Let $d\geq 4$ and let $u$ be a global solution to the focusing mass-critical nonlinear Schrödinger equation $iu_t+Δu=-|u|^{\frac 4d}u$ with spherically symmetric $H_x^1$ initial data and mass equal to that of the ground state $Q$. We prove that if $u$ does not scatter then, up to phase rotation and scaling, $u$ is the solitary wave $e^{it}Q$. Combining this result with that of Merle \cite{merle2}, we obtain that in dimensions $d\geq 4$, the only spherically symmetric minimal-mass blowup solutions are, up to phase rotation and scaling, the pseudo-conformal ground state and the solitary wave.
In this paper we prove rigidity for blowup solutions to the focusing, mass-critical nonlinear Schr{ö}dinger equation in dimensions $2 \leq d \leq 15$ with mass equal to the mass of the soliton. We prove that the only such solutions are the solitons and the pseudoconformal transformation of the solitons.
In this paper, we study the $H^s$-stability of the log-log blowup regime (which has been completely described in a series of recent works by Merle and Raphael) for the focusing mass-critical nonlinear Schrödinger equations $i\partial_tu+Δu+|u|^\frac4du=0$ in $\mathbb{R}^d$ with $d\geq3$. We aim to extend the result in [Colliander and Raphael, Rough blowup solutions to the $L^2$ critical NLS, Math. Anna., 345(2009), 307-366.] for dimension two to the higher dimensions cases $d\geq3$, where we use the bootstrap argument in the above paper and the commutator estimates in [M. Visan and X. Zhang, O…
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…
Consider a branch of unstable solitons of NLS whose linearized operators have one pair of simple real eigenvalues in addition to the zero eigenvalue. Under radial symmetry and standard assumptions, solutions to initial data from a neighbourhood of the branch either converge to a soliton, or exit a larger neighbourhood of the branch transversally. The qualitative dynamic near a branch of unstable solitons is irrespective of whether blowup eventually occurs, which has practical implications for the description of blowup of NLS with supercritical nonlinearity.
Rough log-log blowup solutions to mass-critical NLS in higher dimensions $d\geq 3$
Institute of Applied Physics and Computational Mathematics（应用物理与计算力学研究所）；School of Mathematics and Physics, University of Science and Technology Beijing（北京科技大学数学与物理学院）
We prove there exist solutions to the focusing cubic nonlinear Schrödinger equation in three dimensions that blowup on a circle, in the sense of L^2 concentration on a ring, bounded H^1 norm outside any surrounding toroid, and growth of the global H^1 norm with the log-log rate. Analogous behaviour occurs in any higher dimension. That is, there exists data for which the corresponding evolution by cubic NLS explodes on a set of co-dimension two. To simplify the exposition, the proof is presented in dimension three, with remarks to indicate the adaptations in higher dimension.
We consider fractional NLS with focusing power-type nonlinearity $$i \partial_t u = (-Δ)^s u - |u|^{2 σ} u, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^N,$$ where $1/2< s < 1$ and $0 < σ< \infty$ for $s \geq N/2$ and $0 < σ\leq 2s/(N-2s)$ for $s < N/2$. We prove a general criterion for blowup of radial solutions in $\mathbb{R}^N$ with $N \geq 2$ for $L^2$-supercritical and $L^2$-critical powers $σ\geq 2s/N$. In addition, we study the case of fractional NLS posed on a bounded star-shaped domain $Ω\subset \mathbb{R}^N$ in any dimension $N \geq 1$ and subject to exterior Dirichlet conditions. In…
We consider the blowup rate for blowup solutions to $L^2$-critical, focusing NLS with a harmonic potential and a rotation term. Under a suitable spectral condition we prove that there holds the "$\log$-$\log$ law" when the initial data is slightly above the ground state. We also construct minimal mass blowup solutions near the ground state level with distinct blowup rates.
