Global existence and time decay of the non-cutoff Boltzmann equation with hard potential

This paper is concerned with the Cauchy problem on the Boltzmann equation without angular cutoff assumption for hard potential in the whole space. When the initial data is a small perturbation of a global Maxwellian, the global existence of solution to this problem is proved in unweighted Sobolev spaces $H^N\left({\mathbb{R}}_{x,v}^6\right)$ with $N\ge 2$. But if we want to obtain the optimal temporal decay estimates, we need to add the velocity weight function, in this case the global existence and the optimal temporal decay estimate of the Boltzmann equation are all established. Meanwhile, we further gain a more accurate energy estimate, which can guarantee the validity of the assumption in Chen et al. (0000).     

大尺寸电阻加热式碳化硅晶体生长热场设计与优化

大尺寸低缺陷碳化硅(SiC)单晶体是功率器件和射频(RF)器件的重要基础材料,物理气相传输(physical vapor transport, PVT)法是目前生长大尺寸SiC单晶体的主要方法。获得大尺寸高品质晶体的核心是通过调节组分、温度、压力实现气相组分在晶体生长界面均匀定向结晶,同时尽可能减小晶体的热应力。本文对电阻加热式8英寸(1英寸=2.54 cm)碳化硅大尺寸晶体生长系统展开热场设计研究。首先建立描述碳化硅原料受热分解热质输运及其多孔结构演变、系统热输运的物理和数学模型,进而使用数值模拟方法研究加热器位置、加热器功率和辐射孔径对温度分布的影响及其规律,并优化热场结构。数值模拟结果显示,通过优化散热孔形状、保温棉的结构等设计参数,电阻加热式大尺寸晶体生长系统在晶锭厚度变化、多孔介质原料消耗的情况下均能达到较低的晶体横向温度梯度和较高的纵向温度梯度。     

On pointwise decay rates of time‐periodic solutions to the Navier–Stokes equation

We study the existence of a time‐periodic solution with pointwise decay properties to the Navier–Stokes equation in the whole space. We show that if the time‐periodic external force is sufficiently small in an appropriate sense, then there exists a time‐periodic solution $\{u,p\}$ of the Navier–Stokes equation such that $|{?}^j{u(t,x)|=O(|x|}^{1?n?j})$ and $|{?}^j{p(t,x)|=O(|x|}^{?n?j})(j=0,1,\dots )$ uniformly in $t\in \mathbb{R}$ as $|x|\to \infty$. Our solution decays faster than the time‐periodic Stokes fundamental solution and the faster decay of its spatial derivatives of higher order is also described.     

An analysis of over-relaxation in a kinetic approximation of systems of conservation laws

The over-relaxation approach is an alternative to the Jin–Xin relaxation method in order to apply the equilibrium source term in a more precise way. This is also a key ingredient of the lattice Boltzmann method for achieving second-order accuracy. In this work, we provide an analysis of the over-relaxation kinetic scheme. We compute its equivalent equation, which is particularly useful for devising stable boundary conditions for the hidden kinetic variables.