arXiv subjects
Publications and source records attributed to Changxing Miao.
In this paper, we improve some known uniqueness results of weak solutions for the 3D Navier-Stokes equations. The proof uses the Fourier localization technique and the losing derivative estimates.
We prove the global well-posedness and scattering for the defocusing $H^{\frac12}$-subcritical (that is, $2<γ<3$) Hartree equation with low regularity data in $\mathbb{R}^d$, $d\geq 3$. Precisely, we show that a unique and global solution exists for initial data in the Sobolev space $H^s\big(\mathbb{R}^d\big)$ with $s>4(γ-2)/(3γ-4)$, which also scatters in both time directions. This improves the result in \cite{ChHKY}, where the global well-posedness was established for any $s>\max\big(1/2,4(γ-2)/(3γ-4)\big)$. The new ingredients in our proof are that we make use of an interaction Morawetz estimate for the smoothed out solution $Iu$, instead of an interaction Morawetz estimate for the solution $u$, and that we make careful analysis of the monotonicity property of the multiplier $m(ξ)\cdot < ξ>^p$. As a byproduct of our proof, we obtain that the $H^s$ norm of the solution obeys the uniform-in-time bounds.
We consider the fifth order Kadomtsev-Petviashvili I (KP-I) equation as $\partial_tu+α\partial_x^3u+\partial^5_xu+\partial_x^{-1}\partial_y^2u+uu_x=0,$ while $α\in \mathbb{R}$. We introduce an interpolated energy space $E_s$ to consider the well-posedeness of the initial value problem (IVP) of the fifth order KP-I equation. We obtain the local well-posedness of IVP of the fifth order KP-I equation in $E_s$ for $0<s\leq1$. To obtain the local well-posedness, we present a bilinear estimate in the Bourgain space in the framework of the interpolated energy space. It crucially depends on the dyadic decomposed Strichartz estimate, the fifth order dispersive smoothing effect and maximal estimate. Key words: The fifth order KP-I equation, Bourgain space, Dyadic decomposed Strichartz estimate, Dispersive smoothing effect, Maximal estimate.
We improve and extend some known regularity criterion of weak solution for the 3D viscous Magneto-hydrodynamics equations by means of the Fourier localization technique and Bony's para-product decomposition.
This paper is devoted to the study of the Cauchy problem of incompressible magneto-hydrodynamics system in framework of Besov spaces. In the case of spatial dimension $n\ge 3$ we establish the global well-posedness of the Cauchy problem of incompressible magneto-hydrodynamics system for small data and the local one for large data in Besov space $\dot{B}^{\frac np-1}_{p,r}(\mr^n)$, $1\le p<\infty$ and $1\le r\le\infty$. Meanwhile, we also prove the weak-strong uniqueness of solutions with data in $\dot{B}^{\frac np-1}_{p,r}(\mr^n)\cap L^2(\mr^n)$ for $\frac n{2p}+\frac2r>1$. In case of $n=2$, we establish the global well-posedness of solutions for large initial data in homogeneous Besov space $\dot{B}^{\frac2p-1}_{p,r}(\mr^2)$ for $2< p<\infty$ and $1\le r<\infty$.
We establish global existence, scattering for radial solutions to the energy-critical focusing Hartree equation with energy and $\dot{H}^1$ norm less than those of the ground state in $\mathbb{R}\times \mathbb{R}^d$, $d\geq 5$.
We consider the defocusing, $\dot{H}^1$-critical Hartree equation for the radial data in all dimensions $(n\geq 5)$. We show the global well-posedness and scattering results in the energy space. The new ingredient in this paper is that we first take advantage of the term $\displaystyle - \int_{I}\int_{|x|\leq A|I|^{1/2}}|u|^{2}Δ\Big(\frac{1}{|x|}\Big)dxdt$ in the localized Morawetz identity to rule out the possibility of energy concentration, instead of the classical Morawetz estimate dependent of the nonlinearity.
We first give the local well-posedness of strong solutions to the Cauchy problem of the 3D two-fluid MHD equations, then study the blow-up criterion of the strong solutions. By means of the Fourier frequency localization and Bony's paraproduct decomposition, it is proved that strong solution $(u,b)$ can be extended after $t=T$ if either $u\in L^q_T(\dot B^{0}_{p,\infty})$ with $\frac{2}{q}+\frac{3}{p}\le 1$ and $b\in L^1_T(\dot B^{0}_{\infty,\infty})$, or $(ω, J)\in L^q_T(\dot B^{0}_{p,\infty})$ with $\frac{2}{q}+\frac{3}{p}\le 2$, where $ω(t)=\na\times u $ denotes the vorticity of the velocity and $J=\na\times b$ the current density.
In the paper, the large time behavior of solutions of the Cauchy problem for the one dimensional fractal Burgers equation $u_t+(-\partial^2_x)^{α/2} u+uu_x=0$ with $α\in (1,2)$ is studied. It is shown that if the nondecreasing initial datum approaches the constant states $u_\pm$ ($u_-<u_+$) as $x\to \pm\infty$, respectively, then the corresponding solution converges toward the rarefaction wave, {\it i.e.} the unique entropy solution of the Riemann problem for the nonviscous Burgers equation.
We show a new Bernstein's inequality which generalizes the results of Cannone-Planchon, Danchin and Lemarié-Rieusset. As an application of this inequality, we prove the global well-posedness of the 2D quasi-geostrophic equation with the critical and super-critical dissipation for the small initial data in the critical Besov space, and local well-posedness for the large initial data.
In this paper we consider some dissipative versions of the modified Korteweg de Vries equation $u_t+u_{xxx}+|D_x|^αu+u^2u_x=0$ with $0<α\leq 3$. We prove some well-posedness results on the associated Cauchy problem in the Sobolev spaces $H^s({\Bbb R})$ for $s>1/4-α/4$ on the basis of the $[k; Z]-$multiplier norm estimate obtained by Tao in \cite{Tao} for KdV equation.
We study the blow-up criterion of smooth solutions to the 3D MHD equations. By means of the Littlewood-Paley decomposition, we prove a Beale-Kato-Majda type blow-up criterion of smooth solutions via the vorticity of velocity only, i. e. $\sup_{j\in\Z}\int_0^T\|Δ_j(\na\times u)\|_\infty dt$, where $Δ_j$ is a frequency localization on $|ξ|\approx 2^j$.
This paper studies the Cauchy problem for the nonlinear fractional power dissipative equation $u_t+(-\triangle)^αu= F(u)$ for initial data in the Lebesgue space $L^r(\mr^n)$ with $\ds r\ge r_d\triangleq{nb}/({2α-d})$ or the homogeneous Besov space $\ds\dot{B}^{-σ}_{p,\infty}(\mr^n)$ with $\dsσ=(2α-d)/b-n/p$ and $1\le p\le \infty$, where $α>0$, $F(u)=f(u)$ or $Q(D)f(u)$ with $Q(D)$ being a homogeneous pseudo-differential operator of order $d\in[0,2α)$ and $f(u)$ is a function of $u$ which behaves like $|u|^bu$ with $b>0$.
In this paper, we consider the Klein-Gordon-Schrödinger system with the higher order Yukawa coupling in $ \mathbb{R}^{1+1} $, and prove the local and global wellposedness in $L^2\times H^{1/2}$. The method to be used is adapted from the scheme originally by Colliander J., Holmer J., Tzirakis N. \cite{CoHT06} to use the available $L^2$ conservation law of $u$ and control the growth of $n$ via the estimates in the local theory.