相关论文: Local well-posedness for the Maxwell-Schr\"odinger…

The primary objective of this paper is to investigate the well-posedness theories associated with the discrete nonlinear Schr\"odinger equation and Klein-Gordon equation. These theories encompass both local and global well-posedness, as…

The local and global well-posedness for the one dimensional fourth-order nonlinear Schr\"odinger equation are established in the modulation space $M^{s}_{2,q}$ for $s\geq \frac12$ and $2\leq q <\infty$. The local result is based on the…

We study the Derivative Nonlinear Schr\"odinger equation for general initial conditions in weighted Sobolev spaces that can support bright solitons (but excluding spectral singularities). We prove global well-posedness and give a full…

The Cauchy problem for the derivative nonlinear Schr\"odinger equation with periodic boundary condition is considered. Local well-posedness for periodic initial data u_0 in the space ^H^s_r, defined by the norms ||u_0||_{^H^s_r}=||<xi>^s…

This work studies the local well-posedness of the initial-value problem for the nonlinear sixth-order Boussinesq equation $u_{tt}=u_{xx}+\beta u_{xxxx}+u_{xxxxxx}+(u^2)_{xx}$, where $\beta=\pm1$. We prove local well-posedness with initial…

We consider the initial value problem of the fifth order modified KdV equation on the Sobolev spaces. \partial_t u - \partial_x^5u + c_1\partial_x^3(u^3) + c_2u\partial_x u\partial_x^2 u + c_3uu\partial_x^3 u =0, u(x,0)= u_0(x) where $…

We consider the Vlasov--Poisson equation on $\mathbb{R}^n \times \mathbb{R}^n$ with $n \ge 3$. We prove local well-posedness in $H^{s}(\mathbb{R}^n \times \mathbb{R}^n)$ with $s> n/2-1/4$, for initial distribution $f_{0} \in…

We establish a comprehensive local wellposedness theory for the quasilinear Maxwell system with interfaces in the space of piecewise $H^m$-functions for $m \geq 3$. The system is equipped with instantaneous and piecewise regular material…

We prove that the Cauchy problem for the two-dimensional Zakharov system is locally well-posed for initial data which are localized perturbations of a line solitary wave. Furthermore, for this Zakharov system, we prove weak convergence to a…

We prove that the Yang-Mills equations in the Lorenz gauge (YM-LG) is locally well-posed for data below the energy norm, in particular, we can take data for the gauge potential $A$ and the associated curvature $F$ in $H^s\times H^{s-1}$ and…

We introduce a Littlewood-Paley characterization of modulation spaces and use it to give an alternative proof of the algebra property, somehow implicitly contained in Sugimoto (2011), of the intersection $M^s_{p,q}(\mathbb{R}^d) \cap…

In this work I study the well-posedness of the Cauchy problem associated with the coupled Schr\"odinger equations {with quadratic nonlinearities}, which appears modeling problems in nonlinear optics. I obtain the local well-posedness for…

We proved the local well-posedness for the power-type nonlinear semi-relativistic or half-wave equation (NLHW) in Sobolev spaces. Our proofs mainly bases on the contraction mapping argument using Strichartz estimate. We also apply the…

We prove that the periodic modified Benjamin-Ono equation is locally well-posed in the energy space $H^{1/2}$. This ensures the global well-posedness in the defocusing case. The proof is based on an $X^{s,b}$ analysis of the system after…

We construct local (in time) strong solutions in {$H^s(\mathbb{R}^3)$, $s>3/2$} and global weak solutions with finite energy for both the Pauli-Darwin and the Pauli-Poisswell systems. These are the first rigorous results on local and global…

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 prove that the Cauchy problem for the Schr\"odinger-Korteweg-de Vries system is locally well-posed for the initial data belonging to the Sovolev spaces $L^2(\R)\times H^{-{3/4}}(\R)$. The new ingredient is that we use the $\bar{F}^s$…

In this paper, we study the probabilistic local well-posedness of the cubic Schr\"odinger equation (cubic NLS): \[ (i\partial_{t} + \Delta) u = \pm |u|^{2} u \text{ on } [0,T) \times \mathbb{R}^{d}, \] with initial data being a Wiener…

In this paper we consider the local well-posedness theory for the quadratic nonlinear Schr\"odinger equation with low regularity initial data in the case when the nonlinearity contains derivatives. We work in 2+1 dimensions and prove a…

Although standard quantum mechanics has some non-local features, the probability current of the Schr\"odinger equation is locally conserved, and this allows minimal electromagnetic coupling. For some important extensions of the…