We establish quantitative blow-up criteria below the scaling threshold for radially symmetric solutions to the defocusing nonlinear Schrödinger equation with nonlinearity $|u|^6u$. This provides to our knowledge the first generic results distinguishing potential blow-up solutions of the defocusing equation from many of the known examples of blow-up in the focusing case. Our main tool is a quantitative version of a result showing that uniform bounds on $L^2$-based critical Sobolev norms imply scattering estimates.
As another application of our techniques, we establish a variant which allows for slow growth in the critical norm. We show that if the critical Sobolev norm on compact time intervals is controlled by a slowly growing quantity depending on the Stricharz norm, then the solution can be extended globally in time, with a corresponding scattering estimate.
Painlevé \uppercase\expandafter{\romannumeral34\relax} and collisionless shock in the defocusing NLS equation with step-like initial data in the transition regions
We study a defocusing semilinear wave equation, with a power nonlinearity $|u|^{p-1}u$, defined outside the unit ball of $\mathbb{R}^{n}$, $n\ge3$, with Dirichlet boundary conditions. We prove that if $p>n+4$ and the initial data are nonradial perturbations of large radial data, there exists a global smooth solution. The solution is unique among energy class solutions satisfying an energy inequality. The main tools used are the Penrose transform and a pointwise decay estimate for the exterior linear wave equation perturbed with a large, time dependent potential.
We show that all supercritical monic focusing NLS in one space dimension exhibit asymptotic stability of perturbed standing waves provided the perturbations are chosen on a small hypersuface in a suitable space.
In this paper we prove that the non-linear Fourier transform of the defocusing NLS equation on the circle is linear up to terms which are one order smoother.
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.
We prove the existence of orbitally stable ground states to NLS with a partial confinement together with qualitative and symmetry properties. This result is obtained for nonlinearities which are $L^2$-supercritical, in particular we cover the physically relevant cubic case. The equation that we consider is the limit case of the cigar-shaped model in BEC.
In this paper we prove approximation properties of the solutions of the defoucsing NLS equation on the circle by nearly linear flows. In addition we show that spatially periodic solutions of the defocusing NLS equation evolving in fractional Sobolev spaces $H^s$ with $s\geq 1$ remain bounded for all times.
We consider the following family of Cauchy problems: {equation*} i\partial_t u= Δu - u|u|^α, (t,x) \in \R \times \R^d {equation*} $$u(0)=φ\in H^1(\R^d)$$ where $0<α<\frac 4{d-2}$ for $d\geq 3$ and $0<α<\infty$ for $d=1,2$. We prove that the $L^r$-norms of the solutions decay as $t\to \pm \infty$, provided that $2<r<\frac{2d}{d-2}$ when $d\geq 3$ and $2<r<\infty$ when $d=1,2$. In particular we extend previous results obtained by Ginibre and Velo for $d\geq 3$ and by Nakanishi for $d=1,2$, where the same decay results are proved under the extra assumption $α>\frac 4d$.
We revisit the proof of global well-posedness and scattering for the defocusing energy-critical NLS in three space dimensions in light of recent developments. This result was obtained previously by Colliander, Keel, Staffilani, Takaoka, and Tao.
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 prove $H^{1}$ scattering for a defocusing NLS on the line with fully variable coefficients. The result is proved by adapting the Kenig--Merle scheme to a non translation invariant setting. In addition, we give an abstract version of the scheme which can be applied to other operators.
We prove the existence and the invariance of a Gibbs measure associated to the defocusing sub-quintic Nonlinear Schroedinger equations on the disc of the plane $\R^2$. We also prove an estimate giving some intuition to what may happen in 3 dimensions.
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 prove global well-posedness and scattering for the defocusing, cubic NLS on $\mathbb{R}^3$ with initial data in $H^s(\mathbb{R}^3)$ for $s>49/74$. The proof combines the ideas of resonance decomposition in \cite{CKSTT4} and linear-nonlinear decomposition in \cite{ben1}\cite{roy} together with the idea of large time iteration.
We prove global wellposedness in the energy space of the defocusing cubic nonlinear Schroedinger and Gross-Pitaevskii equations on the exterior of a non-trapping domain in dimension 3. The main ingredient is a Strichartz estimate obtained combining a semi-classical Strichartz estimate with a smoothing effect on exterior domains.
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.
In a case study for integrable PDEs, we construct real analytic, canonical coordinates for the defocusing NLS equation on the circle, specifically taylored towards the needs in perturbation theory. They are defined in neighbourhoods of families of finite dimensional invariant tori and are shown to satisfy together with their derivatives tame estimates. When expressed in these coordinates, the dNLS Hamiltonian is in normal form up to order three.
We show that if a solution of the defocusing cubic NLS in 3d remains bounded in the homogeneous Sobolev norm of order 1/2 in its maximal interval of existence, then the interval is infinite and the solution scatters. No radial assumption is made.
