On a quadratic nonlinear Schr\"odinger equation: sharp well-posedness and ill-posedness

We study the initial value problem of the quadratic nonlinear Schr\"odinger equation $$iu_t+u_{xx}=u\bar{u},$$ where u:\R\times \R\to \C. We prove that it's locally well-posed in $H^s(\R)$ when $s\geq -\dfrac{1}{4}$ and ill-posed when $s< -\dfrac{1}{4}$, which improve the previous work in \cite{KPV}. Moreover, we consider the problem in the following space, $$H^{s,a}(\R)={u:\|u\|_{H^{s,a}}\triangleq (\displaystyle\int (|\xi|^s\chi_{\{|\xi|>1\}}+|\xi|^a\chi_{\{|\xi|\leq 1\}})^2|\hat{u}(\xi)|^2 d\xi)^{{1/2}}<\infty}$$ for $s\leq 0, a\geq 0$. We establish the local well-posedness in $H^{s,a}(\R)$ when $s\geq -\dfrac{1}{4}-\dfrac{1}{2}a$ and $a<\dfrac{1}{2}$. Also we prove that it's ill-posed in $H^{s,a}(\R)$ when $s\dfrac{1}{2}$. It remains the cases on the line segment: $a=\dfrac{1}{2}$, $-\dfrac{1}{2}\leq s\leq 0$ open in this paper.Comment: 26 page

Global Attractor for Weakly Damped Forced KdV Equation in Low Regularity on T

In this paper we consider the long time behavior of the weakly damped, forced Korteweg-de Vries equation in the Sololev spaces of the negative indices in the periodic case. We prove that the solutions are uniformly bounded in \dot{H}^s(\T) for $s>-\dfrac{1}{2}$. Moreover, we show that the solution-map possesses a global attractor in \dot{H}^s(\T) for $s>-\dfrac{1}{2}$, which is a compact set in H^{s+3}(\T).Comment: 34 page

Global well-posedness for the Benjamin equation in low regularity

In this paper we consider the initial value problem of the Benjamin equation $$\partial_{t}u+\nu \H(\partial^2_xu) +\mu\partial_{x}^{3}u+\partial_xu^2=0,$$ where $u:\R\times [0,T]\mapsto \R$, and the constants $\nu,\mu\in \R,\mu\neq0$. We use the I-method to show that it is globally well-posed in Sobolev spaces $H^s(\R)$ for $s>-3/4$. Moreover, we use some argument to obtain a good estimative for the lifetime of the local solution, and employ some multiplier decomposition argument to construct the almost conserved quantities.Comment: 29 page

Global well-posedness of the generalized KP-II equation in anisotropic Sobolev spaces

In this paper, we consider the Cauchy problem for the generalized KP-II equation \begin{eqnarray*} u_{t}-|D_{x}|^{\alpha}u_{x}+\partial_{x}^{-1}\partial_{y}^{2}u+\frac{1}{2}\partial_{x}(u^{2})=0,\alpha\geq4. \end{eqnarray*} The goal of this paper is two-fold. Firstly, we prove that the problem is locally well-posed in anisotropic Sobolev spaces H^{s_{1},\>s_{2}}(\R^{2}) with s_{1}>\frac{1}{4}-\frac{3}{8}\alpha, s_{2}\geq 0 and \alpha\geq4. Secondly, we prove that the problem is globally well-posed in anisotropic Sobolev spaces H^{s_{1},\>0}(\R^{2}) with -\frac{(3\alpha-4)^{2}}{28\alpha}<s_{1}\leq0. and \alpha\geq4. Thus, our global well-posedness result improves the global well-posedness result of Hadac (Transaction of the American Mathematical Society, 360(2008), 6555-6572.) when 4\leq \alpha\leq6.Comment: We correct some misprints. arXiv admin note: substantial text overlap with arXiv:1709.01983, arXiv:1712.0933

Hardy space theory on spaces of homogeneous type via orthonormal wavelet bases

In this paper, using the remarkable orthonormal wavelet basis constructed recently by Auscher and Hyt\"onen, we establish the theory of product Hardy spaces on spaces ${\widetilde X} = X_1\times X_2\times\cdot \cdot\cdot\times X_n$, where each factor $X_i$ is a space of homogeneous type in the sense of Coifman and Weiss. The main tool we develop is the Littlewood--Paley theory on $\widetilde X$, which in turn is a consequence of a corresponding theory on each factor space. We define the square function for this theory in terms of the wavelet coefficients. The Hardy space theory developed in this paper includes product~$H^p$, the dual \cmo^p of $H^p$ with the special case \bmo = \cmo^1, and the predual \vmo of $H^1$. We also use the wavelet expansion to establish the Calder\'on--Zygmund decomposition for product $H^p$, and deduce an interpolation theorem. We make no additional assumptions on the quasi-metric or the doubling measure for each factor space, and thus we extend to the full generality of product spaces of homogeneous type the aspects of both one-parameter and multiparameter theory involving the Littlewood--Paley theory and function spaces. Moreover, our methods would be expected to be a powerful tool for developing wavelet analysis on spaces of homogeneous type

