相关论文: Global regularity for the energy-critical NLS on $…

We develop the existence, uniqueness, continuity, stability, and scattering theory for energy-critical nonlinear Schr\"odinger equations in dimensions $n \geq 3$, for solutions which have large, but finite, energy and large, but finite,…

We consider the Gross--Pitaevskii equation on $\R^4$ and the cubic-quintic nonlinear Schr\"odinger equation (NLS) on $\R^3$ with non-vanishing boundary conditions at spatial infinity. By viewing these equations as perturbations to the…

We study the defocusing energy-critical inhomogeneous nonlinear Schr\"odinger equation \[ i\partial_tu+\Delta u=|x|^{-b}|u|^{\frac{4-2b}{d-2}}u, \qquad (t,x)\in\R\times\R^d, \] with initial data $u_0\in\dot H_x^1(\R^d)$, where $d\ge 3$ and…

We study the nonlinear Schr\"odinger equation (NLS) with bounded initial data which does not vanish at infinity. Examples include periodic, quasi-periodic and random initial data. On the lattice we prove that solutions are polynomially…

We consider a Schr{\"o}dinger equation with a nonlinearity which is a general perturbation of a power'' nonlinearity. We construct a profile decomposition adapted to this nonlinearity.We also prove global existence and scattering in a…

We consider the Cauchy problem for the defocusing nonlinear Schr\"odinger equations (NLS) on the real line with a special subclass of almost periodic functions as initial data. In particular, we prove global existence of solutions to NLS…

The energy-critical defocusing nonlinear Schr\"odinger equation on 3-dimensional rectangular tori is considered. We prove that the global well-posedness result for the standard torus of Ionescu and Pausader extends to this class of…

In this article, we prove the existence of global weak solutions to the three-dimensional focusing energy-critical nonlinear Schr\"odinger (NLS) equation in the non-radial case. Furthermore, we prove the weak-strong uniqueness for some…

We obtain global well-posedness, scattering, and global $L^{\frac{2(n+2)}{n-2}}_{t,x}$ spacetime bounds for energy-space solutions to the energy-critical nonlinear Schr\"odinger equation in $\R_t\times \R^n_x$, $n\geq 5$.

We present numerical simulations of the defocusing nonlinear Schrodinger (NLS) equation with an energy supercritical nonlinearity. These computations were motivated by recent works of Kenig-Merle and Kilip-Visan who considered some energy…

We prove almost sure global existence and scattering for the energy-critical nonlinear Schr\"odinger equation with randomized spherically symmetric initial data in $H^s(\mathbb{R}^4)$ with $\frac56<s<1$. We were inspired to consider this…

A refined trilinear Strichartz estimate for solutions to the Schr\"odinger equation on the flat rational torus T^3 is derived. By a suitable modification of critical function space theory this is applied to prove a small data global…

We consider two classes of defocusing energy-supercritical nonlinear Schr\"odinger equations in dimensions $d\geq 5$. We prove that if the solution $u$ is apriorily bounded in the critical Sobolev space, that is, $u\in L_t^\infty \dot…

We consider the supercritical inhomogeneous nonlinear Schr\"odinger equation (INLS) $$i\partial_t u+\Delta u+|x|^{-b}|u|^{2\sigma}u=0,$$ where $(2-b)/N<\sigma<(2-b)/(N-2)$ and $0<b<\min\{2,N\}$. We prove a Gagliardo-Nirenberg type estimate…

We prove global existence and stability of solution to the mass-critical stochastic nonlinear Schr\"odinger equation in $d=1$ at $L^2$ regularity. Our construction starts with the existence of solution to the truncated subcritical problem.…

We prove global wellposedness in the energy space of the defocusing cubic nonlinear Schroedinger and Gross-Pitaevskii equations on the exterior of a non-trapping domain in dimension 3. The main ingredient is a Strichartz estimate obtained…

We investigate the Cauchy problem for the nonlinear Schr\"odinger equation with a time-dependent linear damping term. Under non standard assumptions on the loss dissipation, we prove the blow-up in the inter-critical regime, and the global…

We consider the mass-subcritical nonlinear Schr\"odinger equation in all space dimensions with focusing or defocusing nonlinearity. For such equations with critical regularity $s_c\in(\max\{-1,-\frac{d}{2}\},0)$, we prove that any solution…

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…

We consider a class of defocusing energy-supercritical nonlinear Schr\"odinger equations in four space dimensions. Following a concentration-compactness approach, we show that for $1<s_c<3/2$, any solution that remains bounded in the…