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Journal ArticleDOI

Profiles and Quantization of the Blow Up Mass for Critical Nonlinear Schrödinger Equation

TL;DR: In this paper, the authors consider finite time blow up solutions to the critical nonlinear Schrodinger equation, and prove that the solution splits in two parts: the first part corresponds to the singular part and accumulates a quantized amount of L2 mass at the blow up point, the second part correspond to the regular part and has a strong L2 limit at blow up time.
Abstract: We consider finite time blow up solutions to the critical nonlinear Schrodinger equation Open image in new window For a suitable class of initial data in the energy space H1, we prove that the solution splits in two parts: the first part corresponds to the singular part and accumulates a quantized amount of L2 mass at the blow up point, the second part corresponds to the regular part and has a strong L2 limit at blow up time.
Citations
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Journal ArticleDOI
TL;DR: In this paper, the cubic defocusing nonlinear Schrodinger equa- tion on the two dimensional torus is considered and smooth solutions for which the support of the conserved energy moves to higher Fourier modes are presented.
Abstract: We consider the cubic defocusing nonlinear Schrodinger equa- tion on the two dimensional torus We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes This behavior is quantified by the growth of higher Sobolev norms: given any δ � 1 ,K � 1 ,s >1, we construct smooth initial data u0 withu0� H s K at some time T This growth occurs despite the Hamiltonian's bound onu(t)� ˙ H 1 and despite the conservation of the quantityu(t)� L2

299 citations

Journal ArticleDOI
TL;DR: In this article, the critical nonlinear Schrodinger equation with initial condition u(0, x) = u0 was considered and the initial condition was obtained for the case where x = 0.
Abstract: We consider the critical nonlinear Schrodinger equation $iu_{t} = -\Delta u-|u|^{4/N}$ with initial condition u(0, x) = u0

297 citations

Journal ArticleDOI
TL;DR: In this paper, a comprehensive study of the nonlinear Schrodinger equation where u(t, x) is a complex-valued function in spacetime, λ1 and λ2 are nonzero real constants, and the authors address questions related to local and global wellposedness, finite time blowup, and asymptotic behaviour.
Abstract: We undertake a comprehensive study of the nonlinear Schrodinger equation where u(t, x) is a complex-valued function in spacetime , λ1 and λ2 are nonzero real constants, and . We address questions related to local and global well-posedness, finite time blowup, and asymptotic behaviour. Scattering is considered both in the energy space H 1(ℝ n ) and in the pseudoconformal space Σ := {f ∈ H 1(ℝ n ); xf ∈ L 2(ℝ n )}. Of particular interest is the case when both nonlinearities are defocusing and correspond to the -critical, respectively -critical NLS, that is, λ1, λ2 > 0 and , . The results at the endpoint are conditional on a conjectured global existence and spacetime estimate for the -critical nonlinear Schrodinger equation, which has been verified in dimensions n ≥ 2 for radial data in Tao et al. (Tao et al. to appear a,b) and Killip et al. (preprint). As an off-shoot of our analysis, we also obtain a new, simpler proof of scattering in for solutions to the nonlinear Schrodinger equation with , which was fi...

216 citations

Journal ArticleDOI
TL;DR: In this paper, stable finite time blow up regimes for the energy critical co-rotational Wave Map with the S 2 target in all homotopy classes and for the critical equivariant SO(4) Yang-Mills problem were derived.
Abstract: We exhibit stable finite time blow up regimes for the energy critical co-rotational Wave Map with the S 2 target in all homotopy classes and for the critical equivariant SO(4) Yang-Mills problem. We derive sharp asymptotics on the dynamics at blow up time and prove quantization of the energy focused at the singularity.

197 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that the unique radial positive stationary solution of the focusing wave equation is the sum of a rescaled W concentrating at the origin and a small remainder which is continuous with respect to the time variable in the energy space.
Abstract: Consider the energy-critical focusing wave equation on the Euclidian space. A blow-up type II solution of this equation is a solution which has finite time of existence but stays bounded in the energy space. The aim of this work is to exhibit universal properties of such solutions. Let W be the unique radial positive stationary solution of the equation. Our main result is that in dimension 3, under an appropriate smallness assumption, any type II blow-up radial solution is essentially the sum of a rescaled W concentrating at the origin and a small remainder which is continuous with respect to the time variable in the energy space. This is coherent with the solutions constructed by Krieger, Schlag and Tataru. One ingredient of our proof is that the unique radial solution which is compact up to scaling is equal to W up to symmetries.

186 citations

References
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Journal ArticleDOI
TL;DR: In this article, a constrained minimization method was proposed for the case of dimension N = 1 (Necessary and sufficient conditions) for the zero mass case, where N is the number of dimensions in the dimension N.
Abstract: 1. The Main Result; Examples . . . . . . . . . . . . . . . . . . . . . . . 316 2. Necessary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 319 3. The Constrained Minimization Method . . . . . . . . . . . . . . . . . . 323 4. Further Properties of the Solution . . . . . . . . . . . . . . . . . . . . 328 5. The \"Zero Mass\" Case . . . . . . . . . . . . . . . . . . . . . . . . . 332 6. The Case of Dimension N = 1 (Necessary and Sufficient Conditions) . . . . . 335 Appendix. Technical Results . . . . . . . . . . . . . . . . . . . . . . . . 338

2,385 citations


"Profiles and Quantization of the Bl..." refers background in this paper

  • ...(4) Equation (4) is a standard nonlinear elliptic equation, and from [1] and [6], there is a unique positive solution up to translation Qω(x)....

