Heat Flow for Extrinsic Biharmonic Maps with Small Initial Energy

Let M ^m and N ^{n k} be two compact Riemannian manifolds without boundary. We consider the L ² gradient flow for the energy F(u):= _M |u|². If m 3 or if m= 4 and F(u ₀) is small, we show that the heat flow for extrinsic biharmonic maps exists for all time, and that the solution subconverges to a smooth extrinsic biharmonic map as time goes to infinity.     

The HeatKernel ofthe Sym m etric Space P_n

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The p-Harmonic Heat Flow with Potential into Homogeneous Spaces

In this paper we prove the existence of global weak solutions of the p-harmonic flow with potential between Riemannian manifolds M and N for arbitrary initial data having finite p-energy in the case when the target N is a homogeneous space with a left invariant metric. Received March 17, 1999, Revised September 22, 1999, Accepted October 15, 1999     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-invariants of Riemannian foliations

For a Riemannian foliation on a closed manifold M, we define L ²-spectral sequence Betti numbers and spectral sequence Novikov–Shubin invariants. The spectral sequence of the lift of to the universal covering of M is used in the definitions. These invariants are natural extensions of the L ²-Betti numbers and the Novikov–Shubin invariants of differentiable manifolds. It is shown that these numbers are invariant by foliated homotopy equivalences, and they are computed for several examples.      

Inverse Problems for Differential Forms on Riemannian Manifolds with Boundary

Consider a real-analytic orientable connected complete Riemannian manifold M with boundary of dimension n ≥ 2 and let k be an integer 1 ≤ k ≤ n. In the case when M is compact of dimension n ≥ 3, we show that the manifold and the metric on it can be reconstructed, up to an isometry, from the set of the Cauchy data for harmonic k-forms, given on an open subset of the boundary. This extends a result of [14 Lassas , M. , Uhlmann , G. ( 2001 ). On determining a Riemannian manifold from the Dirichlet-to-Neumann map . Ann. Sci. École Norm. 34 : 771 – 787 .[Web of Science ®] , [Google Scholar]] when k = 0. In the two-dimensional case, the same conclusion is obtained when considering the set of the Cauchy data for harmonic 1-forms. Under additional assumptions on the curvature of the manifold, we carry out the same program when M is complete non-compact. In the case n ≥ 3, this generalizes the results of [13 Lassas , M. , Taylor , M. , Uhlmann , G. ( 2003 ). The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary . Comm. Anal. Geom. 11 : 207 – 221 .[Crossref], [Web of Science ®] , [Google Scholar]] when k = 0. In the two-dimensional case, we are able to reconstruct the manifold from the set of the Cauchy data for harmonic 1-forms.     

Time Asymptotic Behaviour for a One-Velocity Transport Operator with Maxwell Boundary Condition

This paper is concerned with the spectral analysis of a one-velocity transport operator with Maxwell boundary condition in L ₁-space. After a detailed spectral analysis it is shown that the associated Cauchy problem is governed by a C ₀-semigroup. Next, we discuss the irreducibility of the transport semigroup. In particular, we show that the transport semigroup is irreducible. Finally, a spectral decomposition of the solutions into an asymptotic term and a transient one which will be estimated for smooth initial data is given.     

Nonlinear Markov Operators,Discrete Heat Flow,and Harmonic Maps Between Singular Spaces

We develop a theory of harmonic maps f:MN between singular spaces M and N. The target will be a complete metric space (N,d) of nonpositive curvature in the sense of A. D. Alexandrov. The domain will be a measurable space (M,) with a given Markov kernel p(x,dy) on it. Given a measurable map f:MN, we define a new map Pf:MN in the following way: for each xM, the point Pf(x)N is the barycenter of the probability measure p(x,f ^–1(dy)) on N. The map f is called harmonic on DM if Pf=f on D. Our theory is a nonlinear generalization of the theory of Markov kernels and Markov chains on M. It allows to construct harmonic maps by an explicit nonlinear Markov chain algorithm (which under suitable conditions converges exponentially fast). Many smoothing and contraction properties of the linear Markov operator P _M,R carry over to the nonlinear Markov operator P=P _M,N . For instance, if the underlying Markov kernel has the strong Lipschitz Feller property then all harmonic maps will be Lipschitz continuous.     

Large Time Behavior of a Weak Solution for Non-Newtonian Flow

This paper concerns large time behavior of a regular weak solution for non-Newtonian flow equations. It is shown that the decay of the solution is of exponential type when the force term is equal to zero and the domain is bounded. Moreover, the ratio of the enstrophy over the energy has a limit as time tends to infinity, which is an eigenvalue of the Stokes operator.     

热交换多孔圆盘间三阶流体的MHD轴对称流动

在两个具有热交换可渗透的多孔圆盘之间,研究三阶流体的磁流体动力学(MHD)流动.通过适当变换,将偏微分的控制方程转换为常微分方程.采用同伦分析法(HAM)求解转换后的方程.定义了均方残余误差的表达式,并选择了最佳的、收敛的控制参数值.检测了无量纲参数变化时的无量纲速度和温度场.列表显示表面摩擦因数和Nusselt数,并分析了无量纲参数的影响.     

