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 dimensions $d\geq 3$ for the mass-critical NLS and by Kenig and Merle, \cite{merlekenig}, in the energy-critical case. In a subsequent paper we shall use this compactness result to establish global existence and scattering in $L^2_x(\R^d)$ for the defocusing NLS in three and higher dimensions with spherically symmetric data.
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…
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.
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 Cauchy problem of nonlinear Schrödinger equations (NLS) with almost periodic functions as initial data. We first prove that, given a frequency set $\pmbω =\{ω_j\}_{j = 1}^\infty$, NLS is local well-posed in the algebra $\mathcal{A}_{\pmbω}(\mathbb R)$ of almost periodic functions with absolutely convergent Fourier series. Then, we prove a finite time blowup result for NLS with a nonlinearity $|u|^p$, $p \in 2\mathbb{N}$. This elementary argument presents the first instance of finite time blowup solutions to NLS with generic almost periodic initial data.
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 …
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 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 $d\ge 3$.
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\|…
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 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 …
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) …
In the previous papers in this series, the global regularity conjecture for wave maps from two-dimensional Minkowski space $\R^{1+2}$ to hyperbolic space $\H^m$ was reduced to the problem of constructing a minimal-energy blowup solution which is almost periodic modulo symmetries in the event that the conjecture fails. In this paper, we show that this problem can be reduced further, to that of showing that solutions at the critical energy which are either frequency-delocalised, spatially-dispersed, or spatially-delocalised have bounded ``entropy''. These latter facts will be demonstrated in the…
We consider the focussing energy-critical inhomogeneous nonlinear Schrödinger equation: $$ iu_t + Δu + g|u|^2u = 0, u(0)= φ\in \dot{H}^1,\;\; 0 \le g_i \le |x|g \le g_s.$$ On the road map of Kenig-Merle \cite{km} we show the global well-posedness and scattering of radial solutions under energy condition $$E_g(φ) < E_g(Q),\;\;\mbox{and}\;\; g_s\|φ\|_{\dot{H}^1}^2 < \|Q\|_{\dot{H}^1}^2,$$ where $Q$ is the solution of $ΔQ + |x|^{-1}Q^3 = 0$, together with scaling condition $|g(x)| + |x||\nabla g(x)| \lesssim |x|^{-1}$, variational condition $g_s(2-g_i) \le 1$, and rigidity condition $-g(x) \le x\…
Classification of minimal-mass blowup solutions to the mass-critical Hartree equation at the $L^2$-critical level
Graduate School of China Academy of Engineering Physics（中国工程物理研究院研究生院）；School of Mathematics and Physics, University of Science and Technology Beijing（北京科技大学数理学院）；Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）
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 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 global regularity problem for defocusing nonlinear Schrödinger systems $$ i \partial_t + Δu = (\nabla_{{\bf R}^m} F)(u) + G $$ on Galilean spacetime ${\bf R} \times {\bf R}^d$, where the field $u\colon {\bf R}^{1+d} \to {\bf C}^m$ is vector-valued, $F\colon {\bf C}^m \to {\bf R}$ is a smooth potential which is positive, phase-rotation-invariant, and homogeneous of order $p+1$ outside of the unit ball for some exponent $p >1$, and $G: {\bf R} \times {\bf R}^d \to {\bf C}^m$ is a smooth, compactly supported forcing term. This generalises the scalar defocusing nonlinear Schrödinge…
We consider the cubic and quintic nonlinear Schrödinger equations (NLS) under the $\mathbb{R}^{d}$ and $\mathbb{T}^{d}$ energy-supercritical setting. Via a newly developed unified scheme, we prove the unconditional uniqueness for solutions to NLS at critical regularity for all dimensions. Thus, together with [18,19], the unconditional uniqueness problems for $H^{1}$-critical and $H^{1}$-supercritical cubic and quintic NLS are completely and uniformly resolved at critical regularity for these domains. One application of our theorem is to prove that defocusing blowup solutions of the type in [54…
We consider the 2-dimensional focusing mass critical NLS with an inhomogeneous nonlinearity: $i\partial_tu+Δu+k(x)|u|^{2}u=0$. From standard argument, there exists a threshold $M_k>0$ such that $H^1$ solutions with $\|u\|_{L^2}<M_k$ are global in time while a finite time blow up singularity formation may occur for $\|u\|_{L^2}>M_k$. In this paper, we consider the dynamics at threshold $\|u_0\|_{L^2}=M_k$ and give a necessary and sufficient condition on $k$ to ensure the existence of critical mass finite time blow up elements. Moreover, we give a complete classification in the energy class of t…
We analyze the asymptotic behavior of a $2$-dimensional integral current which is almost minimizing in a suitable sense at a singular point. Our analysis is the second half of an argument which shows the discreteness of the singular set for the following three classes of $2$-dimensional currents: area minimizing in Riemannian manifolds, semicalibrated and spherical cross sections of $3$-dimensional area minimizing cones.
We consider singular solutions of the biharmonic NLS. In the L^2-critical case, the blowup rate is bounded by a quartic-root power law, the solution approaches a self-similar profile, and a finite amount of L^2-norm, which is no less than the critical power, concentrates into the singularity ("strong collapse"). In the L^2-critical and supercritical cases, we use asymptotic analysis and numerical simulations to characterize singular solutions with a peak-type self-similar collapsing core. In the critical case, the blowup rate is slightly faster than a quartic-root, and the self-similar profile…
We consider the half-wave equation $i u_t=Du-|u|^{\frac{2}{3}}u$ in three dimension and in the mass critical. For initial data $u(t_0,x)=u_0(x)\in H^{1/2}_{rad}(\mathbb{R}^3)$ with radial symmetry, we construct a new class of minimal mass blowup solutions with the blow up rate $\|D^{\frac{1}{2}}u(t)\|_2\sim\frac{C(u_0)}{|t|^{\frac{1}{4}}}$ as $t\rightarrow0^-$.
