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In this paper, we derive some local a priori estimates for the Ricci flow. This gives rise to some strong uniqueness theorems. As a corollary, let g(t) be a smooth complete solution to the Ricci flow on R3, with the canonical Euclidean metric E as initial data, then g(t) is trivial, i.e. g(t) ≡ E.

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We consider numerical boundary conditions for high order finite difference schemes for solving convection-diffusion equations on arbitrary geometry. The two main difficulties for numerical boundary conditions in such situations are: (1) the wide stencil of the high order finite difference operator requires special treatment for a few ghost points near the boundary; (2) the physical boundary may not coincide with grid points in a Cartesian mesh and may intersect with the mesh in an arbitrary fashion. For purely convection equations, the so-called inverse Lax-Wendroff procedure, in which we convert the normal derivatives into the time derivatives and tangential derivatives along the physical boundary by using the equations, have been quite successful. In this paper, we extend this methodology to convection-diffusion equations. It turns out that this extension is non-trivial, because totally different boundary treatments are needed for the diffusion-dominated and the convection-dominated regimes. We design a careful combination of the boundary treatments for the two regimes and obtain a stable and accurate boundary condition for general convection-diffusion equations. We provide extensive numerical tests for one- and two-dimensional problems involving both scalar equations and systems, including the compressible Navier-Stokes equations, to demonstrate the good performance of our numerical boundary conditions.

We shall be concerned with the indicator$p$of an analytic functional μ on a complex manifold$U$: $$p(\varphi ) = \overline {\mathop {\lim }\limits_{t \to + \infty } } \frac{l}{t}\log \left| {\mu (e^{t\varphi } )} \right|,$$ where ϕ is an arbitrary analytic function on$U$. More specifically, we shall consider the smallest upper semicontinuous majorant$p$^{$J$}of the restriction of$p$to a subspace £ of the analytic functions. An obvious problem is then to characterize the set of functions$p$^{$J$}which can occur as regularizations of indicators. In the case when$U$=$C$^{$n$}and £ is the space of all linear functions on$C$^{$n$}, this set can be described more easily as the set of functions(0.1) $$\mathop {\lim }\limits_{\theta \to \zeta } \overline {\mathop {\lim }\limits_{t \to + \infty } } \frac{l}{t}\log \left| {u(t\theta )} \right|$$ of$n$complex variables ζ∈$C$^{$n$}where$u$is an entire function of exponential type in$C$^{$n$}. We hall prove that a function in$C$^{$n$}is of the form (0.1) for some entire function$u$of exponential type if and only if it is plurisubharmonic and positively homogeneous of order one (Theorem 3.4). The proof is based on the characterization given by Fujita and Takeuchi of those open subsets of complex projective$n$-space which are Stein manifolds.