We propose an approach to nonlinear evolution equations with large and decaying external potentials that addresses the question of controlling globally-in-time the nonlinear interactions of localized waves in this setting. This problem arises when studying localized perturbations around (possibly non-decaying) special solutions of evolution PDEs, and trying to control the projection onto the continuous spectrum of the nonlinear radiative interactions. One of our main tools is the Fourier transform adapted to the Schrödinger operator $H=-Δ+V$, which we employ at a nonlinear level. As a first step we analyze the spatial integral of the product of three generalized eigenfunctions of $H$, and determine the precise structure of its singularities. This leads to study bilinear operators with certain singular kernels, for which we derive product estimates of Coifman-Meyer type. This analysis can then be combined with multilinear harmonic analysis tools and the study of oscillations to obtain (distorted Fourier space analogues of) weighted estimates for dispersive and wave equations. As a first application we consider the nonlinear Schrödinger equation in $3$d in the presence of large decaying potential with no bound states, and with a $u^2$ non-linearity. The main difficulty is that a quadratic nonlinearity in $3$d is critical with respect to the Strauss exponent; moreover, this nonlinearity has non-trivial fully coherent interactions even when $V=0$. We prove quantitative global-in-time bounds and scattering for small solutions.
In this paper, we establish bilinear and gradient bilinear Strichartz estimates for Schrödinger operators in 2 dimensional compact manifolds with boundary. Using these estimates, we can infer the local well-posedness of cubic nonlinear Schrödinger equation in $H^s$ for every $s>{2/3}$ on such manifolds.
We prove polynomial upper bounds on the growth of solutions to 2d cubic NLS where the Laplacian is confined by the harmonic potential. Due to better bilinear effects our bounds improve on those available for the $2d$ cubic NLS in the periodic setting: our growth rate for a Sobolev norm of order s=2k, $k\in \mathbb{N}$, is $t^{2(s-1)/3+\varepsilon}$. In the appendix we provide an direct proof, based on integration by parts, of bilinear estimates associated with the harmonic oscillator.
In this article, we study long time dynamics for defocusing cubic NLS on three dimensional product space. First, we apply the decoupling method in Bourgain-Demeter \cite{BD} to establish a bilinear Strichartz estimate. Moreover, we prove global well-posedness for defocusing, cubic NLS on three dimensional product space with rough initial data ($H^s$, $s>\frac{5}{6}$) based on I-method and the bilinear estimate. At last, we discuss the growth of higher Sobolev norm problem which is tightly linked to the weak turbulence phenomenon.
We study the cubic non linear Schrödinger equation (NLS) on compact surfaces. On the sphere $\mathbb{S}^2$ and more generally on Zoll surfaces, we prove that, for $s>1/4$, NLS is uniformly well-posed in $H^s$, which is sharp on the sphere. The main ingredient in our proof is a sharp bilinear estimate for Laplace spectral projectors on compact surfaces. On étudie l'équation de Schrödinger non linéaire (NLS) sur une surface compacte.Sur la sphère $\mathbb{S}^2$ et plus généralement sur toute surface de Zoll, on démontre que pour $s>1/4$, NLS est uniformément bien posée dans $H^s$, ce qui est opt…
We prove a bilinear Strichartz type estimate for irrational tori via a decoupling type argument, \cite{bourgain2014proof}, recovering and generalizing the result of \cite{de2006global}. As a corollary, we derive a global well-posedness result for the cubic defocusing NLS on two dimensional irrational tori with data of infinite energy.
We establish local and global well-posedness for the initial value problem associated to the one-dimensional Schrodinger-Debye (SD) system for data in the Sobolev spaces with low regularity. To obtain local results we prove two new sharp bilinear estimates for the coupling terms of this system in the continuous and periodic cases. Concerning global results, in the continuous case, the system is shown to be globally well-posed in $H^s\times H^s, -3/14< s< 0$. For initial data in Sobolev spaces with high regularity ($H^s\times H^s, s>5/2$), Bidégaray \cite{Bidegaray} proved that there are one-pa…
We prove bilinear estimates for the Schrödinger equation on 3D domains, with Dirichlet boundary conditions. On non-trapping domains, they match the $\mathbb{R}^3$ case, while on bounded domains they match the generic boundary less manifold case. As an application, we obtain global well-posedness for the defocusing cubic NLS for data in $H^s_0(Ω)$, $1<s\leq 3$, with $Ω$ any bounded domain with smooth boundary.
