Monica Visan

The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions

We obtain global well-posedness, scattering, and global

Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions

L^{\frac{2(n+2)}{n-2}}_{t,x}

Minimal-mass blowup solutions of the mass-critical NLS

spacetime bounds for energy-space solutions to the energy-critical nonlinear Schrodinger equation in

The Nonlinear Schrödinger Equation with Combined Power-Type Nonlinearities

\R_t\times \R^n_x

THE DEFOCUSING ENERGY-SUPERCRITICAL NONLINEAR WAVE EQUATION IN THREE SPACE DIMENSIONS

,

The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions

n\geq 5

A Counterexample to Dispersive Estimates for Schrödinger Operators in Higher Dimensions

.

Energy-Supercritical NLS: Critical [Hdot] s -Bounds Imply Scattering

We establish global well-posedness and scattering for solutions to the defocusing mass-critical (pseudoconformal) nonlinear Schrodinger equation iut+ u = |u|4/nu for large, spherically symmetric, Lx(R ) initial data in dimensions n ≥ 3. After using the concentration-compactness reductions in [32] to reduce to eliminating blow-up solutions that are almost periodic modulo scaling, we obtain a frequency-localized Morawetz estimate and exclude a mass evacuation scenario (somewhat analogously to [10], [23], [36]) in order to conclude the argument.

Energy-Critical NLS with Quadratic Potentials

Abstract We consider the minimal mass m 0 required for solutions to the mass-critical nonlinear Schrödinger (NLS) equation iut + 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 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, [Keraani S.: On the blow-up phenomenon of the critical nonlinear Schrödinger equation. J. Funct. Anal. 235 (2006), 171–192], in dimensions 1, 2 and Begout and Vargas, [Begout P., Vargas A.: Mass concentration phenomena for the L 2-critical nonlinear Schrödinger equation, preprint], in dimensions d ≥ 3 for the mass-critical NLS and by Kenig and Merle, [Kenig C., Merle F.: Global well-posedness, scattering, and blowup for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, preprint], in the energy-critical case. In a subsequent paper we shall use this compactness result to establish global existence and scattering in for the defocusing NLS in three and higher dimensions with spherically symmetric data.

On the Blowup for the

We undertake a comprehensive study of the nonlinear Schrödinger 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 Schrödinger 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 Schrödinger equation with , which was first obtained by Ginibre and Velo (1985).