Combining the usual energy functional with a higher-order conserved quantity originating from integrability theory, we show that the black soliton is a local minimizer of a quantity that is conserved along the flow of the cubic defocusing NLS equation in one space dimension. This unconstrained variational characterization gives an elementary proof of the orbital stability of the black soliton with respect to perturbations in $H^2(\mathbb{R})$.
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 $x\in\R\times\T^3$.
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.
We consider the radial defocusing nonlinear Schrödinger equations $iu_t+Δu=|u|^{p}u$ with supercritical exponent $p>4$ in four space dimensions, and prove that any radial solution that remains bounded in the critical Sobolev space must be global and scatter.
The defocusing NLS-equation $\mathrm{i} u_t = -u_{xx} + 2|u|^2u$ on the circle admits a global nonlinear Fourier transform, also known as Birkhoff map, linearising the NLS-flow. The regularity properties of $u$ are known to be closely related to the decay properties of the corresponding nonlinear Fourier coefficients. In this paper we quantify this relationship by providing two sided polynomial estimates of all integer Sobolev norms $\|u\|_m$, $m\ge 0$, in terms of the weighted norms of the nonlinear Fourier transformed.
We establish global regularity for the logarithmically energy-supercritical wave equation $\Box u = u^5 \log(2+u^2)$ in three spatial dimensions for spherically symmetric initial data, by modifying an argument of Ginibre, Soffer and Velo \cite{gsv} for the energy-critical equation. This example demonstrates that critical regularity arguments can penetrate very slightly into the supercritical regime.
We consider a class of defocusing energy-supercritical nonlinear Schrödinger equations in four space dimensions. Following a concentration-compactness approach, we show that for $1<s_c<3/2$, any solution that remains bounded in the critical Sobolev space $\dot{H}_x^{s_c}(\R^4)$ must be global and scatter. Key ingredients in the proof include a long-time Strichartz estimate and a frequency-localized interaction Morawetz inequality.
We consider the defocusing quintic nonlinear Schrödinger equation in four space dimensions. We prove that any solution that remains bounded in the critical Sobolev space must be global and scatter. We employ a space-localized interaction Morawetz inequality, the proof of which requires us to overcome the logarithmic failure in the double Duhamel argument in four dimensions.
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.
In this work, we consider the energy-supercritical defocusing cubic nonlinear wave equation in dimension d=5 for radially symmetric initial data. We prove that an a priori bound in the critical space implies global well-posedness and scattering. The main tool that we use is a frequency localized version of the classical Morawetz inequality, inspired by recent developments in the study of the mass and energy critical nonlinear Schrödinger equation.
We consider the defocusing nonlinear wave equation $u_{tt}-Δu + |u|^p u=0$ in the energy-supercritical regime p>4. For even values of the power p, we show that blowup (or failure to scatter) must be accompanied by blowup of the critical Sobolev norm. An equivalent formulation is that solutions with bounded critical Sobolev norm are global and scatter. The impetus to consider this problem comes from recent work of Kenig and Merle who treated the case of spherically-symmetric solutions.
We prove $H^1$ scattering for defocusing NLS with a delta potential and mass-supercritical nonlinearity, hence extending in an inhomogeneous setting the classical 1-D scattering results first proved by Nakanishi in the translation invariant case.
We consider the defocusing supercritical generalized Korteweg-de Vries (gKdV) equation $\partial_t u+\partial_x^3u-\partial_x(u^{k+1})=0$, where $k>4$ is an even integer number. We show that if the initial data $u_0$ belongs to $H^1$ then the corresponding solution is global and scatters in $H^1$. Our method of proof is inspired on the compactness method introduced by C. Kenig and F. Merle.
We consider the defocusing nonlinear wave equation $u_{tt}-Δu + |u|^p u=0$ with spherically-symmetric initial data in the regime $\frac4{d-2}<p<\frac4{d-3}$ (which is energy-supercritical) and dimensions $3\leq d\leq 6$; we also consider $d\geq 7$, but for a smaller range of $p>\frac4{d-2}$. The principal result is that blowup (or failure to scatter) must be accompanied by blowup of the critical Sobolev norm. An equivalent formulation is that maximal-lifespan solutions with bounded critical Sobolev norm are global and scatter.
