An Estimate for a Fundamental Solution of a Parabolic Equation with Drift on a Riemannian Manifold

We construct a fundamental solution for a parabolic equation with drift on a Riemannian manifold of nonpositive curvature. We obtain some estimates for this fundamental solution that depend on the conditions on the drift field.     

Perturbed Parabolic Equation on a Riemannian Manifold

We construct a fundamental solution of an equation with perturbed diffusion operator on a manifold of nonnegative curvature.     

The Logarithmic Gradient of the Kernel of the Heat Equation with Drift on a Riemannian Manifold

For a parabolic equation with drift on a Riemannian manifold of positive curvature we obtain a representation for the logarithmic gradient in the form of the sum of two vector fields one of which is known and the other is bounded. The drift field is assumed to be of sufficiently rapid decay at infinity.     

具有边界摄动的抛物型方程的可解性

本文研究了具有边界摄动的抛物型方程.利用可解性的讨论,得到了原问题的摄动解.     

紧致黎曼流形上一类非线性热方程的梯度估计

In this paper,we study gradient estimates for the nonlinear heat equation ut-△u =au log u,on compact Riemannian manifold with or without boundary.We get a Hamilton type gradient estimate for the positi...     

黎曼流形上一类非线性抛物方程的Hamilton 梯度估计

本文我们得到了黎曼流形上一类非线性抛物方程的局部Hamilton梯度估计. 利用这个局部估计,我们得到了一个Harnack型不等式和一个Liouville型定理.     

Behavior of the Double-Layer Potential for a Parabolic Equation on a Manifold

We prove that, similarly to the double-layer potential in , the double-layer potential constructed in a Riemann manifold of nonpositive sectional curvature has a jump in passing through the surface where its density is defined.     

Riemann流形上的抛物型Monge-Ampère方程

This paper deals with a class of parabolic Monge-Ampère equation on Riemannian manifolds. The existence and uniqueness of the solution to the first initial-boundary value problem for the equation are established.     

L1-Uniqueness of the Fokker-Planck Equation on a Riemannian Manifold

Potential Analysis - In this paper, we obtain a necessary and sufficient condition for L∞-uniqueness of Sturm-Liouville operator $a(x)\frac {d^{2}}{dx^{2}} + b(x) \frac d{dx} -V$ on an open...     

Nash Embedding,Shape Operator and Navier-Stokes Equation on a Riemannian Manifold

What is the suitable Laplace operator on vector fields for the Navier-Stokes equation on a Riemannian manifold? In this note, by considering Nash embedding, we will try to elucidate different aspects of different Laplace operators such as de Rham-Hodge Laplacian as well as Ebin-Marsden’s Laplacian. A probabilistic representation formula for Navier-Stokes equations on a general compact Riemannian manifold is obtained when de Rham-Hodge Laplacian is involved.     

黎曼流形上薛定谔方程的Harnack估计

推导了薛定谔方程正解的一种新的整体梯度估计和Harnack不等式,推广了一些有关热方程的结论,并且得到了一个关于薛定谔算子的刘维尔定理.     

Semigroup Domination on a Riemannian Manifold with Boundary

We discuss the semigroup domination on a Riemannian manifold with boundary. Our main interest is the Hodge–Kodaira Laplacian for differential forms. We consider two kinds of boundary conditions; the absolutely boundary condition and the relative boundary condition. Our main tool is the square field operator. We also develop a general theory of semigroup commutation.     

Construction of an Inertial Manifold of a Parabolic Equation with a Monotonic Nonlinear Part

Sufficient conditions are obtained for the existence of an inertial manifold of a parabolic equation with a monotonic nonlinear part in a bounded region of not more than three dimensions. The existence of an inertial manifold is proved for a reaction diffusion equation viewed on a two-dimensional torus.     

Parameter Estimation with the Augmented Lagrangian Method for a Parabolic Equation

In this paper, we investigate the numerical identification of the diffusion parameters in a linear parabolic problem. The identification is formulated as a constrained minimization problem. By using the augmented Lagrangian method, the inverse problem is reduced to a coupled nonlinear algebraic system, which can be solved efficiently with the preconditioned conjugate gradient method. Finally, we present some numerical experiments to show the efficiency of the proposed methods, even for identifying highly discontinuous parameters.This work was partially supported by the Research Council of Norway, Grant NFR-128224/431.     

抛物型偏微分方程奇异摄动问题的边界层方法

本文讨论抛物型偏微分方程奇异摄动问题,通常,为了使边界层的特性不致丧失,在边界层附近必须减小网格,当网格足够小时需要很大的运算量。我们提出边界层格式,在边界层附近不必取很细的网格,数值例子表明采用中等步长即可满足精度。     

黎曼流形上的Nash不等式

本文通过对满足Nash不等式的黎曼流形的研究,证明了对任一完备的Ricci曲率非负的n维黎曼流形,若它满足Nash不等式,且Nash常数大于最佳Nash常数,则它微分同胚于Rn．     

局部对称空间中具有平行平均曲率向量的黎曼叶层的Pinching定理

本文利用Nakagawa和Takagi的计算散度的方法,求出局部对称空间中具有平行平均曲率向量的黎曼叶状结构${\cal F}$上向量场的散度,并证明了其上的整体Pinching定理.     

Biharmonic Submanifolds in a Riemannian Manifold with Non-Positive Curvature

In this paper, we show that, for every biharmonic submanifold (M, g) of a Riemannian manifold (N, h) with non-positive sectional curvature, if ${\int_M\vert \eta \vert^2 v_g < \infty}$ , then (M, g) is minimal in (N, h), i.e., ${\eta\equiv0}$ , where η is the mean curvature tensor field of (M, g) in (N, h). This result gives an affirmative answer under the condition ${\int_M\vert \eta \vert^2 v_g < \infty}$ to the following generalized Chen’s conjecture: every biharmonic submanifold of a Riemannian manifold with non-positive sectional curvature must be minimal. The conjecture turned out false in case of an incomplete Riemannian manifold (M, g) by a counter example of Ou and Tang (in The generalized Chen’s conjecture on biharmonic sub-manifolds is false, a preprint, 2010).