相关论文: Low-regularity Schr\"{o}dinger maps
We study the initial value problem for the nonlinear Schr\"odinger equation. We will prove that the blow-up of the L^{2}-norm of solutions with suitable initial data. We impose a condition related to the sign of the data but put no…
We formulate the half-wave maps problem with target $S^2$ and prove global regularity in sufficiently high spatial dimensions for a class of small critical data in Besov spaces.
We prove global well-posedness and scattering in $H^1$ for the defocusing nonlinear Schr\"{o}dinger equations \begin{equation*} \begin{cases} &(i\partial_t+\Delta_\g)u=u|u|^{2\sigma}; &u(0)=\phi, \end{cases} \end{equation*} on the…
We investigate nonregular elliptic problems with boundary conditions of higher orders. We prove that these problems are Fredholm on appropriate pairs of inner product H\"ormander spaces that form a two-sided refined Sobolev scale. We also…
The first target of this article is the local well-posedness question for 1D quasilinear Schr\"odinger equations with cubic nonlinearities. The study of this class of problems, in all dimensions, was initiated in pioneering work of…
We show new local $L^p$-smoothing estimates for the Schr\"odinger equation with initial data in modulation spaces via decoupling inequalities. Furthermore, we probe necessary conditions by Knapp-type examples for space-time estimates of…
The purpose of this work is to evidence a pathological set of initial data for which the regularized solutions by convolution experience a norm-inflation mechanism, in arbitrarily short time. The result is in the spirit of the construction…
We consider the Schr\"odinger-Debye system in $\mathbb{R}^n$, for $n=3,4$. Developing on previously known local well-posedness results, we start by establishing global well-posedness in $H^1(\mathbb{R}^3)\times L^2(\mathbb{R}^3)$ for a…
In the prototypical setting of non-Euclidean geometry, the 2-dimensional Real Hyperbolic space $\mathbb{H}^2$, we consider the Carleson's problem for the Schr\"odinger equation and improve the best known result until now by proving that the…
We consider short-range mass-subcritical nonlinear Schr\"odinger equations and we show that the corresponding solutions with initial data in $\Sigma$ scatter in $H^1$. Hence we up-grade the classical scattering result proved by Yajima and…
We address the existence of global solutions to the initial value problem for the integrable nonlocal derivative nonlinear Schr\"{o}dinger equation in weighted Sobolev space $H^{2}(\mathbb{R})\cap H^{1,1}(\mathbb{R})$. The key to prove this…
In this paper, we study local well-posedness for the Navier-Stokes \linebreak equations with arbitrary initial data in homogeneous Sobolev spaces $\dot{H}^s_p(\mathbb{R}^d)$ for $d \geq 2, p > \frac{d}{2},\ {\rm and}\ \frac{d}{p} - 1 \leq s…
We consider semilinear Schr\"odinger equations with nonlinearity that is a polynomial in the unknown function and its complex conjugate, on $\mathbb{R}^d$ or on the torus. Norm inflation (ill-posedness) of the associated initial value…
The existence of local unique mild solutions to the Navier-Stokes equations in the whole space with an initial tempered distribution datum in critical homogeneous or inhomogeneous Sobolev spaces is shown. Especially, the case when the…
We prove the local well-posedness of the initial boundary value problem for the nonlinear quadratic Schr\"odinger equation under low initial-boundary regularity assumption via the boundary integral operator method introduced by…
In this paper we consider the Schr\"odinger equation with power-like nonlinearity and confining potential or without potential. This equation is known to be well-posed with data in a Sobolev space $\H^{s}$ if $s$ is large enough and…
This paper is dedicated to the study of the derivative nonlinear Schr\"odinger equation on the real line. The local well-posedness of this equation in the Sobolev spaces is well understood since a couple of decades, while the global…
We consider optimization problems in the fractional order Sobolev spaces $H^s(\Omega)$, $s\in (0,1)$, with sparsity promoting objective functionals containing $L^p$-pseudonorms, $p\in (0,1)$. Existence of solutions is proven. By means of a…
We prove local well-posedness results for the semi-linear wave equation for data in $H^\gamma$, $0 < \gamma < \frac{n-3}{2(n-1)}$, extending the previously known results for this problem. The improvement comes from an introduction of a…
In this work we consider the initial value problem (IVP) associated to the two dimensional Zakharov-Kuznetsov equation $$\left. \begin{array}{rl} u_t+\partial_x^3 u+\partial_x \partial_y^2 u +u \partial_x u &\hspace{-2mm}=0,\qquad\qquad…