相关论文: Global well-posedness below energy space for the 1…

In this paper, we study the Cauchy problem of the 3-dimensional (3D) generalized incompressible Navier-Stokes equations (gNS) in Triebel-Lizorkin space $\dot{F}^{-\alpha,r}_{q_\alpha}(\mathbb{R}^3)$ with…

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 $…

We study the Gross-Pitaevskii equation involving a nonlocal interaction potential. Our aim is to give sufficient conditions that cover a variety of nonlocal interactions such that the associated Cauchy problem is globally well-posed with…

We study the Cauchy problem of the Schr\"odinger-Korteweg-de Vries system. First, we establish the local well-posedness results, which improve the results of Corcho, Linares (2007). Moreover, we obtain some ill-posedness results, which show…

Considering the Cauchy problem for the Korteweg-de Vries-Burgers equation \begin{eqnarray*} u_t+u_{xxx}+\epsilon |\partial_x|^{2\alpha}u+(u^2)_x=0, \ u(0)=\phi, \end{eqnarray*} where $0<\epsilon,\alpha\leq 1$ and $u$ is a real-valued…

The Cauchy problem for the classical Dirac-Klein-Gordon system in two space dimensions is globally well-posed for L^2 Schoedinger data and wave data in H^{1/2} \times H^{-1/2}. In the case of smooth data there exists a global smooth…

The Schroedinger equation with the nonlinearity concentrated at a single point proves to be an interesting and important model for the analysis of long-time behavior of solutions, such as the asymptotic stability of solitary waves and…

For $n\geq 3$, we study the Cauchy problem for the fourth order nonlinear Schr\"{o}dinger equations, for which the existence of the scattering operators and the global well-posedness of solutions with small data in Besov spaces…

We consider a Cauchy problem of energy-critical fractional Schr\"odinger equation with Hartree nonlinearity below the energy space. Using a method of randomization of functions on $\mathbb{R}^d$ associated with the Wiener decomposition,…

We prove that the Cauchy problem for the dispersion generalized Benjamin-Ono equation where $0<\alpha \leq 1$ \begin{eqnarray*} \left\{ \begin{array}{l} \partial_t u+|\partial_x|^{1+\alpha}\partial_x u+uu_x=0,\\ u(x,0)=u_0(x), \end{array}…

We prove that the Cauchy problem associated to the Zakharov-Schulman system $iu_t+L_1u=uv$, $L_2v=L_3(|u|^2)$ is locally well-posed for given initial data in Sobolev spaces $H^s(R^n)$, $s\geq n/4$, for n =2,3. Here, L_j denote second order…

We prove local in time well-posedness for a large class of quasilinear Hamiltonian, or parity preserving, Schr\"odinger equations on the circle. After a paralinearization of the equation, we perform several paradifferential changes of…

The unique global strong solution in the Chemin-Lerner type space to the Cauchy problem on the Boltzmann equation for hard potentials is constructed in perturbation framework. Such solution space is of critical regularity with respect to…

We prove the local well-posedness for the two-dimensional Zakharov-Kuznetsov equation in $H^s(\mathbb{R}^2)$, for $s\in [1,2]$, on the background of an $L^\infty(\mathbb{R}^3)$-function $\Psi(t,x,y)$, with $\Psi(t,x,y)$ satisfying some…

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 article, we study the low-regularity Cauchy problem of a one dimensional quadratic Schrodinger system with coupled parameter $\alpha\in (0, 1)$. When $\frac{1}{2}<\alpha<1$,we prove the global well-posedness in $H^s(\mathbb{R})$…

We consider the Cauchy problem for the defocusing energy-critical stochastic nonlinear wave equations (SNLW) with an additive stochastic forcing on $\mathbb{R}^{d}$ and $\mathbb{T}^{d}$ with $d \geq 3$. By adapting the probabilistic…

We use a method developed by Strauss to obtain global wellposedness results in the mild sense for the small data Cauchy problem in modulation spaces $M_{p,q}^s(\mathbb{R}^d)$, where $q=1$ and $s\geq0$ or $q\in(1,\infty]$ and…

We consider the Cauchy problem for the nonlinear Schr\"odinger equation on $\mathbb{R}^d$, where the initial data is in $\dot{H}^1(\mathbb{R}^d)\cap L^p(\mathbb{R}^d)$. We prove local well-posedness for large ranges of $p$ and discuss some…

In this paper we consider the Cauchy problem for 2D viscous shallow water system in Besov spaces. We firstly prove the local well-posedness of this problem in $B^s_{p,r}(\mathbb{R}^2)$, $s>max\{1,\frac{2}{p}\}$, $1\leq p,r\leq \infty$ by…