arXiv subjects

Jianhua Huang

Publications and source records attributed to Jianhua Huang.

33 records · Page 2Linked to original sources

arXiv2020

Exponential mixing for the fractional Magneto-Hydrodynamic equations with degenerate stochastic forcing

We establish the existence, uniqueness and exponential attraction properties of an invariant measure for the MHD equations with degenerate stochastic forcing acting only in the magnetic equation. The central challenge is to establish time asymptotic smoothing properties of the associated Markovian semigroup corresponding to this system. Towards this aim we take full advantage of the characteristics of the advective structure to discover a novel Hörmander-type condition which only allows for several noises in the magnetic direction.

math.PR↗

arXiv2020

Ergodicity and exponential mixing of the real Ginzburg-Landau equation with a degenerate noiss

In this paper, we establish the existence, uniqueness and attraction properties of an invariant measure for the real Ginzburg-Landau equation in the presence of a degenerate stochastic forcing acting only in four directions. The main challenge is to establish time asymptotic smoothing properties of the Markovian dynamics corresponding to this system. To achieve this, we propose a condition which only requires four noises

math.PR↗

arXiv2019

Well-posedness for Stochastic Generalized Fractional Benjamin-Ono Equation

This paper is devoted to the Cauchy problem for the stochastic generalized Benjamin-Ono equation. By using the Bourgain spaces and Fourier restriction method and the assumption that $u_{0}$ is $\mathcal{F}_{0}$-measurable, we prove that the Cauchy problem for the stochastic generalized Benjamin-Ono equation is locally well-posed for the initial data $u_{0}(x,w)\in L^{2} (Ω; H^{s}(\mathbb{R}))$ with $s\geq\frac{1}{2}-\fracα{4}$, where $0< α\leq 1.$ In particular, when $u_{0}\in L^{2}(Ω; H^{\frac{α+1}{2}}(\mathbb{R}))\cap L^{\frac{2(2+3α)}α}(Ω; L^{2}(\mathbb{R}))$, we prove that there exists a unique global solution $u\in L^{2}(Ω; H^{\frac{α+1}{2}}(\mathbb{R}))$ with $0< α\leq 1.$

math.AP↗

arXiv2018

New lower bounds on the radius of spatial analyticity for the KdV equation

The radius of spatial analyticity for solutions of the KdV equation is studied. It is shown that the analyticity radius does not decay faster than $t^{-1/4}$ as time $t$ goes to infinity. This improves the works [Selberg, da Silva, Lower bounds on the radius of spatial analyticity for the KdV equation, Annales Henri Poincaré, 2017, 18(3): 1009-1023] and [Tesfahun, Asymptotic lower bound for the radius of spatial analtyicity to solutions of KdV equation, arXiv preprint arXiv:1707.07810, 2017]. Our strategy mainly relies on a higher order almost conservation law in Gevrey spaces, which is inspired by the $I-$method.

math.AP↗

arXiv2018

Random data Cauchy problem for the wave equation on compact manifold

Inspired by the work of Burq and Tzvetkov (Invent. math. 173(2008), 449-475.), firstly, we construct the local strong solution to the cubic nonlinear wave equation with random data for a large set of initial data in $H^{s}(M)$ with $s\geq \frac{5}{14}$, where M is a three dimensional compact manifold with boundary, moreover, our result improves the result of Theorem 2 in (Invent. math. 173(2008), 449-475.); secondly, we construct the local strong solution to the quintic nonlinear wave equation with random data for a large set of initial data in $H^{s}(M)$ with $s\geq\frac{1}{6}$, where M is a two dimensional compact boundaryless manifold; finally, we construct the local strong solution to the quintic nonlinear wave equation with random data for a large set of initial data in $H^{s}(M)$ with $s\geq \frac{23}{90}$, where M is a two dimensional compact manifold with boundary.

math.AP↗

arXiv2017

The Cauchy problem for two dimensional generalized Kadomtsev-Petviashvili-I equation in anisotropic Sobolev spaces