Periodic waves of the one-dimensional cubic defocusing NLS equation are considered. Using tools from integrability theory, these waves have been shown in [Bottman, Deconinck, and Nivala, 2011] to be linearly stable and the Floquet-Bloch spectrum of the linearized operator has been explicitly computed. We combine here the first four conserved quantities of the NLS equation to give a direct proof that cnoidal periodic waves are orbitally stable with respect to subharmonic perturbations, with period equal to an integer multiple of the period of the wave. Our result is not restricted to the period…
Scattering for Defocusing NLS with Inhomogeneous Nonlinear Damping and Nonlinear Trapping Potential
CNRS and Institut de Mathématiques de Toulouse（法国国家科学研究中心和图卢兹数学研究所；UMR5219，图卢兹大学；法国国家科学研究中心，图卢兹第三国立高等理工学院）；UMR5219, Université de Toulouse；CNRS, UPS IMT
We consider the energy-supercritical nonlinear wave equation $u_{tt}-Δu+|u|^2u=0$ with defocusing cubic nonlinearity in dimension $d=5$ with no radial assumption on the initial data. We prove that a uniform-in-time {\it a priori} bound on the critical norm implies that solutions exist globally in time and scatter at infinity in both time directions. Together with our earlier works in dimensions $d\geq 6$ with general data and dimension $d=5$ with radial data, the present work completes the study of global well-posedness and scattering in the energy-supercritical regime for the cubic nonlineari…
In this paper, we consider the defocusing cubic nonlinear wave equation $u_{tt}-Δu+|u|^2u=0$ in the energy-supercritical regime, in dimensions $d\geq 6$, with no radial assumption on the initial data. We prove that if a solution satisfies an a priori bound in the critical homogeneous Sobolev space throughout its maximal interval of existence, that is, $u\in L_t^\infty(\dot{H}_x^{s_c}\times\dot{H}_x^{s_c-1})$, then the solution is global and it scatters. Our analysis is based on the methods of the recent works of Kenig-Merle \cite{KenigMerleSupercritical} and Killip-Visan \cite{KillipVisanSuper…
We prove that the scattering operators and wave operators are well-defined in the energy space for the system of defocusing Schrödinger equations $$ \begin{cases} i\partial_t u_μ+ Δu_μ- \sum_{μ,ν=1 }^N β_{μν}|u_ν|^{p+1}|u_μ|^{p-1}u_μ=0, \quad\quad μ=1,\dots,N,\\(u_μ(0,\cdot))_{μ=1}^N= (u_{μ,0})_{μ=1}^N \in H^1(\mathbb R^d)^N. \end{cases} $$ with $N\geq 2$, $β_{μν} \geq 0$, $β_{μμ}\neq 0$ for $p>2 $ if $d=1$, $p>1 $ if $d=2$ and $ 1 \leq p < 2$ if $d=3$.
We show global wellposedness for the defocusing cubic nonlinear Schrödinger equation (NLS) in $H^1(\mathbb{R}) + H^{3/2+}(\mathbb{T})$, and for the defocusing NLS with polynomial nonlinearities in $H^1(\mathbb{R}) + H^{5/2+}(\mathbb{T})$. This complements local results for the cubic NLS and global results for the quadratic NLS in this hybrid setting.
We consider radial solutions of a mass supercritical monic NLS and we prove the existence of a set, which looks like a hypersurface, in the space of finite energy functions, invariant for the flow and formed by solutions which converge to ground states.
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…
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…
We consider the Cauchy problem for the defocusing nonlinear Schrödinger equations (NLS) on the real line with a special subclass of almost periodic functions as initial data. In particular, we prove global existence of solutions to NLS with limit periodic functions as initial data under some regularity assumption.
In this paper, using $I$-method, we establish $H^s_x\times H^s_x$ scattering theories for the following Cauchy problem \begin{equation*} \left\{ \begin{array}{lll} iu_t+Δu=λ|v|^2u,\quad iv_t+Δv=μ|u|^2v,\quad x\in \mathbb{R}^3,\ t>0,\\ u(x,0)=u_0(x),\quad v(x,0)=v_0(x),\quad x\in \mathbb{R}^3. \end{array}\right. \end{equation*} Here $λ>0$, $μ>0$, $(u_0,v_0)\in H^s_x(\mathbb{R}^3)\times H^s_x(\mathbb{R}^3)$ and $\frac{1}{2}<s<1$.
Multi-soliton solutions, i.e. solutions behaving as the sum of N given solitons as $t\to +\infty$, were constructed in previous works for the L2 critical and subcritical (NLS) and (gKdV) equations. In this paper, we extend the construction of multi-soliton solutions to the L2 supercritical case both for (gKdV) and (NLS) equations, using a topological argument to control the direction of instability.
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 derive the defocusing cubic Gross-Pitaevskii (GP) hierarchy in dimension $d=3$, from an $N$-body Schrödinger equation describing a gas of interacting bosons in the GP scaling, in the limit $N\rightarrow\infty$. The main result of this paper is the proof of convergence of the corresponding BBGKY hierarchy to a GP hierarchy in the spaces introduced in our previous work on the well-posedness of the Cauchy problem for GP hierarchies, \cite{chpa2,chpa3,chpa4}, which are inspired by the solutions spaces based on space-time norms introduced by Klainerman and Machedon in \cite{klma}. We note that i…
In this paper we study the existence and stability of normalized standing waves for the nonlinear Schrödinger equation on a general starlike graph with potentials. Under general assumptions on the graph and the potential, we show the existence of orbitally stable standing waves when the nonlinearity is $L^{2}$-critical and supercritical.
This paper is devoted to the existence of non-trivial bound states of prescribed mass for the mass-supercritical nonlinear Schrödinger equation on compact metric graphs. The investigation is based upon a general variational principle which combines the monotonicity trick and a min-max theorem with second order information, and upon the blow-up analysis of bound states with prescribed mass and bounded Morse index.