We present a new method for obtaining sharp asymptotics of solutions of the defocussing nonlinear Schrödinger (NLS) equation, based on dbar methods and under essentially minimal regularity assumptions on initial data.
We study the existence and stability of standing waves for a system of nonlinear Schrödinger equations with quadratic interaction in dimensions $d\leq 3$. We also study the characterization of finite time blow-up solutions with minimal mass to the system under mass resonance condition in dimension $d=4$. Finite time blow-up solutions with minimal mass are showed to be (up to symmetries) pseudo-conformal transformations of a ground state standing wave.
We study minimal mass blow-up solutions of the focusing $L^2$ critical nonlinear Schrödinger equation with inverse-square potential, \[ i\partial_t u + Δu + \frac{c}{|x|^2}u+|u|^{\frac{4}{N}}u = 0, \] with $N\geqslant 3$ and $0<c<\frac{(N-2)^2}{4}$. We first prove a sharp global well-posedness result: all $H^1$ solutions with a mass (i.e. $L^2$ norm) strictly below that of the ground states are global. Note that, unlike the equation in free space, we do not know if the ground state is unique in the presence of the inverse-square potential. Nevertheless, all ground states have the same, minimal…
We consider the $L^2$-critical nonlinear Schrödinger equation (NLS) with the delta potential $$i\partial_tu +\partial^2_x u + μδu +|u|^{4}u=0, \, \, t\in \R, \, x\in \R , $$ where $ μ\in \R$, and $δ$ is the Dirac delta distribution at $x=0$. Local well-posedness theory together with sharp Gagliardo-Nirenberg inequality and the conservation laws of mass and energy implies that the solution with mass less than $\|Q\|_{2}$ is global existence in $H^1(\R)$, where $Q$ is the ground state of the $L^2$-critical NLS without the delta potential (i.e. $μ=0$). We are interested in the dynamics of the sol…
We study the nonlinear-damping continuation of singular solutions of the critical and supercritical NLS. Our simulations suggest that for generic initial conditions that lead to collapse in the undamped NLS, the solution of the weakly-damped NLS $$ iψ_t(t,\X)+Δψ+|ψ|^{p-1}ψ+iδ|ψ|^{q-1}ψ=0,\qquad0<δ\ll 1, $$ is highly asymmetric with respect to the singularity time, and the post-collapse defocusing velocity of the singular core goes to infinity as the damping coefficient $δ$ goes to zero. In the special case of the minimal-power blowup solutions of the critical NLS, the continuation is a minimal…
Asymmetric minimal mass blow-up for the quintic NLS on a star graph
Sapienza Università di Roma（罗马大学）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Università degli Studi di Pisa（比萨大学）；Waseda University（早稻田大学）；Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与信息学研究所）
In the last twenty years, there have been significant advances in the study of the blow-up phenomenon for the critical generalized Korteweg-de Vries equation, including the determination of sufficient conditions for blowup, the stability of blowup in a refined topology and the classification of minimal mass blowup. Exotic blow-up solutions with a continuum of blow-up rates and multi-point blow-up solutions were also constructed. However, all these results, as well as numerical simulations, involve the bubbling of a solitary wave going at infinity at the blow-up time, which means that the blow-…
Once one knows that singularities occur, one naturally wonders what the singularities are like. For minimal varieties the first answer, already known to Federer-Fleming in 1959, is that they weakly resemble cones. For mean curvature flow, by the combined work of Huisken, Ilmanen, and White, singularities weakly resemble shrinkers. Unfortunately, the simple proofs leave open the possibility that a minimal variety or a mean curvature flow looked at under a microscope will resemble one blowup, but under higher magnification, it might (as far as anyone knows) resemble a completely different blowup…
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.
We consider the focusing nonlinear Schrödinger equation with the critical inverse square potential. We give the first proof of the uniqueness of the ground state solution. Consequently, we obtain a sharp Hardy-Gagliardo-Nirenberg interpolation inequality. Moreover, we provide a complete characterization for the minimal mass blow-up solutions to the time dependent problem.
We consider the mass-critical generalized Korteweg--de Vries equation $$(\partial_t + \partial_{xxx})u=\pm \partial_x(u^5)$$ for real-valued functions $u(t,x)$. We prove that if the global well-posedness and scattering conjecture for this equation failed, then, conditional on a positive answer to the global well-posedness and scattering conjecture for the mass-critical nonlinear Schrödinger equation $(-i\partial_t + \partial_{xx})u=\pm (|u|^4u)$, there exists a minimal-mass blowup solution to the mass-critical generalized KdV equation which is almost periodic modulo the symmetries of the equat…
We consider the nonlinear Schrödinger equation with pure power nonlinearity on a general compact metric graph, and in particular its stationary solutions with fixed mass. Since the graph is compact, for every value of the mass there is a constant solution. Our scope is to analyze (in dependence of the mass) the variational properties of this solution, as a critical point of the energy functional: local and global minimality, and (orbital) stability. We consider both the subcritical regime and the critical one, in which the features of the graph become relevant. We describe how the above proper…