The goal of this paper is three-fold. Firstly, we prove that the Cauchy problem for generalized KP-I equation \begin{eqnarray*} u_{t}+|D_{x}|^α\partial_{x}u+\partial_{x}^{-1}\partial_{y}^{2}u+\frac{1}{2}\partial_{x}(u^{2})=0,α\geq4 \end{eqnarray*} is locally well-posed in the anisotropic Sobolev spaces$ H^{s_{1},\>s_{2}}(\R^{2})$ with $s_{1}>-\frac{α-1}{4}$ and $s_{2}\geq 0$. Secondly, we prove that the problem is globally well-posed in $H^{s_{1},\>0}(\R^{2})$ with $s_{1}>-\frac{(α-1)(3α-4)}{4(5α+3)}$ if $4\leq α\leq5$. Finally, we prove that the problem is globally well-posed in $H^{s_{1},\>0}(\R^{2})$ with $s_{1}>-\frac{α(3α-4)}{4(5α+4)}$ if $α>5$. Our result improves the result of Saut and Tzvetkov (J. Math. Pures Appl. 79(2000), 307-338.) and Li and Xiao (J. Math. Pures Appl. 90(2008), 338-352.).

math.AP↗

arXiv2017

Random data Cauchy problem for a generalized KdV equation in the supercritical case

We consider the Cauchy problem for a generalized KdV equation \begin{eqnarray*} u_{t}+\partial_{x}^{3}u+u^{7}u_{x}=0, \end{eqnarray*} with random data on \R. Kenig, Ponce, Vega(Comm. Pure Appl. Math.46(1993), 527-620)proved that the problem is globally well-posed in H^{s}(\R)$ with s> s_{crit}=\frac{3}{14}, which is the scaling critical regularity indices. Birnir, Kenig, Ponce, Svanstedt, Vega(J. London Math. Soc. 53 (1996), 551-559.) proved that the problem is ill-posed in the sense that the time of existence T and the continuous dependence cannot be expressed in terms of the size of the data in the H^{\frac{3}{14}}-norm. In this present paper, we prove that almost sure local in time well-posedness holds in H^{s}(\R) with s>\frac{17}{112}, whose lower bound is below \frac{3}{14}. The key ingredients are the Wiener randomization of the initial data and probabilistic Strichartz estimates together with some important embedding Theorems.

math.AP↗

arXiv2017

The Cauchy problem for the Ostrovsky equation with positive dispersion

This paper is devoted to studying the Cauchy problem for the Ostrovsky equation \begin{eqnarray*} \partial_{x}\left(u_{t}-β\partial_{x}^{3}u +\frac{1}{2}\partial_{x}(u^{2})\right) -γu=0, \end{eqnarray*} with positive $β$ and $γ$. This equation describes the propagation of surface waves in a rotating oceanic flow. We first prove that the problem is locally well-posed in $H^{-\frac{3}{4}}(\R)$. Then we reestablish the bilinear estimate, by means of the Strichartz estimates instead of calculus inequalities and Cauchy-Schwartz inequalities. As a byproduct, this bilinear estimate leads to the proof of the local well-posedness of the problem in $H^{s}(\R)$ for $ s>-\frac{3}{4}$, with help of a fixed point argument.

math.AP↗

arXiv2015

Sharp well-posedness and ill-posedness of the Cauchy problem for the higher-order KdV

In this paper, we investigate the Cauchy problem for the higher-order KdV-type equation \begin{eqnarray*} u_{t}+(-1)^{j+1}\partial_{x}^{2j+1}u + \frac{1}{2}\partial_{x}(u^{2}) = 0,j\in N^{+},x\in\mathbf{T}= [0,2πλ) \end{eqnarray*} with low regularity data and $λ\geq 1$. Firstly, we show that the Cauchy problem for the periodic higher-order KdV equation is locally well-posed in $H^{s}(\mathbf{T})$ with $s\geq -j+\frac{1}{2},j\geq2.$ By using some new Strichartz estimate and some new function spaces, we also show that the Cauchy problem for the periodic higher-order KdV equation is ill-posed in $H^{s}(\mathbf{T})$ with $s<-j+\frac{1}{2},j\geq2$ in the sense that the solution map is $C^{3}.$ The result of this paper improves the result of \cite{H} with $j\geq2$.

math.AP↗

arXiv2015

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 $x\in \mathbf{T}=\mathbf{R}/2π$ 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.)

math.AP↗

arXiv2014

The Cauchy problem for the Ostrovsky equation with negative dispersion at the critical regularity

