@article{22035, abstract = {We prove that solutions of the cubic nonlinear Schr\"odinger equation on $\Bbb{R}^2$ can be approximated by a finite-dimensional Hamiltonian system, uniformly on bounded sets of initial data. This is despite the wealth of non-compact symmetries: scaling, translation, and Galilei boosts. Complementing this approximation result, we show that all solutions of the finite-dimensional Hamiltonian system we use can be approximated by the full PDE. A key ingredient in these results is the development of a general methodology for transfering uniform global space-time bounds to suitable Fourier truncations of dispersive PDE models. As an application, we prove symplectic non-squeezing (in the sense of Gromov) for the cubic NLS on $\Bbb{R}^2$. This is the first symplectic non-squeezing result for a Hamiltonian PDE in infinite volume. It is also the first unconditional symplectic non-squeezing result in a scaling-critical setting. Finally, we discuss implications of non-squeezing on the nature of scattering.}, author = {Killip, Rowan and Visan, Monica and Zhang, Xiaoyi}, issn = {1080-6377}, journal = {American Journal of Mathematics}, number = {2}, pages = {613--680}, publisher = {Johns Hopkins University Press}, title = {{Finite-dimensional approximation and non-squeezing for the cubic nonlinear Schrödinger equation on ℝ2}}, doi = {10.1353/ajm.2021.0014}, volume = {143}, year = {2021}, } @article{22087, abstract = {We prove that solutions of the cubic nonlinear Schrödinger equation on $\Bbb{R}^2$ can be approximated by a finite-dimensional Hamiltonian system, uniformly on bounded sets of initial data. This is despite the wealth of non-compact symmetries: scaling, translation, and Galilei boosts. Complementing this approximation result, we show that all solutions of the finite-dimensional Hamiltonian system we use can be approximated by the full PDE. A key ingredient in these results is the development of a general methodology for transfering uniform global space-time bounds to suitable Fourier truncations of dispersive PDE models. As an application, we prove symplectic non-squeezing (in the sense of Gromov) for the cubic NLS on $\Bbb{R}^2$. This is the first symplectic non-squeezing result for a Hamiltonian PDE in infinite volume. It is also the first unconditional symplectic non-squeezing result in a scaling-critical setting. Finally, we discuss implications of non-squeezing on the nature of scattering.}, author = {Killip, Rowan and Visan, Monica and Zhang, Xiaoyi}, issn = {1080-6377}, journal = {American Journal of Mathematics}, number = {2}, pages = {613--680}, publisher = {Johns Hopkins University Press}, title = {{Finite-dimensional approximation and non-squeezing for the cubic nonlinear Schrödinger equation on ℝ2}}, doi = {10.1353/ajm.2021.0014}, volume = {143}, year = {2021}, } @article{22041, abstract = {We consider the defocusing energy-critical nonlinear Schr\"odinger equation in the exterior of a smooth compact strictly convex obstacle in three dimensions. For the initial-value problem with Dirichlet boundary condition we prove global well-posedness and scattering for all initial data in the energy space.}, author = {Killip, Rowan and Visan, Monica and Zhang, Xiaoyi}, issn = {1080-6377}, journal = {American Journal of Mathematics}, number = {5}, pages = {1193--1346}, publisher = {Johns Hopkins University Press}, title = {{Quintic NLS in the exterior of a strictly convex obstacle}}, doi = {10.1353/ajm.2016.0039}, volume = {138}, year = {2016}, } @article{22023, abstract = {We consider the focusing energy-critical nonlinear Schrödinger equation iut + ∆u = −|u| 4 d−2 u in dimensions d ≥ 5. We prove that if a maximal-lifespan solution u : I × Rd → C obeys supt∈I k∇u(t)k2 < k∇Wk2, then it is global and scatters both forward and backward in time. Here W denotes the ground state, which is a stationary solution of the equation. In particular, if a solution has both energy and kinetic energy less than those of the ground state W at some point in time, then the solution is global and scatters. We also show that any solution that blows up with bounded kinetic energy must concentrate at least the kinetic energy of the ground state. Similar results were obtained by Kenig and Merle in [17, 18] for spherically symmetric initial data and dimensions d = 3, 4, 5.}, author = {Killip, Rowan and Visan, Monica}, issn = {1080-6377}, journal = {American Journal of Mathematics}, number = {2}, pages = {361--424}, publisher = {Johns Hopkins University Press}, title = {{The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher}}, doi = {10.1353/ajm.0.0107}, volume = {132}, year = {2010}, } @article{22089, abstract = {We consider the focusing energy-critical nonlinear Schr\"odinger equation $iu_t+\Delta u = - |u|^{4\over{d-2}}u$ in dimensions $d\geq 5$. We prove that if a maximal-lifespan solution $u\colon \ I\times {\Bbb R}^d\to {\Bbb C}$ obeys $\sup_{t\in I}\|\nabla u(t)\|_2<\|\nabla W\|_2$, then it is global and scatters both forward and backward in time. Here $W$ denotes the ground state, which is a stationary solution of the equation. In particular, if a solution has both energy and kinetic energy less than those of the ground state $W$ at some point in time, then the solution is global and scatters. We also show that any solution that blows up with bounded kinetic energy must concentrate at least the kinetic energy of the ground state. Similar results were obtained by Kenig and Merle for spherically symmetric initial data and dimensions $d=3,4,5$. }, author = {Killip, Rowan and Visan, Monica}, issn = {1080-6377}, journal = {American Journal of Mathematics}, number = {2}, pages = {361--424}, publisher = {Johns Hopkins University Press}, title = {{The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher}}, doi = {10.1353/ajm.0.0107}, volume = {132}, year = {2010}, } @article{22057, abstract = { We obtain global well-posedness, scattering, uniform regularity, and global L6t,x spacetime bounds for energy-space solutions to the defocusing energy-critical nonlinear Schrödinger equation in ℝ×ℝ4. Our arguments closely follow those of Colliander, Keel, et al., though our derivation of the frequency-localized interaction Morawetz estimate is somewhat simpler. As a consequence, our method yields a better bound on the L6t,x-norm. }, author = {Ryckman, E and Visan, Monica}, issn = {1080-6377}, journal = {American Journal of Mathematics}, number = {1}, pages = {1--60}, publisher = {Johns Hopkins University Press}, title = {{Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in R 1+4}}, doi = {10.1353/ajm.2007.0004}, volume = {129}, year = {2007}, }