Xiaoyi Zhang

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

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.

Minimal-mass blowup solutions of the mass-critical NLS

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.

The Nonlinear Schrödinger Equation with Combined Power-Type Nonlinearities

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).

Remarks of global wellposedness of liquid crystal flows and heat flows of harmonic maps in two dimensions

We consider the Cauchy problem to the two-dimensional incompressible liquid crystal equation and the heat flows of harmonic maps equation. Under a natural geometric angle condition, we give a new proof of the global well-posedness of smooth solutions for a class of large initial data in energy space. This result was originally obtained by Ding-Lin in \cite{DingLin} and Lin-Lin-Wang in \cite{LinLinWang}. Our main technical tool is a rigidity theorem which gives the coercivity of the harmonic energy under certain angle condition. Our proof is based on a frequency localization argument combined with the concentration-compactness approach which can be of independent interest.

Exploding solutions for a nonlocal quadratic evolution problem

We consider a nonlinear parabolic equation with fractional diffusion which arises from modelling chemotaxis in bacteria. We prove the wellposedness, continuation criteria, and smoothness of local solutions. In the repulsive case we prove global wellposedness in Sobolev spaces. Finally in the attractive case, we prove that for a class of smooth initial data the L-x(infinity)-norm of the corresponding solution blows up in finite time. This solves a problem left open by Biler and Woyczynski [8].

Energy-Critical NLS with Quadratic Potentials

We consider the defocusing -critical nonlinear Schrödinger equation in all dimensions (n ≥ 3) with a quadratic potential . We show global well-posedness for radial initial data obeying ∇u 0(x), xu 0(x) ∈ L 2. In view of the potential V, this is the natural energy space. In the repulsive case, we also prove scattering. We follow the approach pioneered by Bourgain and Tao in the case of no potential; indeed, we include a proof of their results that incorporates a couple of simplifications discovered while treating the problem with quadratic potential.

Stability and Unconditional Uniqueness of Solutions for Energy Critical Wave Equations in High Dimensions

In this paper we establish a complete local theory for the energy-critical nonlinear wave equation (NLW) in high dimensions ℝ × ℝ d with d ≥ 6. We prove the stability of solutions under the weak condition that the perturbation of the linear flow is small in certain space-time norms. As a by-product of our stability analysis, we also prove local well-posedness of solutions for which we only assume the smallness of the linear evolution. These results provide essential technical tools that can be applied towards obtaining the extension to high dimensions of the analysis of Kenig and Merle [17] of the dynamics of the focusing (NLW) below the energy threshold. By employing refined paraproduct estimates we also prove unconditional uniqueness of solutions for d ≥ 6 in the natural energy class. This extends an earlier result by Planchon [26].

On the Blowup for the

We consider the focusing mass‐critical nonlinear Schrodinger equation and prove that blowup solutions to this equation with initial data in

L^2

H^s(\R^d)

‐Critical Focusing Nonlinear Schrödinger Equation in Higher Dimensions below the Energy Class

,