    [...]

  • ...Let ( x−x(t) R(t0) ) ∈ Supp(ψ) ⊂ [1, 2], then:∣∣∣∣x− x(t)R(t0) ∣∣∣∣ ≥ 1 implies ∣∣∣∣x− x(t)λ(t) ∣∣∣∣ = ∣∣∣∣x− x(t)R(t0) ∣∣∣∣ R(t0)λ(t) ≥ 100b(t) ....

    [...]

  • ...Let ( x−x(t) R(t0) ) ∈ Supp(ψ) ⊂ [1, 2], then: ∣∣∣∣x− x(t) R(t0) ∣∣∣∣ ≥ 1 implies ∣∣∣∣x− x(t) λ(t) ∣∣∣∣ = ∣∣∣∣x− x(t) R(t0) ∣∣∣∣ R(t0) λ(t) ≥ 100 b(t) ....

    [...]

  • ...(78) Remark 9 From now on, parameter η is fixed small enough so that above estimates hold, and so is a = √ η. Proof of Lemma 4 We first claim from support property of ψ: ∀t ∈ [t0, T ), QS(t, x) = 0 ie u(t, x) = ũ(t, x) for x− x(t) R(t0) ∈ Supp(ψ)....

    [...]

BookDOI
01 Jan 2004
TL;DR: In this article, the authors present a basic framework to understand structural properties and long-time behavior of standing wave solutions and their relationship to a mean field generation and acoustic wave coupling.
Abstract: Basic Framework.- The Physical Context.- Structural Properties.- Rigorous Theory.- Existence and Long-Time Behavior.- Standing Wave Solutions.- Blowup Solutions.- Asymptotic Analysis near Collapse.- Numerical Observations.- Supercritical Collapse.- Critical Collapse.- Perturbations of Focusing NLS.- Coupling to a Mean Field.- Mean Field Generation.- Gravity-Capillary Surface Waves.- The Davey-Stewartson System.- Coupling to Acoustic Waves.- Langmuir Oscillations.- The Scalar Model.- Progressive Waves in Plasmas.

1,658 citations


"Profiles and Quantization of the Bl..." refers background or result in this paper

  • ...The behavior in R of the right-hand side is a consequence of (74), or more specifically as exhibited in [14] by the fact that in the region |y |≤ A(t), a good approximation of the solution e(t) is the universal radiation ζb(t) of Lemma 1. From (75), we now may estimate the size of the region in space for which this approximation is good, which will turn out to be much smaller than the one suggested at a formal level in [ 19 ]....

    [...]

  • ...• On the other hand, numerical simulations, [7], and formal arguments, [ 19 ], suggest...

    [...]

Journal ArticleDOI
TL;DR: In this paper, the existence of standing wave solutions of nonlinear Schrodinger equations was studied and sufficient conditions for nontrivial solutionsu ∈W¯¯¯¯1,2(ℝ�姫 n ) were established.
Abstract: This paper concerns the existence of standing wave solutions of nonlinear Schrodinger equations. Making a standing wave ansatz reduces the problem to that of studying the semilinear elliptic equation: (*) $$ - \Delta u + b(x)u = f(x, u), x \in \mathbb{R}^n .$$ The functionf is assumed to be “superlinear”. A special case is the power nonlinearityf(x, z)=∥z∥ s−1 z where 1

1,467 citations

Journal ArticleDOI
TL;DR: In this article, the uniqueness of the positive, radially symmetric solution to the differential equation Δu−u+up=0 (with p>1) in a bounded or unbounded annular region in Rn for all n ≥ 1, with the Neumann boundary condition on the inner ball and the Dirichlet boundary condition decaying to zero in the case of an unbounded region, was established.
Abstract: We establish the uniqueness of the positive, radially symmetric solution to the differential equation Δu−u+up=0 (with p>1) in a bounded or unbounded annular region in Rn for all n≧1, with the Neumann boundary condition on the inner ball and the Dirichlet boundary condition on the outer ball (to be interpreted as decaying to zero in the case of an unbounded region). The regions we are interested in include, in particular, the cases of a ball, the exterior of a ball, and the whole space. For p=3 and n=3, this a well-known result of Coffman, which was later extended by McLeod & Serrin to general n and all values of p below a certain bound depending on n. Our result shows that such a bound on p is not needed. The basic approach used in this work is that of Coffman, but several of the principal steps in the proof are carried out with the help of Sturm's oscillation theory for linear second-order differential equations. Elementary topological arguments are widely used in the study.

1,338 citations

Journal ArticleDOI
TL;DR: In this paper, a sharp sufficient condition for global existence for the nonlinear Schrodinger equation is obtained for the case σ = 2/N. This condition is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of Nirenberg and Gagliardo.
Abstract: A sharp sufficient condition for global existence is obtained for the nonlinear Schrodinger equation $$\begin{array}{*{20}c} {(NLS)} & {2i\phi _t + \Delta \phi + \left| \phi \right|^{2\sigma } \phi = 0,} & {x \in \mathbb{R}^N } & {t \in \mathbb{R}^ + } \\ \end{array} $$ in the case σ=2/N. This condition is in terms of an exact stationary solution (nonlinear ground state) of (NLS). It is derived by solving a variational problem to obtain the “best constant” for classical interpolation estimates of Nirenberg and Gagliardo.

1,255 citations