二维可压缩Euler方程轴对称解的生命区间的下界

对于二维可压缩的Ｅｕｌｅｒ方程，当其初值满足ρ｜ｔ＝０＝ρ，ｄｉｖｖ｜ｔ＝０＝０，ｒｏｔｖ｜ｔ＝０＝εω１００时（０＜ε１），证明了对充分小的ε，存在常数ｂ＞０，使得其经典解的生命区间Ｔεｂｌｎ｜ｌｎε｜ε２．与文［５］中ｌｉｍε０＋ε２Ｔε＝＋∞的结论相比，本文的结果更为精确．     

三维不可压缩Euler方程的H1(R~3)轴对称解

在适当的初值条件下，对三维不可压缩Ｅｕｌｅｒ方程组，我们证明了Ｈ１（Ｒ３）轴对称解的存在性．     

三维可压等熵Euler方程光滑解的整体存在性

研究了三维可压等熵Euler方程Cauchy问题光滑解的整体存在性.如果初值是一个常状态的小扰动并且初速度的旋度等于零,证明了三维可压等熵Euler方程Cauchy问题光滑解的整体存在性.     

二维瞬态热传导问题的无单元Galerkin法分析

采用无单元Galerkin(element-free Galerkin, EFG)法求解具有混合边界条件的二维瞬态热传导问题.首先采用二阶向后微分公式离散热传导方程的时间变量,将该问题转化为与时间无关的混合边值问题;然后采用罚函数法处理Dirichlet边界条件,建立了二维瞬态热传导问题的无单元Galerkin法;最后基于移动最小二乘近似的误差结果,详细推导了无单元Galerkin法求解二维瞬态热传导问题的误差估计公式.给出的数值算例表明计算结果与解析解或已有数值解吻合较好,该方法具有较高的计算精度和较好的收敛性.     

二维无界区域中不可压非牛顿流体力学方程组的整体吸引子

研究一类二维无界区域中的等热双极不可压粘性非牛顿流体力学方程组，通过证明和相应的解半群的紧性，得到整体吸引子的存在性。     

3D轴对称Boussinesq方程强解的存在唯一性

给出了三维有旋轴对称Boussinesq方程当区域离开轴时初边值问题强解的存在唯一性.通过证明速度场u及密度▽ρ在区域离开轴时类似二维情形的L⁴估计,得到u,u_t及ρ,ρ_t的先验估计,最后由Galerkin方法证得其强解的存在唯一性.由此将三维轴对称Navier-Stoke方程的相应结果推广到了三维轴对称Boussinesq方程情形.     

Global Existence and Convergence Rates of Smooth Solutions for the 3-D Compressible Magnetohydrodynamic Equations Without Heat Conductivity

In this paper, we are concerned with the global existence and convergence rates of the smooth solutions for the compressible magnetohydrodynamic equations without heat conductivity, which is a hyperbolic-parabolic system. The global solutions are obtained by combining the local existence and a priori estimates if H³-norm of the initial perturbation around a constant states is small enough and its L¹-norm is bounded. A priori decay-in-time estimates on the pressure, velocity and magnetic field are used to get the uniform bound of entropy. Moreover, the optimal convergence rates are also obtained.     

Deformation and Symmetry in the Inviscid SQG and the 3D Euler Equations

The global regularity problem concerning the inviscid SQG and the 3D Euler equations remains an outstanding open question. This paper presents several geometric observations on solutions of these equations. One observation stems from a relation between what we call Eulerian and Lagrangian deformations and reflects the alignment of the stretching directions of these deformations and the tangent direction of the level curves for the SQG equation. Various spatial symmetries in solutions to the 3D Euler equations are exploited. In addition, two observations on the curvature of the level curves of the SQG equation are also included.     

带非线性阻尼项的三维可压缩欧拉方程组的整体解

研究了三维空间中带非线性阻尼项的可压缩欧拉方程组的初值问题.利用能量估计和傅立叶分析的方法,在初值是常状态附近的一个H~3∩L~1中的小扰动时获得了初值问题的解整体存在,并得到了解在大时间的L~2,L~∞衰减率分别为t~(-3/4),t~(-3/2),将线性阻尼的情形推广到了非线性阻尼的情形.     

一类热传导方程的整体解及解的逐点估计

研究长柱体区域中的热传导方程.通过构造辅助函数,利用Hopf极值原理,得到整体解的充分条件,并给出解的逐点估计与空间衰减估计.     

Local Structure of The Set of Steady-State Solutions to The 2d Incompressible Euler Equations

It is well known that the incompressible Euler equations can be formulated in a very geometric language. The geometric structures provide very valuable insights into the properties of the solutions. Analogies with the finite-dimensional model of geodesics on a Lie group with left-invariant metric can be very instructive, but it is often difficult to prove analogues of finite-dimensional results in the infinite-dimensional setting of Euler’s equations. In this paper we establish a result in this direction in the simple case of steady-state solutions in two dimensions, under some non-degeneracy assumptions. In particular, we establish, in a non-degenerate situation, a local one-to-one correspondence between steady-states and co-adjoint orbits.     

流体力学方程组一类人为解构造方法

针对柱对称二维流体力学方程组,基于考虑方程右端附加源项的人为构造解方法,构造出一类统一形式的人为解.此类形式的人为解,对验证多维流体力学应用程序的正确性有重要的作用.同时将该类统一形式的人为解应用到PPM格式的程序,验证了构造的人为解的可行性.