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2014年07月07日

【期刊论文】Global well-posedness for axisymmetric Boussinesq system with horizontal viscosity

Changxing Miao, Xiaoxin Zheng

J. Math. Pures Appl.,2014,101(2): 842–872

2014年05月15日

摘要

Under the assumption that the support of the axisymmetric initial data $\\rho_{0}(r,z)$ does not intersect the axis $(Oz)$, we prove the global well-posedness for this system with axisymmetric initial data. We first show the growth of the quantity $\\frac\\rho r$ for large time by taking advantage of characteristic of transport equation. This growing property together with the horizontal smoothing effect enables us to establish $H^1$-estimate of the velocity via the $L^2$-energy estimate of velocity and the Maximum principle of density. Based on this, we further establish the estimate for the quantity $\\|\\omega(t)\\|_{\\sqrt{\\mathbb{L}}}:=\\sup_{2\\leq p<\\infty}\\frac{\\norm{\\omega(t)}_{L^p(\\mathbb{R}^3)}}{\\sqrt{p}}<\\infty$ which implies $\\|\\nabla u(t)\\|_{\\mathbb{L}^{\\frac{3}{2}}}:=\\sup_{2\\leq p<\\infty}\\frac{\\norm{\\nabla u(t)}_{L^p(\\mathbb{R}^3)}}{p\\sqrt{p}}<\\infty$. However, this regularity for the flow admits forbidden singularity since $ \\mathbb{L}$ (see \\eqref{eq-kl} for the definition) seems be the minimum space for the gradient vector field $u(x,t)$ ensuring uniqueness of flow. To bridge this gap, we exploit the space-time estimate about $ \\sup_{2\\leq p<\\infty}\\int_0^t\\frac{\\|\\nabla u(\\tau)\\|_{L^p(\\mathbb{R}^3)}}{\\sqrt{p}}\\mathrm{d}\\tau<\\infty$ by making good use of the horizontal smoothing effect and micro-local techniques. The global well-posedness for the large initial data is achieved by establishing a new type space-time logarithmic inequality.

关键词: Boussinesq system,, horizontal viscosity,, anisotropic inequality,, global well-posedness.,

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2018年10月18日

【期刊论文】The study of charge injection and spin polarization in ferromagnetic metal–polymer–ferromagnetic metal structure

Hua Zhao, Zhong-Wang Liu,, Shao-Bo Chen, Liu-An Chang

Physica B,2014,450(1):111-115

2014年05月10日

摘要

By using extended SSH Hamiltonian plus long-range electron correlation Hamiltonian model, we calculated charge injection and spin polarization in a ferromagnetic metal/polymer/ferromagnetic metal structure. We adjust relative chemical potential between the ferromagnetic materials and the polymer to control charge transfer. It is found that when spin orientations of two ferromagnetic materials are parallel to each other, spin-polarized single polaron can be formed in the polymer, but when the spin orientations of two ferromagnetic materials are antiparallel to each other, bipolaron is formed and that spin polarization is found to be zero inside the polymer. The influence of the long-range electronic correlation on these polarons in the polymer is discussed

关键词: Spin polarization,, Charge injection,, Long-range electron correlation,, Bipolaron,, Spin splitting

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2019年05月03日

【期刊论文】The 2D incompressible Boussinesq equations with general critical dissipation

苗长兴等

SIAM J. Math. Anal.,2014,46(5): 3426 –345

2014年06月05日

摘要

This paper aims at the global regularity problem concerning the 2D incompressible Boussinesq equations with general critical dissipation. The critical dissipation refers to $\alpha +\beta=1$ when $\Lambda^\alpha \equiv (-\Delta)^{\frac{\alpha}{2}}$ and $\Lambda^\beta$ represent the fractional Laplacian dissipation in the velocity and the temperature equations, respectively. We establish the global regularity for the general case with $\alpha+\beta=1$ and $0.9132\approx \alpha_0<\alpha<1$. The cases when $\alpha=1$ and when $\alpha=0$ were previously resolved by Hmidi, Keraani and Rousset \cite{HKR1,HKR2}. The global existence and uniqueness is achieved here by exploiting the global regularity of a generalized critical surface quasi-gesotrophic equation as well as the regularity of a combined quantity of the vorticity and the temperature.

关键词: Boussinesq equations,, critical dissipation,, global regularity

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2019年05月03日

【期刊论文】The defocusing energy critical wave equation with a cubic convolution,

Changxing Miao, Jungyong Zhang, Jiqiang Zheng

Indiana University Mathematics Journal ,2014,63(4):993-1015

2014年08月20日

摘要

In this paper, we study the theory of the global well-posedness and scattering for the energy-critical wave equation with a cubic convolution nonlinearity $u_{tt}-\Delta u+(|x|^{-4}\ast|u|^2)u=0$ in spatial dimension $d \geq 5$. The main difficulties are the absence of the classical finite speed of propagation (i.e. the monotonic local energy estimate on the light cone), which is a fundamental property to show the global well-posedness and then to obtain scattering for the wave equations with the local nonlinearity $u_{tt}-\Delta u+|u|^\frac4{d-2}u=0$. To compensate it, we resort to the extended causality and utilize the strategy derived from concentration compactness ideas. Then, the proof of the global well-posedness and scattering is reduced to show the nonexistence of the three enemies: finite time blowup; soliton-like solutions and low-to-high cascade. We will utilize the Morawetz estimate, the extended causality and the potential energy concentration to preclude the above three enemies.

关键词: wave-Hartree equation,, Concentration compactness,, Morawetz estimate,, Scattering

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