We consider $L^2$-critical focusing nonlinear Schroedinger equations with Hartree type nonlinearity $$i \pr_t u = -\DD u - \big (Φ\ast |u|^2 \big) u \quad {in $\RR^4$},$$ where $Φ(x)$ is a perturbation of the convolution kernel $|x|^{-2}$. Despite the lack of pseudo conformal invariance for this equation, we prove the existence of critical mass finite-time blowup solutions $u(t,x)$ that exhibit the pseudo-conformal blowup rate $$ \| \nabla u(t) \|_{L^2_x} \sim \frac{1}{|t|} \quad {as} \quad t \nearrow 0 . $$ Furthermore, we prove the finite-codimensional stability of this conformal blow up, by…
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）
We consider the Cauchy problem for the biharmonic (i.\,e.~fourth-order) NLS with focusing nonlinearity given by $i \partial_t u = Δ^2 u - μΔu -|u|^{2 σ} u$ for $(t,x) \in [0,T) \times \mathbb{R}^d$, where $0 < σ<\infty$ for $d \leq 4$ and $0 < σ\leq 4/(d-4)$ for $d \geq 5$; and $μ\in \mathbb{R}$ is some parameter to include a possible lower-order dispersion. In the mass-supercritical case $σ> 4/d$, we prove a general result on finite-time blowup for radial data in $H^2(\mathbb{R}^d)$ in any dimension $d \geq 2$. Moreover, we derive a universal upper bound for the blowup rate for suitable $4/d …
In this paper we study dynamical properties of blowup solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. We establish a profile decomposition and a compactness lemma related to the equation. As a result, we obtain the $L^2$-concentration and the limiting profile with minimal mass of blowup solutions.
We consider the focusing mass-critical nonlinear Schrödinger equation and prove that blowup solutions to this equation with initial data in $H^s(\R^d)$, $s > s_0(d)$ and $d\geq 3$, concentrate at least the mass of the ground state at the blowup time. This extends recent work by J. Colliander, S. Raynor, C. Sulem, and J. D. Wright, \cite{crsw}, T. Hmidi and S. Keraani, \cite{hk}, and N. Tzirakis, \cite{tzirakis}, on the blowup of the two-dimensional and one-dimensional mass-critical focusing NLS below the energy space to all dimensions $d\ge 3$.
In this paper we study dynamical properties of blowup solutions to the focusing intercritical (mass-supercritical and energy-subcritical) nonlinear fourth-order Schrödinger equation. We firstly establish the profile decomposition of bounded sequences in $\dot{H}^{γ_{\text{c}}} \cap \dot{H}^2$. We also prove a compactness lemma and a variational characterization of ground states related to the equation. As a result, we obtain the $\dot{H}^{γ_{\text{c}}}$-concentration of blowup solutions with bounded $\dot{H}^{γ_{\text{c}}}$-norm and the limiting profile of blowup solutions with critical $\dot{…
We prove finite time supercritical blowup in a parity-time-symmetric system of the two coupled nonlinear Schrödinger (NLS) equations. One of the equations contains gain and the other one contains dissipation such that strengths of the gain and dissipation are equal. We address two cases: in the first model all nonlinear coefficients (i.e. the ones describing self-action and non-linear coupling) correspond to attractive (focusing) nonlinearities, and in the second case the NLS equation with gain has attractive nonlinearity while the NLS equation with dissipation has repulsive (defocusing) nonli…
We consider the focusing cubic nonlinear Schrödinger equation with inverse-square potential in three space dimensions. We identify a sharp threshold between scattering and blowup, establishing a result analogous to that of Duyckaerts, Holmer, and Roudenko for the standard focusing cubic NLS. We also prove failure of uniform space-time bounds at the threshold.
We prove an infinite dimensional version of the Arnold-Liouville theorem for integrable non-linear PDEs: In a case study we consider the {\em focusing} NLS equation with periodic boundary conditions.
