相关论文: Global well-posedness for Schr\"odinger equations …

We consider the Schr\"odinger equations with arbitrary (large) power non-linearity on the three-dimensional torus. We construct non-trivial probability measures supported on Sobolev spaces and show that the equations are globally well-posed…

We show that the Benjamin-Ono equation is globally well-posed in $H^s(\R)$ for $s \geq 1$. This is despite the presence of the derivative in the non-linearity, which causes the solution map to not be uniformly continuous in $H^s$ for any…

Let T be a homogeneous tree and L the Laplace operator on T. We consider the semilinear Schrodinger equation associated to L with a power-like nonlinearity F of degree d. We first obtain dispersive estimates and Strichartz estimates with no…

We prove global well-posedness for the cubic nonlinear Schr\"odinger equation for periodic initial data in the mass-critical dimension $d=2$ for initial data of arbitrary size in the defocusing case and data below the ground state threshold…

In this paper, we introduce a novel first-order derivative for functions on a lattice graph, which extends the discrete Laplacian and generalizes the theory of discrete PDEs on lattices. First, we establish the well-posedness of generalized…

We investigate the local and global well-posedness of the kinetic derivative nonlinear Schr\"odinger equation (KDNLS) on $\mathbb{R}$, described by \[ i\partial_t u + \partial_x^2 u = i\alpha \partial_x (|u|^2 u) + i\beta \partial_x…

We obtain the global well-posedness for Schr\"odinger equations of higher orders in weighted $L^2$ spaces. This is based on weighted $L^2$ Strichartz estimates for the corresponding propagator with higher-order dispersion. Our method is…

Well-posedness and global attractor are established for 2D damped driven nonlinear Schr\"odinger equation with almost periodic pumping in a bounded region. The key role is played by a novel application of the energy equation.

For $s \in (\frac{1}{2},1]$ we investigate well-posedness of the equation \[ \left ( i \partial_t + (-\Delta)^{s} \right ) u = \left (|D|^{1-2s} |u|^2 \right)\ |D|^{2s-1} u \] under small initial data…

We consider the quadratic derivative nonlinear Schr\"odinger equation (dNLS) on the circle. In particular, we develop an infinite iteration scheme of normal form reductions for dNLS. By combining this normal form procedure with the…

In this paper, we consider the defocusing mass-supercritical, energy-subcritical nonlinear Schr\"odinger equation, $$ i\partial_{t}u+\Delta u= |u|^p u, \quad (t,x)\in \mathbb R^{d+1}, $$ with $p\in (\frac4d,\frac4{d-2})$. We prove that…

We study the Cauchy problem for a generalized derivative nonlinear Schr\"odinger equation with the Dirichlet boundary condition. We establish the local well-posedness results in the Sobolev spaces $H^1$ and $H^2$. Solutions are constructed…

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 paper we consider the local well-posedness theory for the quadratic nonlinear Schr\"odinger equation with low regularity initial data in the case when the nonlinearity contains derivatives. We work in 2+1 dimensions and prove a…

We consider the nonlinear Schr\"odinger equation with multiplicative spatial white noise and an arbitrary polynomial nonlinearity on the two-dimensional full space domain. We prove global well-posedness by using a gauge-transform introduced…

We study the Derivative Nonlinear Schr\"odinger equation for general initial conditions in weighted Sobolev spaces that can support bright solitons (but excluding spectral singularities). We prove global well-posedness and give a full…

The local and global well-posedness for the one dimensional fourth-order nonlinear Schr\"odinger equation are established in the modulation space $M^{s}_{2,q}$ for $s\geq \frac12$ and $2\leq q <\infty$. The local result is based on the…

In this paper we study the Cauchy problem for the elliptic and non-elliptic derivative nonlinear Schr\"odinger equations in higher spatial dimensions ($n\geq 2$) and some global well-posedness results with small initial data in critical…

We prove new local and global well-posedness results for the cubic one-dimensional nonlinear Schr\"odinger equation in modulation spaces. Local results are obtained via multilinear interpolation. Global results are proven using conserved…

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…