Related papers: Global Well-posedness for the Fourth-order Nonline…

We prove that, the initial value problem associated to u_{t} + i\alphau_{xx} + \beta u_{xxx} + i\gamma |u|^{2}u = 0, x,t \in R, is locally well-posed in Sobolev spaces H^{s} for s>-1/4.

In this paper, we study local well-posedness theory of the Cauchy problem for Schr\"{o}dinger-KdV system in Sobolev spaces $H^{s_1}\times H^{s_2}$. We obtain the local well-posedness when $s_1\geq 0$, $\max\{-3/4,s_1-3\}\leq s_2\leq…

In this paper we prove global well - posedness and scattering for the focusing, energy - critical nonlinear Schr\"odinger initial value problem in four dimensions. Previous work proved this in five dimensions and higher using the double…

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 study the stochastic complex Ginzburg-Landau equation (SCGL) with an additive space-time white noise forcing on the two-dimensional torus. This equation is singular and thus we need to renormalize the nonlinearity in order to give proper…

We consider the cubic nonlinear fourth-order Schr\"odinger equation \[ i\partial_t u - \Delta^2 u + \mu \Delta u = \pm |u|^2 u, \quad \mu \geq 0 \] on $\mathbb{R}^N, N \geq 5$ with random initial data. We prove almost sure local…

In this paper, we develop the well-posedness theory and uncover the noise-regularization effect on scattering for the stochastic Zakharov system in dimensions $d \geq 4$ and beyond the energy space. Our focus is particularly directed at the…

In this paper we present a method to study global regularity properties of solutions of large-data critical Schrodinger equations on certain noncompact Riemannian manifolds. We rely on concentration compactness arguments and a global…

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.

This paper is the first part of a trilogy dedicated to a proof of global well-posedness and scattering of the (4+1)-dimensional mass-less Maxwell-Klein-Gordon equation (MKG) for any finite energy initial data. The main result of the present…

We continue our study for the stochastic defocusing mass crtical nonlinear Schr\"odinger equation with conservative multiplicative noise, and show that it is globally well-posed for arbitrary initial data in $L_{\omega}^{\infty}L_{x}^{2}$.…

In this paper we discuss global well - posedness and scattering for some initial value problems that are $L^{2}$ supercritical and $\dot{H}^{1}$ subcritical, with radial data. We prove global well - posedness and scattering for radial data…

We consider the two-dimensional nonlinear Schr\"odinger equation with point interaction and we establish a local well-posedness theory, including blow-up alternative and continuous dependence on the initial data in the energy space. We…

We show the global well-posedness for the two-dimensional Zakharov-Kuznetsov equation in $H^{s}({\mathbb{R}^2})$ when $\frac{11}{13}<s<1$ via the I-method. Additionally, local well-posedness for the symmetrized ZK equation in $…

This article constitutes the final and main part of a three-paper sequence, whose goal is to prove global well-posedness and scattering of the energy critical Maxwell-Klein-Gordon equation (MKG) on $\mathbb{R}^{1+4}$ for arbitrary finite…

We prove global wellposedness and scattering for the Mass-critical homogeneous fourth-order Schrodinger equation in high dimensions n>4, for general L^2 initial data in the defocusing case, and for general initial data with Mass less than…

We prove global well-posedness of the short-pulse equation with small initial data in Sobolev space $H^2$. Our analysis relies on local well-posedness results of Sch\"afer & Wayne, the correspondence of the short-pulse equation to the…

We consider a nonlinear fourth order in space partial differential equation arising in the context of the modeling of nonlinear acoustic wave propagation in thermally relaxing viscous fluids. We use the theory of operator semigroups in…

In this paper, we prove that there exists some small $\varepsilon_*>0$, such that the derivative nonlinear Schr\"{o}dinger equation (DNLS) is global well-posedness in the energy space, provided that the initial data $u_0\in H^1(\mathbb{R})$…

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…