Abstract:
In this paper, a new algorithm based on differential geometry viewpoint to solve the 3D rotating Navier-Stokes equations with complex Boundary is proposed, which is called Bi-parallel algorithm. For xample, it can be applied to passage flow between two blades in impeller and circulation flow through aircrafts with complex geometric shape of boundary. Assume that a domain in $R^3$ can be decomposed into a series sub-domain, which is called "flow layer", by a series smooth surface $\Im_k, k=1,...,M$. Applying differential geometry method, the 3D Navier-Stokes operator can be split into two kind of operator: the "Membrane Operator" on the tangent space at the surface $\Im_k$ and the "Bending Operator" along the transverse direction. The Bending Operators are approximated by the finite different quotients and restricted the 3D Naver-Stokes equations on the interface surface $\Im_k$, a Bi-Parallel algorithm can be constructed along two directors: "Bending" direction and "Membrane" directors. The advantages of the method are that: (1) it can improve the accuracy of approximate solution caused of irregular mesh nearly the complex boundary; (2) it can overcome the numerically effects of boundary layer, whic is a good boundary layer numerical method; (3) it is sufficiency to solve a two dimensional sub-problem without solving 3D sub-problem.

Abstract:
In this paper, the
boundary layer equations (abbreviation BLE) for exterior flow around an obstacle
are established using semi-geodesic coordinate system (S-coordinate) based on the curved two dimensional surface of the
obstacle. BLE are nonlinear partial differential equations on unknown normal
viscous stress tensor and pressure on the obstacle and the existence of
solution of BLE is proved. In addition a dimensional split method for dimensional
three Navier-Stokes equations is established by applying several 2D-3C partial
differential equations on two dimensional manifolds to approach 3D
Navier-Stokes equations. The examples for the exterior flow around spheroid and
ellipsoid are presents here.

Abstract:
This paper is mainly based on the integrated framework for disaster risk and risk field theory, building an event risk field model and using this model for group events. Then, the paper describes group events for the concept, analyzes the characteristics and types of mass incidents and constructs the risk field model according to the risk of field theory. Finally, the paper gives a case triggered by taxi transport strike incident and puts forward the mass event control strategy according to the use of the risk field model.

Abstract:
This paper deals with an inverse problem of determining a nonlinear source term in a quasilinear diffusion equation with overposed final observations. Applying integral identity methods, data compatibilities are deduced by which the inverse source problem here is proved to be reasonable and solvable. Furthermore, with the aid of an integral identity that connects the unknown source terms with the known data, a conditional stability is established.

Abstract:
In this paper A Geometrical Design Method for Blade's surface $\Im$ in the impeller is provided here $\Im$ is a solution to a coupling system consisting of the well-known Navier-Stokes equations and a four order elliptic boundary value problem . The coupling system is used to describe the relations between solutions of Navier-Stokes equations and the geometry of the domain occupied by fluids, and also provides new theory and methods for optimal geometric design of the boundary of domain mentioned above. This coupling system is the Eular-Lagrange equations of the optimal control problem which is describing a new principle of the geometric design for the blade's surface of an impeller. The control variable is the surface of the blade and the state equations are Navier-Stokes equations with mixed boundary conditions in the channel between two blades. The objective functional depending on the geometry shape of blade's surface describes the dissipation energy of the flow and the power of the impeller. First we prove the existence of a solution of the optimal control problem. Then we use a special coordinate system of the Navier-Stokes equations to derive the objective functional which depends on the surface $\Theta$ explicitly. We also show the weakly continuity of the solution of the Navier-Stokes equations with respect to the geometry shape of the blade's surface.

Abstract:
\noindent{\bf Abstract:} In this paper, the authors propose the concept of stream surface and stream layer. By using classical tensor calculus, the authors derive 3D Navier-Stokes Equations(NSE) in the stream layer undersemigeodesic coordinate system, Navier-Stokes equation on the stream surface and 2-D Navier-Stokes equations on a two dimensional manifold. After introducing stream function on the stream surface, a nonlinear initial-boundary value problem satisfied by stream function is obtained,existence and uniqueness of its solution are proved. Based on this theory the authors propose a new method called ``dimension split method" to solve 3D NSE.

Abstract:
To construct the embedded subbundles of free bundles and the embedded free subbundles of any module bundles, by using embedded relation of module bundles. It is indicated that any module bundles can becomes a embedded subbundles of free bundles,and that there is a free bundles (or projective bundles) is a embedded subbundles of any module bundles also. In particular, the condition which projective bundles change into free bundles is given.

Abstract:
In this paper, three general principles for constructing approximate inertial manifolds are provided, under which the associate approximate inertial form of origin problem, which is a finite dimensional ordinary differential equation, is well-possed and its solution will approximate the genuine solution at some degree. At last, for some kinds of approximate inertial manifolds and a family of approximate inertial manifolds, we indicate that the principles given here are suitable.

Abstract:
A optimum ffote element nonlinear Galerkin algorithm is presented for thetwo-dimensional nonstationary Navier-Stokes equations. The standard finite elemellt Galerkin algorithIn consists in solving a nonlinear equation on the fine gridfinite elemellt space Xh' The optimum finite element nonlinear Galerkin algorithm consists in solving a nonlinear subproblem on a coarse grid finite elementspace XH(H > h) and solving a linear subproblem on a fine grid incremental finite element space Wh = (I - RH)Xh- If H is chosen such that H = O(h1/2),then two algorithms are of the c0nvergence rate of same order. However, sinceH >> h, dimXH << dimXh, the optimum finite elemellt nonlinear Galerkin algorithm can save a large amoullt of comPutational time. Finally, we give thenumerical test which shows the correctness of theoretical analysis.