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- ROWAN KILLIP, XIAOYI ZHANG
- 2008

We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schrödinger equation iut +∆u = ±|u| 4/d u for large spherically symmetric L 2 x (R d) initial data in dimensions d ≥ 3. In the focusing case we require that the mass is strictly less than that of the ground state. As a consequence, we obtain that in the focusing… (More)

- XIAOYI ZHANG
- 2006

We consider the minimal mass m 0 required for solutions to the mass-critical nonlinear Schrödinger (NLS) equation iut + ∆u = 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… (More)

- TERENCE TAO, XIAOYI ZHANG
- 2008

We establish global well-posedness and scattering for solutions to the defocusing mass-critical (pseudoconformal) nonlinear Schrödinger equation iut + ∆u = |u| 4/n u for large spherically symmetric L 2 x (R n) initial data in dimensions n ≥ 3. After using the reductions in [32] to reduce to eliminating blowup solutions which are almost periodic modulo… (More)

- XIAOYI ZHANG
- 2005

We undertake a comprehensive study of the nonlinear Schrödinger equation iut + ∆u = λ 1 |u| p 1 u + λ 2 |u| p 2 u, where u(t, x) is a complex-valued function in spacetime Rt × R n x , λ 1 and λ 2 are nonzero real constants, and 0 < p 1 < p 2 ≤ 4 n−2. We address questions related to local and global well-posedness, finite time blowup, and asymptotic… (More)

Let d ≥ 4 and let u be a global solution to the focusing mass-critical nonlinear Schrödinger equation iut + ∆u = −|u| 4 d u with spherically symmetric H 1 x 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… (More)

- ROWAN KILLIP, XIAOYI ZHANG
- 2006

We consider the defocusing ˙ H 1-critical nonlinear Schrödinger equation in all dimensions (n ≥ 3) with a quadratic potential V (x) = ± 1 2 |x| 2. 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… (More)

- DONG LI, XIAOYI ZHANG
- 2008

We consider a nonlocal aggregation equation with non-linear diffusion which arises from the study of biological aggrega-tion dynamics. As a degenerate parabolic problem, we prove the well-posedness, continuation criteria and smoothness of local solutions. For compactly supported nonnegative smooth initial data we prove that the gradient of the solution… (More)

Physical rehabilitation is an important process for patients recovering after surgery. In this paper, we propose and develop a framework to monitor on-bed range of motion exercises that allows physical therapists to evaluate patient adherence to set exercise programs. Using a dense pressure sensitive bedsheet, a sequence of pressure maps are produced and… (More)