Related papers: A priori estimates for the derivative nonlinear Sc…
We introduce a general framework of low regularity integrators which allows us to approximate the time dynamics of a large class of equations, including parabolic and hyperbolic problems, as well as dispersive equations, up to arbitrary…
We prove dispersive estimates for linear Schroedinger equations in two space dimensions. The potential is assumed to be real-valued with some polynomial decay (faster than a negative third power), and zero energy is assumed to be a regular…
In this paper, we continue some investigations on the periodic NLSE started by Lebowitz, Rose and Speer and by Bourgain with the addition of a distributional multiplicative potential. We prove that the equation is globally wellposed for a…
We present analytical results and numerical simulations for a class of nonlinear dispersive equations in two spatial dimensions. These equations are of (derivative) nonlinear Schr\"odinger type and have recently been obtained in \cite{DLS}…
We consider an initial- and boundary- value problem for the nonlinear Schr\"odinger equation with homogeneous Dirichlet boundary conditions in the one space dimension case. We discretize the problem in space by a central finite difference…
In dimensions greater than or equal to 3, we prove that the Schroedinger map initial-value problem is globally well-posed for small data in the critical Besov space.
In this paper we prove that the 1D Schr\"odinger equation with derivative in the nonlinear term is globally well-posed in $H^{s}$, for $s>\frac12$ for data small in $L^{2}$. To understand the strength of this result one should recall that…
We use the scale of Besov spaces B^\alpha_{\tau,\tau}(O), \alpha>0, 1/\tau=\alpha/d+1/p, p fixed, to study the spatial regularity of the solutions of linear parabolic stochastic partial differential equations on bounded Lipschitz domains…
In this paper, we consider the derivative nonlinear Schr\"odinger (DNLS) equation. While the existence theory has been intensely studied, properties like dispersive estimates for the solutions have not yet been investigated. Here we address…
In this paper, we establish pointwise Schauder estimates for solutions of nonlocal fully nonlinear elliptic equations by perturbative arguments. A key ingredient is a recursive Evans-Krylov theorem for nonlocal fully nonlinear translation…
We prove a priori and a posteriori H\"older bounds and Schauder $C^{1,\alpha}$ estimates for continuous solutions of degenerate elliptic equations with variable coefficients of the form $$ \mathrm{div}\left(|u|^a A\nabla…
In this article we prove short time local well-posedness in low-regularity Sobolev spaces for large data general quasilinear Schr\"odinger equations with a non-trapping assumption. These results represent improvements over the small data…
We provide here global Schauder-type estimates for a chain of integro-partial differential equations (IPDE) driven by a degenerate stable Ornstein-Uhlenbeck operator possibly perturbed by a deterministic drift, when the coefficients lie in…
We analyse a class of time discretizations for solving the nonlinear Schr\"odinger equation with non-smooth potential and at low-regularity on an arbitrary Lipschitz domain $\Omega \subset \mathbb{R}^d$, $d \le 3$. We show that these…
We prove some $L^{\infty}$ a priori estimates as well as existence and stability theorems for the weak solutions of the complex Hessian equations in domains of $C^n$ and on compact K\"ahler manifolds. We also show optimal $L^p$…
We introduce a new non-resonant low-regularity integrator for the cubic nonlinear Schr\"odinger equation (NLSE) allowing for long-time error estimates which are optimal in the sense of the underlying PDE. The main idea thereby lies in…
In this article, by applying the well known method for dealing with $p$-Laplace type elliptic boundary value problems, the authors establish a sharp estimate for the decreasing rearrangement of the gradient of solutions to the Dirichlet and…
The method for computation of conditional probability density function for the nonlinear Schr\"odinger equation with additive noise is developed. We present in a constructive form the conditional probability density function in the limit of…
We demonstrate the systematic derivation of a class of discretizations of nonlinear Schr{\"o}dinger (NLS) equations for general polynomial nonlinearity whose stationary solutions can be found from a reduced two-point algebraic condition. We…
For nonlinear Schroedinger equations with a power nonlinearity, a new approach to derive the conservation law of the momentum and the pseudo conformal conservation law is obtained. Since this approach does not contain approximating…