放/吸热流体在两个充满多孔材料的竖直平行板之间作不稳定的自然对流

在一个由两块无限竖直平行板组成的管道中,充满着多孔的介质材料,使用Darcy模型(Brinkman模型的推广)的动量方程,连同能量方程,计算不可压缩、粘性、放/吸热流体在该管道中的不稳定自然对流,即Couette流动.流动是由于边界平板有不对称的加热,以及作加速运动所引起.选用合理的无量纲参数,对控制方程进行简化,通过Laplace变换进行解析求解,得到闭式的速度和温度分布曲线解,随后导出表面摩擦力和传热率.发现在竖直管道中的不同剖面,流体的流动及温度分布曲线随着时间而增加,且在运动平板附近更高.特别是,流体的速度和温度随着平板间距的增加而增加,但是,表面摩擦力和热传导率随着平板间距的增加而减小.     

The Blow-up Locus of Heat Flows for Harmonic Maps

Abstract Let M and N be two compact Riemannian manifolds. Let u _k (x, t) be a sequence of strong stationary weak heat flows from M×R ⁺ to N with bounded energies. Assume that u _k→u weakly in H ^{1, 2}(M×R ⁺, N) and that Σ^t is the blow-up set for a fixed t > 0. In this paper we first prove Σ^t is an H ^m−2-rectifiable set for almost all t∈R ⁺. And then we prove two blow-up formulas for the blow-up set and the limiting map. From the formulas, we can see that if the limiting map u is also a strong stationary weak heat flow, Σ^t is a distance solution of the (m− 2)-dimensional mean curvature flow [1]. If a smooth heat flow blows-up at a finite time, we derive a tangent map or a weakly quasi-harmonic sphere and a blow-up set ∪_t<0Σ^t× {t}. We prove the blow-up map is stationary if and only if the blow-up locus is a Brakke motion. This work is supported by NSF grant     

凝析油-气两相流偏微分方程组问题的求解及应用

根据凝析气藏开发的实际需要，引入凝析油－气两相拟压力、拟时间函数，建立了凝析气在双重渗透率介质中的不稳定渗流新模型，该模型为一类偏微分方程组的混合问题，本文研究了这类偏微分方程组问题的求解方法。     

时间序列系统建模预测的一种新方法

利用微分方程数值解法对时间序列系统建模作了新的探讨 ,给出了单调型和起伏型时间序列的建模预测方法 .本文给出的方法适用范围广泛 ,尤其对间隔较小的时间序列能获得满意的精度 ,文中最后给以建模实例 .     

A Gradient Estimate for Solutions of the Heat Equation II

The author obtains an estimate for the spatial gradient of solutions of the heat equation, subject to a homogeneous Neumann boundary condition, in terms of the gradient of the initial data. The proof is accomplished via the maximum principle; the main assumption is that the sufficiently smooth boundary be convex.     

均匀自由流动的非牛顿流体中连续表面上的磁流体动力学流动和热传递

分析在平行自由流动的非牛顿黏弹性导电流体中,连续平展表面移动时的稳态流和热传递特性,该流动处于横向均匀磁场作用下.以二阶流体构建它的本构方程,得到了速度分布和温度断面图的数值结果.讨论了诸如黏弹性参数、磁场参数和Prandtl数等不同物理参数对诸种动量和热传递特性的影响,并给出相关图示.     

A Numerical Study of Blowup in the Harmonic Map Heat Flow Using the MMPDE Moving Mesh Method

The numerical solution of the harmonic heat map flow problems with blowup in finite or infinite time is considered using an adaptive moving mesh method. A properly chosen monitor function is derived so that the moving mesh method can be used to simulate blowup and produce accurate blowup profiles which agree with formal asymptotic analysis. Moreover, the moving mesh method has finite time blowup when the underlying continuous problem does. In situations where the continuous problem has infinite time blowup, the moving mesh method exhibits finite time blowup with a blowup time tending to infinity as the number of mesh points increases. The inadequacy of a uniform mesh solution is clearly demonstrated.     

Oldroyd B流体依时性管内流动的变分解析方法

在本文中，研究上随体Ｏｌｄｒｏｙｄ Ｂ流体在水平管内依时性流动，该问题可归结为无量纲速度分量三阶偏微分方程的初边值问题，采用改进的Ｋａｎｔｏｒｏｖｉｃｈ方法，将该方程化为各级近似的二阶常微分方程组的初值问题，通过Ｌａｐｌａｃｅ变换，求得其二阶常微分方程的解析解。在本文中，提出了变分解析的新概念，获得了二级近似变分解析解，其中包括常压力力梯度和周期性压力梯度两种情形，应用计算机符呈处理和Ｌａｐｌａｃ     

An inverse spectral problem for a nonsymmetric differential operator: Reconstruction of eigenvalue problem

In this paper, an algorithm is established to reconstruct an eigenvalue problem from the given data satisfying certain conditions. These conditions are proved to be not only necessary but also sufficient for the given data to coincide with the spectral characteristics corresponding to the reconstructed eigenvalue problem.