On the Real Analyticity of the Scattering Operator for the Hartree Equation

In this paper, we study the real analyticity of the scattering operator for the Hartree equation $i\partial_tu=-\Delta u+u(V*|u|^2)$. To this end, we exploit interior and exterior cut-off in time and space, and combining with the compactness argument to overcome difficulties which arise from absence of good properties for the nonlinear Klein-Gordon equation, such as the finite speed of propagation and ideal time decay estimate. Additionally, the method in this paper allows us to simplify the proof of analyticity of the scattering operator for the nonlinear Klein-Gordon equation with cubic nonlinearity in Kumlin.Comment: 16page

Dynamics for the focusing, energy-critical nonlinear Hartree equation

In \cite{LiMZ:e-critical Har, MiaoXZ:09:e-critical radial Har}, the dynamics of the solutions for the focusing energy-critical Hartree equation have been classified when $E(u_0)<E(W)$, where $W$ is the ground state. In this paper, we continue the study on the dynamics of the radial solutions with the threshold energy. Our arguments closely follow those in \cite{DuyMerle:NLS:ThresholdSolution, DuyMerle:NLW:ThresholdSolution, DuyRouden:NLS:ThresholdSolution, LiZh:NLS, LiZh:NLW}. The new ingredient is that we show that the positive solution of the nonlocal elliptic equation in $L^{\frac{2d}{d-2}}(\R^d)$ is regular and unique by the moving plane method in its global form, which plays an important role in the spectral theory of the linearized operator and the dynamics behavior of the threshold solution.Comment: 53 page

The low regularity global solutions for the critical generalized KdV equation

We prove that the Cauchy problem of the mass-critical generalized KdV equation is globally well-posed in Sobolev spaces $H^s(\R)$ for $s>6/13$. Of course, we require that the mass is strictly less than that of the ground state in the focusing case. The main approach is the "I-method" together with the multilinear correction analysis. Moreover, we use some "partially refined" argument to lower the upper control of the multiplier in the resonant interactions. The result improves the previous works of Fonseca, Linares, Ponce (2003) and Farah (2009).Comment: 27pages, the mistake in the previous version is corrected; using I-method with the resonant decomposition gives an improvement over our previous result

Global well-posedness of the critical Burgers equation in critical Besov spaces

We make use of the method of modulus of continuity \cite{K-N-S} and Fourier localization technique \cite{A-H} to prove the global well-posedness of the critical Burgers equation $\partial_{t}u+u\partial_{x}u+\Lambda u=0$ in critical Besov spaces $\dot{B}^{\frac{1}{p}}_{p,1}(\mathbb{R})$ with $p\in[1,\infty)$, where $\Lambda=\sqrt{-\triangle}$.Comment: 21page

Traveling Wave Solutions of the Generalized b-family Equation with Dispersion

Abstract: This paper introduces a family of evolutionary 1+1 PDEs that describe the balance between convection and stretching in the dynamics of 1D nonlinear waves in fluids. It is reversible in time and parity invariant. In the paper, special solutions are discussed for = 0 and for ∕ = 0, the general solutions are given.When = 3 and = −1 , the paper obtains the exact solutions of the generalized −family equation

Boosting implicit discourse relation recognition with connective-based word embeddings

Abstract(#br)Implicit discourse relation recognition is the performance bottleneck of discourse structure analysis. To alleviate the shortage of training data, previous methods usually use explicit discourse data, which are naturally labeled by connectives, as additional training data. However, it is often difficult for them to integrate large amounts of explicit discourse data because of the noise problem. In this paper, we propose a simple and effective method to leverage massive explicit discourse data. Specifically, we learn connective-based word embeddings ( CBWE ) by performing connective classification on explicit discourse data. The learned CBWE is capable of capturing discourse relationships between words, and can be used as pre-trained word embeddings for implicit discourse relation recognition. On both the English PDTB and Chinese CDTB data sets, using CBWE achieves significant improvements over baselines with general word embeddings, and better performance than baselines integrating explicit discourse data. By combining CBWE with a strong baseline, we achieve the state-of-the-art performance

Global Wellposedness for a Modified Critical Dissipative Quasi-Geostrophic Equation

In this paper we consider the following modified quasi-geostrophic equation \partial_{t}\theta+u\cdot\nabla\theta+\nu |D|^{\alpha}\theta=0, \quad u=|D|^{\alpha-1}\mathcal{R}^{\bot}\theta,\quad x\in\mathbb{R}^2 with $\nu>0$ and $\alpha\in ]0,1[\,\cup \,]1,2[$. When $\alpha\in]0,1[$, the equation was firstly introduced by Constantin, Iyer and Wu in \cite{ref ConstanIW}. Here, by using the modulus of continuity method, we prove the global well-posedness of the system with the smooth initial data. As a byproduct, we also show that for every $\alpha\in ]0,2[$, the Lipschitz norm of the solution has a uniform exponential bound.Comment: In this version we extend the range of $\alpha$ from (0,1) to (0,2), we also show that for every $\alpha\in (0,2)$, the Lipschitz norm of the solution has a uniform exponential bound. 27page

On the well-posedness for the Ideal MHD equations in the Triebel-Lizorkin spaces

In this paper, we prove the local well-posedness for the Ideal MHD equations in the Triebel-Lizorkin spaces and obtain blow-up criterion of smooth solutions. Specially, we fill a gap in a step of the proof of the local well-posedness part for the incompressible Euler equation in \cite{Chae1}.Comment: 16page