We consider the quadractic NLS posed on a bidimensional compact Riemannian manifold $(M, g)$ with $ \partial M \neq \emptyset$. Using bilinear and gradient bilinear Strichartz estimates for Schrödinger operators in two-dimensional compact manifolds proved by J. Jiang in \cite{JIANG} we deduce a new evolution bilinear estimates. Consequently, using Bourgain's spaces, we obtain a local well-posedness result for given data $u_0\in H^s(M)$ whenever $s> \frac{2}{3}$ in such manifolds.
We consider the nonlinear Schrödinger equation (NLS) on a torus of arbitrary dimension. The equation is studied in presence of an external potential field whose time-dependent amplitude is taken as control. Assuming that the potential satisfies a saturation property, we show that the NLS equation is approximately controllable between any pair of eigenstates in arbitrarily small time. The proof is obtained by developing a multiplicative version of a geometric control approach introduced by Agrachev and Sarychev. We give an application of this result to the study of the large time behavior of th…
We consider the energy critical nonlinear Schrodinger equation on generic irrational tori. Using the long-time Strichartz estimates proved in [8], we establish polynomial upper bounds for higher Sobolev norms for solutions with small energy.
We investigate the initial value problem for a defocusing nonlinear Schrödingerequation with exponential nonlinearity. We identify subcritical, critical and supercritical regimes in the energy space. We establish global well-posedness in the subcritical and critical regimes. Well-posedness fails to hold in the supercritical case.
In this paper we generalize a weak sequential result of \cite{fan20182} to any non-scattering solutions in one dimension. No symmetry assumptions are required for the initial data.
We prove scattering below the ground state threshold for an energy-critical inhomogeneous nonlinear Schrödinger equation in three space dimensions. In particular, we extend results of Cho, Hong, and Lee from the radial to the non-radial setting.
We consider the defocusing energy-critical NLS in the exterior of the unit ball in three dimensions. For the initial value problem with Dirichlet boundary condition we prove global well-posedness and scattering with large radial initial data in the Sobolev space $\dot H_0^1$. We also point out that the same strategy can be used to treat the energy-supercritical NLS in the exterior of balls with Dirichlet boundary condition and radial $\dot H_0^1$ initial data.
In this paper, we prove a refined version of a compactness lemma and we use it to establish mass-concentration for the focusing nonlinear Schrödinger equation with an inverse-square potential.
We consider two classes of defocusing energy-supercritical nonlinear Schrödinger equations in dimensions $d\geq 5$. We prove that if the solution $u$ is apriorily bounded in the critical Sobolev space, that is, $u\in L_t^\infty \dot H^{s_c}_x$, then $u$ is global and scatters.
We consider a class of power-type nonlinear Schrödinger equations for which the power of the nonlinearity lies between the mass- and energy-critical exponents. Following the concentration-compactness approach, we prove that if a solution $u$ is bounded in the critical Sobolev space throughout its lifespan, that is, $u\in L_t^\infty \dot{H}_x^{s_c}$, then $u$ is global and scatters.
We consider the mass critical fractional (NLS). We show the existence of travelling waves for all mass below the ground state mass, and give a complete description of the associated profiles in the small mass limit. We therefore recover a situation similar to the one discovered in [Gerard P.; Lenzmann E.; Pocovnicu O.; Raphaël, P., A two soliton with transient turbulent regime for the one dimensional cubic half wave, submitted] for the critical case s = 1, but with a completely different asymptotic profile when the mass vanishes.
For the focusing mass-critical NLS $iu_t+Δu=-|u|^{\frac 4d}u$, it is conjectured that the only global non-scattering solution with ground state mass must be a solitary wave up to symmetries of the equation. In this paper, we settle the conjecture for $H^1_x$ initial data in dimensions $d=2,3$ with spherical symmetry and $d\ge 4$ with certain splitting-spherically symmetric initial data.
We consider the defocusing energy-critical nonlinear Schrödinger equation with inverse-square potential $iu_t = -Δu + a|x|^{-2}u + |u|^4u$ in three space dimensions. We prove global well-posedness and scattering for $a>-\frac14 +\frac1{25}$. We also carry out the variational analysis needed to treat the focusing case.
The energy-critical defocusing nonlinear Schrödinger equation on 3-dimensional rectangular tori is considered. We prove that the global well-posedness result for the standard torus of Ionescu and Pausader extends to this class of manifolds, namely, for any initial data in $H^1$ the solution exists globally in time.
