Related papers: Regularity and pointwise convergence for dispersiv…
In this paper, we study the backward problem of determining initial condition for some class of nonlinear parabolic equations in multidimensional domain where data are given under random noise. This problem is ill-posed, i.e., the solution…
This paper is concerned with the Cauchy problem of the quadratic nonlinear Schr\"{o}dinger equation in $\mathbb{R} \times \mathbb{R}^2$ with the nonlinearity $\eta |u|^2$ where $\eta \in \mathbb{C} \setminus \{0\}$ and low regularity…
We study the existence of positive solutions for nonlocal systems in gradient form and set in the whole $\mathbb R^N$. A quasilinear fractional Schr\"odinger equation, where the leading operator is the $\frac Ns$-fractional Laplacian, is…
In the first part of this paper we prove that the flow associated to a dispersive Burgers equation with a non local term of the form $|D|^{\alpha-1} \partial_x u$, $\alpha \in [1,+\infty[$ is Lipschitz from bounded sets of…
We consider the cubic-quintic nonlinear Schr\"odinger equation in two space dimensions. For this model, X. Cheng established scattering for $H^1$ data with mass strictly below that of the ground state for the cubic NLS. Subsequently, R.…
Considered in this report is the one-dimensional fourth-order dispersive cubic nonlinear Schr\"odinger equation with mixed dispersion. Orbital stability, in the energy space, of a particular standing-wave solution is proved in the context…
We investigate the existence of solutions to the fractional nonlinear Schr\"{o}dinger equation $(-\Delta)^s u = f(u)$ with prescribed $L^2$-norm $\int_{\mathbb{R}^N} |u|^2 \, dx =m$ in the Sobolev space $H^s(\mathbb{R}^N)$. Under fairly…
We consider the semilinear equation $$ \epsilon^{2s} (-\Delta)^s u + V(x)u - u^p = 0, \quad u>0, \quad u\in H^{2s}(\R^N) $$ where $0<s<1,\ 1<p<\frac{N+2s}{N-2s}$, $ V(x)$ is a sufficiently smooth potential with $\inf_\R V(x)> 0$, and…
For the first time, a nonlinear Schr\"odinger equation of the general form is considered, depending on time and two spatial variables, the potential and dispersion of which are specified by two arbitrary functions. This equation naturally…
We consider the Schr\"odinger map initial value problem into the sphere in 2+1 dimensions with smooth, decaying, subthreshold initial data. Assuming an a priori $L^4$ boundedness condition on the solution, we prove that the Schr\"odinger…
This paper is concerned with the regularity of solutions to parabolic evolution equations. We consider semilinear problems on non-convex domains. Special attention is paid to the smoothness in the specific scale $B^r_{\tau,\tau}$,…
Let $T_t^{P_2}f(x)$ denote the solution to the linear Schr\"odinger equation at time $t$, with initial value function $f$, where $P_2 (\xi) = |\xi|^2$. In 1980, Carleson asked for the minimal regularity of $f$ that is required for the…
We present a general blow-up technique to obtain local regularity estimates for solutions, and their derivatives, of second order elliptic equations in divergence form in H\"older spaces with variable exponent. The procedure allows to…
Regularity and singularity of the solutions according to the shape of domains is a challenging research theme in the Boltzmann theory (\cite{Kim11,GKTT1}). In this paper, we prove an H\"older regularity in $C^{0,\frac{1}{2}-}_{x,v}$ for the…
This paper is concerned with the regularity of solutions to linear and nonlinear evolution equations extending our findings in [22] to domains of polyhedral type. In particular, we study the smoothness in the specific scale…
This article deals with the numerical analysis of the Cauchy problem for the Korteweg-de Vries equation with a finite difference scheme. We consider the Rusanov scheme for the hyperbolic flux term and a 4-points $\theta$-scheme for the…
The aim of this paper is to study a concentration-compactness principle for inhomogeneous fractional Sobolev space $H^s (\mathbb{R}^N)$ for $0<s\leq N/2.$ As an application we establish Palais-Smale compactness for the Lagrangian associated…
In this paper, we study the following Schr\"odinger equations with potentials and general nonlinearities \begin{equation*} \left\{\begin{aligned} & -\Delta u+V(x)u+\lambda u=|u|^{q-2}u+\beta f(u), \\ & \int |u|^2dx=\Theta, \end{aligned}…
We consider the mass-supercritical, defocusing, nonlinear Schr{\"o}dinger equation. We prove loss of regularity in arbitrarily short times for regularized initial data belonging to a dense set of any fixed Sobolev space for which the…
We consider the cubic nonlinear Schr\"odinger equation with a spatially rough potential, a key equation in the mathematical setup for nonlinear Anderson localization. Our study comprises two main parts: new optimal results on the…