We study the global behavior of finite energy solutions to the $d$-dimensional focusing nonlinear Schrödinger equation (NLS), $i \partial_t u+Δu+ |u|^{p-1}u=0, $ with initial data $u_0\in H^1,\; x \in R^n$. The nonlinearity power $p$ and the dimension $d$ are such that the scaling index $s=\frac{d}2-\frac2{p-1}$ is between 0 and 1, thus, the NLS is mass-supercritical $(s>0)$ and energy-subcritical $(s<1).$ For solutions with $\ME[u_0]<1$ ($\ME[u_0]$ stands for an invariant and conserved quantity in terms of the mass and energy of $u_0$), a sharp threshold for scattering and blowup is given. Na…
We consider the following nonlinear Schrödinger equation with the double $L^2$-critical nonlinearities \begin{align*} iu_t+Δu+|u|^\frac{4}{3}u+μ\left(|x|^{-2}*|u|^2\right)u=0\ \ \ \text{in $\mathbb{R}^3$,} \end{align*} where $μ>0$ is small enough. Our first goal is to prove the existence and the non-degeneracy of the ground state $Q_μ$. In particular, we develop an appropriate perturbation approach to prove the radial non-degeneracy property and then obtain the general non-degeneracy of the ground state $Q_μ$. We then show the existence of finite time blowup solution with minimal mass $\|u_0\|…
We consider the large data scattering problem for the 2D and 3D cubic-quintic NLS in the focusing-focusing regime. Our attention is firstly restricted to the 2D space, where the cubic nonlinearity is $L^2$-critical. We establish a new type of scattering criterion that is uniquely determined by the mass of the initial data, which differs from the classical setting based on the Lyapunov functional. At the end, we also formulate a solely mass-determining scattering threshold for the 3D cubic-quintic NLS in the focusing-focusing regime.
We study dynamical properties of blowup solutions to the focusing $L^2$-supercritical nonlinear fractional Schrödinger equation \[ i\partial_t u -(-Δ)^s u = -|u|^αu, \quad u(0) = u_0, \quad \text{on } [0,\infty) \times \mathbb{R}^d, \] where $d \geq 2, \frac{d}{2d-1} \leq s <1$, $\frac{4s}{d}<α<\frac{4s}{d-2s}$ and $u_0 \in \dot{H}^{s_{\text{c}}} \cap \dot{H}^s$ is radial with the critical Sobolev exponent $s_{\text{c}}$. To this end, we establish a compactness lemma related to the equation by means of the profile decomposition for bounded sequences in $\dot{H}^{s_{\text{c}}} \cap \dot{H}^s$. …
We study the nonlinear Schrödinger equation with an inverse-square potential in dimensions $3\leq d \leq 6$. We consider both focusing and defocusing nonlinearities in the mass-supercritical and energy-subcritical regime. In the focusing case, we prove a scattering/blowup dichotomy below the ground state. In the defocusing case, we prove scattering in $H^1$ for arbitrary data.
We revisit the scattering result of Holmer and Roudenko on the radial focusing cubic NLS in three space dimensions. Using the radial Sobolev embedding and a virial/Morawetz estimate, we give a simple proof of scattering below the ground state that avoids the use of concentration compactness.
We prove that there exist potentials so that near them the focusing non-linear Schrödinger equation does not admit local Birkhoff coordinates. The proof is based on the construction of a local normal form of the linearization of the equation at such potentials.
We revisit the scattering result of Duyckaerts, Holmer, and Roudenko for the non-radial $\dot H^{1/2}$-critical focusing NLS. By proving an interaction Morawetz inequality, we give a simple proof of scattering below the ground state in dimensions $d\geq 3$ that avoids the use of concentration compactness.
In this paper, the Choquard equation with an inverse-square potential and both focusing and defocusing nonlinearities in the energy-subcritical regime is investigated. For all the cases, the local well-posedness result in $H^1(\mathbb{R}^N)$ is established. Moreover, the global existence result for arbitrary initial values is proved in the defocusing case while a global existence/blowup dichotomy below the ground state is established in the focusing case.