The paper is devoted to the study of a singularly perturbed fractional Schrödinger equations involving critical frequency and critical growth in the presence of a magnetic field. By using variational methods, we obtain the existence of mountain pass solutions $u_{\varepsilon}$ which tend to the trivial solution as $\varepsilon\rightarrow0$. Moreover, we get infinitely many solutions and sign-changing solutions for the problem in absence of magnetic effects under some extra assumptions.
In this short note we present a new proof of the global well-posedness and scattering result for the defocusing energy-critical NLS in four space dimensions obtained previously by Ryckman and Visan. The argument is inspired by the recent work of Dodson on the mass-critical NLS.
In this paper we prove global well-posedness in $H^1$ for the energy-critical defocusing initial-value problem \begin{equation*} (i\partial_t+Δ_x)u=u|u|^2,\qquad u(0)=ϕ, \end{equation*} in the semiperiodic setting $x\in\R\times\T^3$.
Threshold dynamics for the 4$d$ mass-energy double critical NLS
Universidad del Valle（巴拿马谷大学）；The Graduate School of China Academy of Engineering Physics（中国工程物理研究院研究生院）；University of Oregon（俄勒冈大学）；Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）
The paper aims at giving a first insight on the existence/nonexistence of ground states for the $L^2$-critical NLS equation on metric graphs with localized nonlinearity. In particular, we focus on the tadpole graph, which, albeit being a toy model, allows to point out some specific features of the problem, whose understanding will be useful for future investigations.
We study Gibbs measures invariant under the flow of the NLS on the unit disc of $\R^2$. For that purpose, we construct the dynamics on a phase space of limited Sobolev regularity and a wighted Wiener measure invariant by the NLS flow. The density of the measure is integrable with respect to the Wiener measure for sub cubic nonlinear interactions. The existence of the dynamics is obtained in Bourgain spaces of low regularity. The key ingredient are bilinear Strichartz estimates for the free evolution. The bilinear effect in our analysis results from simple properties of the Bessel functions and…
We consider the $L^2$ critical inhomogeneous nonlinear Schrödinger (INLS) equation in $\mathbb{R}^N$ $$ i \partial_t u +Δu +|x|^{-b} |u|^{\frac{4-2b}{N}}u = 0, $$ where $N\geq 1$ and $0<b<2$. We prove that if $u_0\in H^1(\mathbb{R}^N)$ satisfies $E[u_0]<0$, then the corresponding solution blows-up in finite time. This is in sharp contrast to the classical $L^2$ critical NLS equation where this type of result is only known in the radial case for $N\geq 2$.
We establish the classification of minimal mass blow-up solutions of the $L^2$ critical inhomogeneous nonlinear Schrödinger equation \[ i\partial_t u + Δu + |x|^{-b}|u|^{\frac{4-2b}{N}}u = 0, \] thereby extending the celebrated result of Merle from the classic case $b=0$ to the case $0<b<\min\{2,N\}$, in any dimension $N\ge1$.
In this paper we generalize a weak sequential result of \cite{fan20182} to a non-scattering solutions in dimension $d \geq 2$. No symmetry assumptions are required for the initial data. We build on a previous result of \cite{dodson20202} for one dimension.
