Related papers: Sharp local existence and nonlinear smoothing for …
In this paper we are concerned with the well-posedness and the exponential stabilization of the generalized Korteweg-de Vries Burgers equation, posed on the whole real line, under the effect of a damping term. Both problems are investigated…
We consider the cubic nonlinear Schr\"odinger equation (NLS) on $\mathbb{R}^3$ with randomized initial data. In particular, we study an iterative approach based on a partial power series expansion in terms of the random initial data. By…
We study the Cauchy problem for the cubic fractional nonlinear Schr\"odinger equation (fNLS) on the real line and on the circle. In particular, we prove global well-posedness of the cubic fNLS with all orders of dispersion higher than the…
In this paper, we prove global well-posedness and scattering for the defocusing, cubic nonlinear Schr{\"o}dinger equation in three dimensions when $n = 3$ when $u_{0} \in H^{s}(\mathbf{R}^{3})$, $s > 3/4$. To this end, we utilize a…
This article is a survey on the local well-posedness problem for the general EPDiff equation. The main contribution concerns recent results on local existence of the geodesics on $\mathrm{Diff}(\mathbb{T}^{d})$ and…
The possibility of finite-time, dispersive blow up for nonlinear equations of Schroedinger type is revisited. This mathematical phenomena is one of the possible explanations for oceanic and optical rogue waves. In dimension one, the…
In this paper, the global well-posedness of semirelativistic equations with a power type nonlinearity on Euclidean spaces is studied. In two dimensional $H^s$ scaling subcritical case with $1 \leq s \leq 2$, the local well-posedness follows…
We consider the Cauchy problem for dispersion managed nonlinear Schroedinger equations, where the dispersion map is assumed to be periodic and piecewise constant in time. We establish local and global well-posedness results and 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 provide a roadmap to establish improved lower bounds on the decay rate of the uniform radius of analyticity $\sigma(T)$ for a given nonlinear dispersive equation, reducing the problem to the derivation of nonlinear smoothing estimates…
We prove the local-in-time well-posedness and the mass and energy conservation laws for a 3d cubic nonlinear Schroedinger equation with a real-valued potential.
We derive sharp Sobolev embeddings on a class of Sobolev spaces with potential weights without assuming any boundary conditions. Moreover, we consider the Adams-type inequalities for the borderline Sobolev embedding into the exponential…
In this paper, we prove the global well-posedness of defocusing 3D quadratic nonlinear Schr\"odinger equation \begin{align*} i\partial_t u + \frac12\Delta u = |u| u, \end{align*} in its sharp critical weighted space $\mathcal F \dot…
We consider the Cauchy problem for the one-dimensional periodic cubic nonlinear Schr\"odinger equation (NLS) with initial data below L^2. In particular, we exhibit nonlinear smoothing when the initial data are randomized. Then, we prove…
In this paper we study the Cauchy problem associated to the Maxwell-Schr\"odinger system with a defocusing pure-power non-linearity. This system has many applications in physics, for instance in the description of a charged non-relativistic…
We establish well-posedness theory for the 1D mass-subcritical nonlinear Schr\"odinger equation (NLS) having power-type nonlinearity $|u|^{\alpha-1}u$ in a certain modulation spaces $M^{p,p'}(\mathbb{R}),$ where $p'$ is a H\"older conjugate…
We consider the nonlinear Schrodinger equation with a modified spatial dispersion, given either by an homogeneous Fourier multiplier, or by a bounded Fourier multiplier. Arguments based on ordinary differential equations yield ill-posedness…
Relevant physical phenomena are described by nonlinear Schr\"odinger equations with non-vanishing conditions at infinity. This paper investigates the respective 2D and 3D Cauchy problems. Local well-posedness in the energy space for…
We prove the local boundedness and the local H\"older continuity of weak solutions to nonlocal equations with variable orders and exponents under sharp assumptions.
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…