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 a class of the focusing nonlinear Schrödinger equation with inverse-square potential \[ i\partial_t u + Δu -c|x|^{-2}u = - |u|^αu, \quad u(0)=u_0 \in H^1, \quad (t,x)\in \mathbb{R} \times \mathbb{R}^d, \] where $d\geq 3$, $\frac{4}{d}\leq α\leq \frac{4}{d-2}$ and $c\ne 0$ satisfies $c>-λ(d):=-\left(\frac{d-2}{2}\right)^2$. In the mass-critical case $α=\frac{4}{d}$, we prove the global existence and blowup below ground states for the equation with $d\geq 3$ and $c>-λ(d)$. In the mass and energy intercritical case $\frac{4}{d}<α<\frac{4}{d-2}$, we prove the global existence and blowu…
The focusing cubic NLS is a canonical model for the propagation of laser beams. In dimensions 2 and 3, it is known that a large class of initial data leads to finite time blow-up. Now, physical experiments suggest that this blow-up does not always occur. This might be explained by the fact that some physical phenomena neglected by the standard NLS model become relevant at large intensities of the beam. Many ad hoc variants of the focusing NLS equation have been proposed to capture such effects. In this paper, we derive some of these variants from Maxwell's equations and propose some new ones. …
In this paper we study long time dynamics (i.e., scattering and blow-up) of solutions for the focusing NLS with a repulsive inverse-power potential and with initial data lying exactly at the mass-energy threshold, namely, when $E_{V}(u_{0})M(u_{0})=E_{0}(Q)M(Q)$. Moreover, we prove failure of the uniform space-time bounds at the mass-energy threshold.
We mainly consider the focusing biharmonic Schrödinger equation with a large radial repulsive potential $V(x)$: \begin{equation*} \left\{ \begin{aligned} iu_{t}+(Δ^2+V)u-|u|^{p-1}u=0,\;\;(t,x) \in {\bf{R}\times{\bf{R}}^{N}}, u(0, x)=u_{0}(x)\in H^{2}({\bf{R}}^{N}), \end{aligned}\right. \end{equation*} If $N>8$, \ $1+\frac{8}{N}<p<1+\frac{8}{N-4}$ (i.e. the $L^{2}$-supercritical and $\dot{H}^{2}$-subcritical case ), and $\langle x\rangle^β\big(|V(x)|+|\nabla V(x)|\big)\in L^\infty$ for some $β>N+4$, then we firstly prove a global well-posedness and scattering result for the radial data $u_0\in …
For the 3D focusing cubic nonlinear Schrodinger equation, Scattering of $H^1$ solutions inside the (scale invariant) potential well was established by Holmer and Roudenko~\cite{HR2} (radial case) and Duyckaerts, Holmer and Roudenko~\cite{DHR} (general case). In this paper, we extend this result to arbitrary space dimensions and focusing, mass-supercritical and energy-subcritical power nonlinearities, by adapting the method of~\cite{DHR}.
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 focusing cubic NLS in the exterior $Ω$ of a smooth, compact, strictly convex obstacle in three dimensions. We prove that the threshold for global existence and scattering is the same as for the problem posed on Euclidean space. Specifically, we prove that if $E(u_0)M(u_0)<E(Q)M(Q)$ and $\|\nabla u_0\|_2\|u_0\|_2<\|\nabla Q\|_2\|Q\|_2$, the corresponding solution to the initial-value problem with Dirichlet boundary conditions exists globally and scatters to linear evolutions asymptotically in the future and in the past. Here, $Q(x)$ denotes the ground state for the focusing cubi…
In this paper, we consider a 3d cubic focusing nonlinear Schrödinger equation (NLS) with slowing decaying potentials. Adopting the variational method of Ibrahim-Masmoudi-Nakanishi \cite{IMN}, we obtain a condition for scattering. It is actually sharp in some sense since the solution will blow up if it's false. The proof of blow-up part relies on the method of Du-Wu-Zhang \cite{DWZ}
In this paper, we prove the global well-posedness for the focusing, cubic nonlinear Schrödinger equation on the product space $\mathbb{R} \times \mathbb{T}^3$ with initial data below the threshold that arises from the the ground state in the Euclidean setting. The defocusing analogue was discussed and proved in Ionescu-Pausader \cite{IPRT3} (Comm. Math. Phys. 312 (2012), no. 3, 781-831).