We study the focusing energy-critical NLS \begin{align}\label{nls_abstract} i\partial_t u+Δ_{x,y} u=-|u|^{\frac{4}{d-1}} u\tag{NLS} \end{align} on the waveguide manifold $\mathbb{R}_x^d\times\mathbb{T}_y$ with $d\geq 2$. We reveal the somewhat counterintuitive phenomenon that despite the energy-criticality of the nonlinear potential, the long time dynamics of \eqref{nls_abstract} are purely determined by the semivirial-vanishing geometry which possesses an energy-subcritical characteristic. As a starting point, we consider a minimization problem $m_c$ defined on the semivirial-vanishing manifo…
We construct a new class of multi-solitary wave solutions for the mass critical two dimensional nonlinear Schrodinger equation (NLS). Given any integer K>1, there exists a global (for positive time) solution of (NLS) that decomposes asymptotically into a sum of solitary waves centered at the vertices of a K-sided regular polygon and concentrating at a logarithmic rate in large time. This solution blows up in infinite time with logarithmic rate. Using the pseudo-conformal transform, this yields the first example of solution blowing up in finite time with a rate strictly above the pseudo-conform…
We consider the NLS with variable coefficients in dimension $n\ge3$ \begin{equation*} i \partial_t u - Lu +f(u)=0, \qquad Lv=\nabla^{b}\cdot(a(x)\nabla^{b}v)-c(x)v, \qquad \nabla^{b}=\nabla+ib(x), \end{equation*} on $\mathbb{R}^{n}$ or more generally on an exterior domain with Dirichlet boundary conditions, for a gauge invariant, defocusing nonlinearity of power type $f(u)\simeq|u|^{γ-1}u$. We assume that $L$ is a small, long range perturbation of $Δ$, plus a potential with a large positive part. The first main result of the paper is a bilinear smoothing (interaction Morawetz) estimate for the…
We consider the minimal mass $m_0$ required for solutions to the mass-critical nonlinear Schrödinger (NLS) equation $iu_t + Δu = μ|u|^{4/d} u$ to blow up. If $m_0$ is finite, we show that there exists a minimal-mass solution blowing up (in the sense of an infinite spacetime norm) in both time directions, whose orbit in $L^2_x(\R^d)$ is compact after quotienting out by the symmetries of the equation. A similar result is obtained for spherically symmetric solutions. Similar results were previously obtained by Keraani, \cite{keraani}, in dimensions 1, 2 and Begout and Vargas, \cite{begout}, in di…
We prove almost sure global existence and scattering for the energy-critical nonlinear Schrödinger equation with randomized spherically symmetric initial data in $H^s(\mathbb{R}^4)$ with $\frac56<s<1$. We were inspired to consider this problem by the recent work of Dodson--Lührmann--Mendelson, which treated the analogous problem for the energy-critical wave equation.
We consider the defocusing $\dot{H}^{1/2}$-critical nonlinear Schrödinger equation in dimensions $d\geq 5$. In the spirit of Kenig and Merle [Trans. Amer. Math. Soc. 362 (2010), 1937--1962], we combine a concentration-compactness approach with the Lin--Strauss Morawetz inequality to prove that if a solution $u$ is bounded in $\dot{H}^{1/2}$ throughout its lifespan, then $u$ is global and scatters.
We establish global existence for the energy-critical nonlinear Schrödinger equation on $\mathbb{S}^3$. This follows similar lines to the work on $\mathbb{T}^3$ but requires new extinction results for linear solutions and bounds on the first nonlinear iterate at a Euclidean profile that are adapted to the new geometry.
We prove that if $u(t)$ is a log-log blow-up solution, of the type studied by Merle-Raphaël (2001-2005), to the $L^2$ critical focusing NLS equation $i\partial_t u +Δu + |u|^{4/d} u=0$ with initial data $u_0\in H^1(\mathbb{R}^d)$ in the cases $d=1, 2$, then $u(t)$ remains bounded in $H^1$ away from the blow-up point. This is obtained without assuming that the initial data $u_0$ has any regularity beyond $H^1(\mathbb{R}^d)$. As an application of the $d=1$ result, we construct an open subset of initial data in the radial energy space $H^1_{rad}(\mathbb{R}^3)$ with corresponding solutions that bl…
This paper is concerned with the global well-posedness and finite time blowup problem for the 3D focusing energy-critical inhomogeneous NLS. In the previous results \cite{chkl2, chkl3} the authors considered the same problems with the spatial inhomogeneity coefficient $g$ such that $g(x) \sim |x|^{-b}$ for $0 \le b < \frac43$. Here we extend the inhomogeneous index $b$ up to $\frac32$. For this purpose, we improve the local theory and develop a new profile decomposition based on weighted space.
We consider the mass-subcritical nonlinear Schrödinger equation in all space dimensions with focusing or defocusing nonlinearity. For such equations with critical regularity $s_c\in(\max\{-1,-\frac{d}{2}\},0)$, we prove that any solution satisfying $\|\, |x|^{|s_c|}e^{-itΔ} u\|_{L_t^\infty L_x^2} <\infty$ on its maximal interval of existence must be global and scatter.
