {145}: {4859--4867}, 2017). This method is based on Tao's scattering criteria and Morawetz estimates. One describes the threshold using some non-conserved quantities in the spirit of the recent paper by Dinh (Discr. Cont. Dyn. Syst. 40: 6441--6471, 2020). The radial assumption avoids a loss of regularity in Strichartz estimates. The challenge here is to overcome two main difficulties. The first one is the presence of the non-local fractional Laplacian operator. The second one is the presence of a singular weight in the non-linearity. The greater part of this paper is devoted to prove the scattering of global solutions in $H^s(\mathbb{R}^N)$.Comment: 37 pages, misprints corrected and remarks adde

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 $d geq 5$

Blowup, solitary waves and scattering for the fractional nonlinear Schrödinger equation

In this thesis we are concerned with the rigorous analysis for an evolution problem arising in mathematical physics: the nonlinear Schrödinger equation with power-type nonlinearity involving the fractional Laplace operator (fractional NLS). We are particularly interested in the long-time dynamics of this nonlocal equation, and study three basic problems of fundamental importance. First, we shall deduce sufficient criteria for blowup of radial solutions of the focusing problem in the mass-supercritical and mass-critical cases. The conditions are given in terms of inequalities between a combination of the (kinetic) energy and mass of the initial datum, and that of the ground state for the corresponding elliptic equation. Using a new method to deal with the nonlocality of the fractional Laplacian, a localized virial argument enables us to conclude blowup in finite and infinite time, respectively. Second, we consider a special class of nondispersive solutions of the focusing fractional NLS: the traveling solitary waves. Introducing an appropriate variational problem, we establish the existence of their stationary profiles (boosted ground states). In order to deal with the lack of compactness, we use the technique of compactness modulo translations adapted to the fractional Sobolev spaces. In the case of algebraic (even integer-order) nonlinearities, we derive symmetries of boosted ground states with respect to the boost direction, relying on symmetric decreasing rearrangements in Fourier space. Moreover, we show a non-optimal spatial decay of these profiles at infinity. Third and finally, we concentrate on the asymptotics of global solutions of the defocusing problem. To have a full range of Strichartz estimates available, we restrict to the radially symmetric case. We construct the wave operator on the radial subclass of the energy space, and show asymptotic completeness. Thus we infer that any radial solution scatters to a linear solution in infinite time. Similarly to the blowup theory, this is done in the spirit of monotonicity formulae: taking a suitable virial weight and using the favourable sign of the defocusing nonlinearity, we develop a lower bound for the Morawetz action. The resulting decay estimates permit us to build a satisfactory scattering theory in the radial case

Standing waves with prescribed L2L^2-norm to nonlinear Schr\"odinger equations with combined inhomogeneous nonlinearities

In this paper, we are concerned with solutions to the following nonlinear Schr\"odinger equation with combined inhomogeneous nonlinearities, -\Delta u + \lambda u= \mu |x|^{-b}|u|^{q-2} u + |x|^{-b}|u|^{p-2} u \quad \mbox{in} \,\, \R^N, under the $L^2$-norm constraint $$\int_{\R^N} |u|^2 \, dx=c>0,$$ where $N \geq 1$, $\mu =\pm 1$, $2<q<p<{2(N-b)}/{(N-2)^+}$, $0<b<\min\{2, N\}$ and the parameter $\lambda \in \R$ appearing as Lagrange multiplier is unknown. In the mass subcritical case, we establish the compactness of any minimizing sequence to the minimization problem given by the underlying energy functional restricted on the constraint. As a consequence of the compactness of any minimizing sequence, orbital stability of minimizers is derived. In the mass critical and supercritical cases, we investigate the existence, radial symmetry and orbital instability of solutions. Meanwhile, we consider the existence, radial symmetry and algebraical decay of ground states to the corresponding zero mass equation with defocusing perturbation. In addition, dynamical behaviors of solutions to the Cauchy problem for the associated dispersive equation are discussed.Comment: 45 page

On blow up for the energy super critical defocusing {nonlinear Schr\"odinger equations

We consider the energy supercritical {\em defocusing} nonlinear Schr\"odinger equation i\pa_tu+\Delta u-u|u|^{p-1}=0 in dimension $d\ge 5$. In a suitable range of energy supercritical parameters $(d,p)$, we prove the existence of $\mathcal C^\infty$ well localized spherically symmetric initial data such that the corresponding unique strong solution blows up in finite time. Unlike other known blow up mechanisms, the singularity formation does not occur by concentration of a soliton or through a self similar solution, which are unknown in the defocusing case, but via a {\em front mechanism}. Blow up is achieved by {\em compression} for the associated hydrodynamical flow which in turn produces a highly oscillatory singularity. The front blow up profile is chosen among the countable family of {\em $\mathcal C^\infty$} spherically symmetric self similar solutions to the compressible Euler equation whose existence and properties in a suitable range of parameters are established in the companion paper \cite{MRRSprofile}

On a fourth order nonlinear Helmholtz equation

In this paper, we study the mixed dispersion fourth order nonlinear Helmholtz equation $\Delta^2 u -\beta \Delta u + \alpha u= \Gamma|u|^{p-2} u$ in $\mathbb R^N$ for positive, bounded and $\mathbb Z^N$-periodic functions $\Gamma$. Using the dual method of Evequoz and Weth, we find solutions to this equation and establish some of their qualitative properties