In this paper, we investigate the Cauchy problem for the Ostrovsky equation \begin{eqnarray*} \partial_{x}\left(u_{t}-β\partial_{x}^{3}u +\frac{1}{2}\partial_{x}(u^{2})\right) -γu=0, \end{eqnarray*} in the Sobolev space $H^{-3/4}(\R)$. Here $β>0(<0)$ corresponds to the positive (negative) dispersion of the media, respectively. P. Isaza and J. Mej\'ıa (J. Diff. Eqns. 230(2006), 601-681; Nonli. Anal. 70(2009), 2306-2316), K. Tsugawa (J. Diff. Eqns. 247(2009), 3163-3180) proved that the problem is locally well-posed in $H^s(\R)$ when $s>-3/4$ and ill-posed when $s<-3/4$. By using some modified Bourgain spaces, we prove that the problem is locally well-posed in $H^{-3/4}(\R)$ with $β<0$ and $γ>0.$ The new ingredient that we introduce in this paper is Lemmas 2.1-2.6.

math.AP↗

arXiv2013

A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems

Non-convex sparsity-inducing penalties have recently received considerable attentions in sparse learning. Recent theoretical investigations have demonstrated their superiority over the convex counterparts in several sparse learning settings. However, solving the non-convex optimization problems associated with non-convex penalties remains a big challenge. A commonly used approach is the Multi-Stage (MS) convex relaxation (or DC programming), which relaxes the original non-convex problem to a sequence of convex problems. This approach is usually not very practical for large-scale problems because its computational cost is a multiple of solving a single convex problem. In this paper, we propose a General Iterative Shrinkage and Thresholding (GIST) algorithm to solve the nonconvex optimization problem for a large class of non-convex penalties. The GIST algorithm iteratively solves a proximal operator problem, which in turn has a closed-form solution for many commonly used penalties. At each outer iteration of the algorithm, we use a line search initialized by the Barzilai-Borwein (BB) rule that allows finding an appropriate step size quickly. The paper also presents a detailed convergence analysis of the GIST algorithm. The efficiency of the proposed algorithm is demonstrated by extensive experiments on large-scale data sets.

cs.LG↗

arXiv2011

Dynamics of 2D Stochastic non-Newtonian fluids driven by fractional Brownian motion

A 2D Stochastic incompressible non-Newtonian fluids driven by fractional Bronwnian motion with Hurst parameter $H \in (1/2,1)$ is studied. The Wiener-type stochastic integrals are introduced for infinite-dimensional fractional Brownian motion. Four groups of assumptions, including the requirement of Nuclear operator or Hilbert-Schmidt operator, are discussed. The existence and regularity of stochastic convolution for the corresponding additive linear stochastic equation are obtained under each group of assumptions. Mild solution are then obtained for the non-Newtonian systems by the modified fix point theorem in the selected intersection space. When the domain is square, the random dynamical system generated by non-Newtonian systems has a random attractor under some condition on the spectrum distribution of the corresponding differential operator.

math-ph↗

arXiv2011

Dynamics of stochastic non-Newtonian fluids driven by fractional Brownian motion with Hurst parameter $H \in (1/4,1/2)$

In this paper we consider the Stochastic isothermal, nonlinear, incompressible bipolar viscous fluids driven by a genuine cylindrical fractional Bronwnian motion with Hurst parameter $H \in (1/4,1/2)$ under Dirichlet boundary condition on 2D square domain. First we prove the existence and regularity of the stochastic convolution corresponding to the stochastic non-Newtonian fluids. Then we obtain the existence and uniqueness results for the stochastic non-Newtonian fluids. Under certain condition, the random dynamical system generated by non-Newtonian fluids has a random attractor.

math.DS↗

arXiv2008

Goos-Hänchen-Like Shifts in Atom Optics

We consider the propagation of a matter wavepacket of two-level atoms through a square potential created by a super-Gaussian laser beam. We explore the matter wave analog of Goos-Hänchen shift within the framework of atom optics where the roles of atom and light is exchanged with respect to conventional optics. Using a vector theory, where atoms are treated as particles possessing two internal spin components, we show that not only large negative but also large positive Goos-Hänchen shifts can occur in the reflected atomic beam.

quant-ph↗