In this paper, we study the critical norm conjecture for the inter-critical nonlinear Schr{ö}dinger equation with critical index $s_c$ satisfying $\frac{1}{2}<s_c<1$ when $d\geq 5$. Under the assumption of uniform boundedness of the critical norm, we prove the global well-posedness and scattering for the Cauchy problem. We follow the standard `Concentration compactness/Rigidity method' established in \cite{KenigMerle1,KenigMerle2}, and treat three scenarios for the critical element respectively. Moreover, double Duhamel method and interaction Morawetz estimate are applied to exclude the critic…
In this note we study the initial value problem in a critical space for the one dimensional Schrödinger equation with a cubic non-linearity and under some smallness conditions. In particular the initial data is given by a sequence of Dirac deltas with different amplitudes but equispaced. This choice is motivated by a related geometrical problem; the one describing the flow of curves in three dimensions moving in the direction of the binormal with a velocity that is given by the curvature.
We study the non-scattering $L^{2}$ solution $u$ to the radial mass critical nonlinear Schrödinger equation with mass just above the ground state, and show that there exists a time sequence $\{t_{n}\}_{n}$, such that $u(t_{n})$ weakly converges to the ground state $Q$ up to scaling and phase transformation. We also give some partial results on the mass concentration of the minimal mass blow up solution.
In this paper, we study the dynamics of subcritical threshold solutions for focusing energy critical NLS on $\mathbb{R}^d$ ($d\geq 5$) with nonradial data. This problem with radial assumption was studied by T. Duyckaerts and F. Merle in \cite{DM} for $d=3,4,5$ and later by D. Li and X. Zhang in \cite{LZ} for $d \geq 6$. We generalize the conclusion for the subcritical threshold solutions by removing the radial assumption for $d\geq 5$. A key step is to show exponential convergence to the ground state $W(x)$ up to symmetries if the scattering phenomenon does not occur. Remarkably, an interactio…
We consider the radial energy-critical non-linear focusing Schrödinger equation in dimension N=3,4,5. An explicit stationnary solution, W, of this equation is known. In a previous work by C. Carlos and F. Merle, the energy E(W) has been shown to be a threshold for the dynamical behavior of solutions of the equation. In the present article, we study the dynamics at the critical level E(u)=E(W) and classify the corresponding solutions. This gives in particular a dynamical characterization of W.
In this paper, we study the defocusing energy-critical nonlinear Schrödinger equations $$ i\partial_t u + Δu = |u|^{\frac{4}{d-2}} u. $$ When $d=3,4$, we prove the almost sure scattering for the equations with non-radial data in $H_x^s$ for any $s\in\mathbb{R}$. In particular, our result does not rely on any spherical symmetry, size or regularity restrictions.
In this article, we establish in the radial framework the $H^1$-scattering for the critical 2-D nonlinear Schrödinger equation with exponential growth. Our strategy relies on both the a priori estimate derived in \cite{CGT, PV} and the characterization of the lack of compactness of the Sobolev embedding of $H_{rad}^1(\R^2)$ into the critical Orlicz space ${\cL}(\R^2)$ settled in \cite{BMM}. The radial setting, and particularly the fact that we deal with bounded functions far away from the origin, occurs in a crucial way in our approach.
In this paper, we are concerned with the multiplicity of nontrivial solutions for the following class of complex problems $$ (-i\nabla - A(μx))^{2}u= μ|u|^{q-2}u + |u|^{2^{*}-2}u \ \mbox{in} \ Ω, \ \ \ \ u \in H^{1}_{0}(Ω,\mathbb{C}), $$ where $Ω\subset \mathbb{R}^{N} (N \geq 4)$ is a bounded domain with smooth boundary. Using the Lusternik-Schnirelman theory, we relate the number of solutions with the topology of $Ω$.
In this paper, we study the one-dimensional cubic nonlinear Schrödinger equation (NLS) on the circle. In particular, we develop a normal form approach to study NLS in almost critical Fourier-Lebesgue spaces. By applying an infinite iteration of normal form reductions introduced by the first author with Z. Guo and S. Kwon (2013), we derive a normal form equation which is equivalent to the renormalized cubic NLS for regular solutions. For rough functions, the normal form equation behaves better than the renormalized cubic NLS, thus providing a further renormalization of the cubic NLS. We then pr…
We construct pure two-bubbles for the energy-critical focusing nonlinear Schrödinger equation in space dimension $N \geq 7$. The constructed solution is global in (at least) one time direction and approaches a superposition of two stationary states both centered at the origin, with the ratio of their length scales converging to $0$. One of the bubbles develops at scale $1$, whereas the length scale of the other converges to $0$ at rate $|t|^{-\frac{2}{N-6}}$. The phases of the two bubbles form the right angle.