Quantum group structure of the q-deformed WW algebra \WW_q

In this paper the q-deformed $W$ algebra \WW_q is constructed, whose nontrivial quantum group structure is presented.Comment: 7 page

Geometric characterizations of embedding theorems

The embedding theorem arises in several problems from analysis and geometry. The purpose of this paper is to provide a deeper understanding of analysis and geometry with a particular focus on embedding theorems on spaces of homogeneous type in the sense of Coifman and Weiss. We prove that embedding theorems hold on spaces of homogeneous type if and only if geometric conditions, namely the measures of all balls have lower bounds, hold. As applications, our results provide new and sharp previous related embedding theorems for the Sobolev, Besov and Triebel-Lizorkin spaces.Comment: arXiv admin note: text overlap with arXiv:1507.0718

On the Well-posedness of 2-D Incompressible Navier-Stokes Equations with Variable Viscosity in Critical Spaces

In this paper, we first prove the local well-posedness of the 2-D incompressible Navier-Stokes equations with variable viscosity in critical Besov spaces with negative regularity indices, without smallness assumption on the variation of the density. The key is to prove for $p\in(1,4)$ and $a\in\dot{B}_{p,1}^{\frac2p}(\mathbb{R}^2)$ that the solution mapping $\mathcal{H}_a:F\mapsto\nabla\Pi$ to the 2-D elliptic equation $\mathrm{div}\big((1+a)\nabla\Pi\big)=\mathrm{div} F$ is bounded on $\dot{B}_{p,1}^{\frac2p-1}(\mathbb{R}^2)$. More precisely, we prove that $$\|\nabla\Pi\|_{\dot{B}_{p,1}^{\frac2p-1}}\leq C\big(1+\|a\|_{\dot{B}_{p,1}^{\frac2p}}\big)^2\|F\|_{\dot{B}_{p,1}^{\frac2p-1}}.$$ The proof of the uniqueness of solution to (1.2) relies on a Lagrangian approach [15]-[17]. When the viscosity coefficient $\mu(\rho)$ is a positive constant, we prove that (1.2) is globally well-posed

The Cauchy problem for a higher order shallow water type equation on the circle

In this paper, we investigate the Cauchy problem for a higher order shallow water type equation \begin{eqnarray*} u_{t}-u_{txx}+\partial_{x}^{2j+1}u-\partial_{x}^{2j+3}u+3uu_{x}-2u_{x}u_{xx}-uu_{xxx}=0, \end{eqnarray*} where $x\in \mathbf{T}=\mathbf{R}/2\pi$ and $j\in N^{+}.$ Firstly, we prove that the Cauchy problem for the shallow water type equation is locally well-posed in $H^{s}(\mathbf{T})$ with $s\geq -\frac{j-2}{2}$ for arbitrary initial data. By using the $I$-method, we prove that the Cauchy problem for the shallow water type equation is globally well-posed in $H^{s}(\mathbf{T})$ with $\frac{2j+1-j^{2}}{2j+1}<s\leq 1.$ Our results improve the result of A. A. Himonas, G. Misiolek (Communications in partial Differential Equations, 23(1998), 123-139;Journal of Differential Equations, 161(2000), 479-495.)Comment: 3

Global well-posedness of the Cauchy problem for a fifth-order KP-I equation in anisotropic Sobolev spaces

In this paper, we consider the Cauchy problem for the fifth-order KP-I equation \begin{align*} u_t + \partial_x^5u+\partial_x^{-1}\partial_y^2u + \frac{1}{2}\partial_x(u^2)=0. \end{align*} Firstly, we establish the local well-posedness of the problem in the anisotropic Sobolev spaces $H^{s_1, s_2}(\mathbb{R}^2)$ with $s_1>-\frac{9}{8}$ and $s_2\geq 0$. Secondly, we establish the global well-posedness of the problem in $H^{s_1,0}(\mathbb{R}^2)$ with $s_1>-\frac{4}{7}$. Our result improves considerably the results of Saut and Tzvetkov (J. Math.\ Pures Appl.\ 79(2000), 307--338.) and Li and Xiao (J. Math.\ Pures Appl.\ 90(2008), 338--352.) and Guo, Huo and Fang (J. Diff.\ Eqns.\ 263 (2017), 5696--5726).Comment: 54pages. arXiv admin note: substantial text overlap with arXiv:1709.01983, arXiv:1709.0607