In this article, we consider the focusing cubic nonlinear Schrödinger equation(NLS) in the exterior domain outside of a convex obstacle in $\mathbb{R}^3$ with Dirichlet boundary conditions. We revisit the scattering result below ground state of Killip-Visan-Zhang by utilizing Dodson and Murphy's argument and the dispersive estimate established by Ivanovici and Lebeau, which avoids using the concentration compactness. We conquer the difficulty of the boundary in the focusing case by establishing a local smoothing effect of the boundary. Based on this effect and the interaction Morawetz estimate…
In this paper, we consider a class of the focusing inhomogeneous nonlinear Schrödinger equation \[ i\partial_t u + Δu + |x|^{-b} |u|^αu = 0, \quad u(0)=u_0 \in H^1(\mathbb{R}^d), \] with $0<b<\min\{2,d\}$ and $α_\star\leq α<α^\star$ where $α_\star =\frac{4-2b}{d}$ and $α^\star=\frac{4-2b}{d-2}$ if $d\geq 3$ and $α^\star = \infty$ if $d=1,2$. In the mass-critical case $α=α_\star$, we prove that if $u_0$ has negative energy and satisfies either $xu_0 \in L^2$ with $d\geq 1$ or $u_0$ is radial with $d\geq 2$, then the corresponding solution blows up in finite time. Moreover, when $d=1$, we prove …
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 consider the Cauchy problem for the nonlinear Schrödinger equations (NLS) with non-algebraic nonlinearities on the Euclidean space. In particular, we study the energy-critical NLS on $\mathbb{R}^d$, $d=5,6$, and energy-critical NLS without gauge invariance and prove that they are almost surely locally well-posed with respect to randomized initial data below the energy space. We also study the long time behavior of solutions to these equations: (i) we prove almost sure global well-posedness of the (standard) energy-critical NLS on $\mathbb{R}^d$, $d = 5, 6$, in the defocusing case, and (ii) …
We consider the fractional Schrödinger equations with focusing Hartree type nonlinearity. When the energy is negative, we use a Glassey's virial type argument to show the finite time blow-up of solutions.
In this paper, we utilize the method in Dodson-Murphy [4] to establish the radial scattering result for the focusing nonlinear Schrödinger equation with inverse square potential $i\pa_tu-\la u=-|u|^{p-1}u$ in the energy space $H^1_a(\R^d)$ in dimensions $d\geq3$, which extends the result of [10,11] to higher dimensions cases but with radial initial data. The new ingredient is to establish the dispersive estimate for radial function and overcome the weak dispersive estimate when $a<0$.
We study the focusing nonlinear Schrödinger equation in the $L^2$-supercritical regime with finite energy and finite variance initial data. We investigate solutions above the energy (or mass-energy) threshold. In our first result, we extend the known scattering versus blow-up dichotomy above that threshold for finite variance solutions in the energy-subcritical and energy-critical regimes, obtaining scattering and blow-up criteria for solutions with arbitrarily large mass and energy. As a consequence, we characterize the behavior of the ground state initial data modulated by a quadratic phase.…
We establish the scattering of solutions to the focusing mass supercritical nonlinear Schrödinger equation with a repulsive Dirac delta potential \[ i\partial_{t}u+\partial^{2}_{x}u+γδ(x)u+|u|^{p-1}u=0, \quad (t,x)\in {\mathbb R}\times{\mathbb R}, \] at the mass-energy threshold, namely, when $E_γ(u_{0})[M(u_{0})]^σ=E_{0}(Q)[M(Q)]^σ$ where $u_{0}\in H^{1}({\mathbb R})$ is the initial data, $Q$ is the ground state of the free NLS on the real line ${\mathbb R}$, $E_γ$ is the energy, $M$ is the mass and $σ=(p+3)/(p-5)$. We also prove failure of the uniform space-time bounds at the mass-energy thr…
