ON THE WELL POSEDNESS OF INITIAL VALUE PROBLEM FOR EULER EQUATIONS OF INCOMPRESSIBLE INVISCID FLUID(Ⅱ)

The solvability of the Euler equations about incompressible inviscid fluid based on the stratification theory is discussed. And the conditions for the existence of formal solutions and the methods are presented for calculating all kinds of ill-posed initial value problems. Two examples are given as the evidences that the initial problems at the hyper surface does not exist any unique solution. Foundation item: the National Natural Science Foundation of China (19971054) Biography: Shen Zhen (1977−)     

Computations of Optimal Controls for Incompressible Flows

Abstract

This paper is concerned with numerical solutions of optimal control problems for unsteady, viscous, incompressible flows. In general, controls can be of the distributed type (external body force) or Dirichlet type 7lpar;e.g., boundary velocity). Here, wc only consider the former case, although most of what we present is also applicable to the latter. Two different optimization objectives and associated solution methodologies are described. One involves a global-in-time functional, the other a local-in-time functional. Which method is preferred depends on the specific application. Some test computational results are presented.     

Optimal Elastic Design for Prescribed Maximum Deflection

ABSTRACT

For the problems of the optimal elastic design with prescribed maximum deflection, a variational formulation is proposed, with reference to the one-or two-dimensional bending structures.

Necessary optimum conditions are found, and the physical features of the optimal solutions are discussed for the “absolute” minimum cost problems, and, when dealing with beams, for the solutions with piece-wise constant design function.

Some examples are solved by using numerical methods that are directly derived from the variational formulation.     

Optimal Design of Viscoelastic Structures under Forced Steady-State Vibration

ABSTRACT

ABSTRACT With minimum dynamic response as the design criterion, we derive by means of variational analysis a general set of equations governing optimal design of one-dimensional, viscoelastic structures acted on by harmonically varying external loading. The equations are specialized to problems of minimizing transverse vibrational response of beams by attaching optimal, nonuniform cover layers made of a solid, viscoelastic material on the beams. Several numerical solutions to such problems are presented.     

On the formal solution of initial value problem of Navier-Stokes equation

Ｉｎｒｅｆｅｒｅｎｃｅ [1 ] ,ｗｅｄｉｓｃｕｓｓｅｄｔｈｅｉｎｓｔａｂｉｌｉｔｙｏｆＮａｖｉｅｒ＿ＳｔｏｋｅｓｅｑｕａｔｉｏｎｏｎＲ3×Ｒ ,ａｎｄ ,ｉｎｔｈｅｉｌｌｕｓｔｒａｔｉｏｎｏｆｉｔ,ｗｅｐｒｅｓｅｎｔｅｄｓｏｍｅｅｘａｍｐｌｅｓｉｎｔｈｅｆｏｒｍｏｆｅｘａｃｔｓｏｌｕｔｉｏｎ .Ｔｈｅｂａｓｉｓｏｆｔｈｅｃｏｎｓｔｒｕｃｔｉｏｎｏｆｔｈｅｍｉｓｔｈｅ“ｅｑｕａｔｉｏｎｓｅｃｏｎｄａｉｒｅ”ｏｆｔｈｅｅｑｕａｔｉｏｎ .Ｉｎｆａｃｔ,ｔｈｅｓｅｅｘａｃｔｓｏｌｕｔｉｏｎｓａｒｅｓｏｍｅｌｉｍｉｔ…     

Periodic solutions of a class of higher order neutral type equations

In this paper, by ssing the theory of Fourier series, some necessary and sufficient conditions of existence and uniqueness of periodic solutions of a class of higher order neutral type equations are obtained. The main results by Shi Jianguo in “Discussion on the periodic solutions for linear equation of neutral type with constant coefficients” are improved, i. e., the condition |b₀|≠1 instead of the condtion |b₀|<1/2 of Theorem 1 by Shi Jianguo is given. Other theorems by Shi are rebuilt and improved according to the new assumption. Foundation item: the Natural Science Foundation of Yunnan Province, China (97A012G)     

Nontangential Limit of the Weyl m-Functions for the Ergodic Schrödinger Equation

This paper deals with the spectral and qualitative problems associated with the one-dimensional ergodic Schrödinger equation. Let A ₂ be the set of those energies for which the real projective flow admits an invariant linear measure with square integrable density function. On this set we calculate the directional derivative of the Floquet coefficient and prove the existence of a nontangential limit of the Weyl m-functions in the L ¹-topology. In particular, we verify that the known Dieft–Simon inequality for the derivative of the rotation number obtained from Kotani's theory is in fact an equality. In the bounded orbit case we deduce the uniform boundedness of the Weyl m-functions and obtain necessary and sufficient conditions to assure their uniform convergence.     

ON THE WELL POSEDNESS OF INITIAL VALUE PROBLEM FOR EULER EQUATIONS OF INCOMPRESSIBLE INVISCID FLUID(Ⅰ)

The ill posed initial value problem of the Euler equations and the formal solvability of ill posed problem based on stratification theory are discussed. For some ill posed initial value problems, the existence conditions of formal solutions and the methods of how to construct a formal solution are given. Finally, an example is given to discuss the ill posedness of the initial value problem on hyper plane {t=0} in R⁴, and explain that the problem has more than one solution. Foundation item: the National Natural Science Foundation of China (19971054) Biography: Shen Zhen (1977−)     

Variational approach to non-linear boundary value problems for elasto-plastic incompressible bending plate

Non-linear boundary value problems for inelastic isotropic homogeneous incompressible bending plate, within the range of J₂-deformation theory, are considered. An existence of the weak solution of the non-linear problem with clamped boundary condition is obtained in H²(Ω) by using monotone operator theory and Browder-Minty theorem. For linearization of the non-linear problem a monotone iteration scheme is constructed. It is shown that the sequence of potentials obtained from the sequence of approximate solutions (i.e. iterations), is a monotone decreasing one. Convergence of the iteration process in H²-norm is proved by using the convexity argument. Numerical solutions, based on finite-difference scheme, are given for linear bending problems with rigid clamped as well as simply supported boundary conditions. Further numerical examples are presented to illustrate the convergence of approximate solutions and monotonicity of the potentials as applied to the non-linear problems.     

STABILITY OF SYSTEM OF TWO-DIMENSIONAL NON-HYDROSTATIC REVOLVING FLUIDS

Applying the theory of stratification, it is proved that the system of the two-dimensional non-hydrostatic revolving fluids is unstable in the two-order continuous function class. The construction of solution space is given and the solution approach is offered. The sufficient and necessary conditions of the existence of formal solutions are expressed for some typical initial and boundary value problems and the calculating formulae to formal solutions are presented in detail.     

Duality among Optimal Design Problems for Torsional Vibration

ABSTRACT

Four types of mass and frequency optimization problems are stated for free torsional vibration of thin-walled cylinders subject to constraints on wall thickness and frequencies of vibration. It is shown, using Pontryagin's method, that the mathematical structure of all four problems is similar and leads to identical classes of optimal thickness distributions. These duality relations are used in an example to construct an optimal frequency solution from the solutions for both maximum and minimum mass problems. General relations among the governing parameters for the four problems are stated. The results of Grinev and Filippov and of Thermann for the abnormal optimization problems are verfied as a specific limiting example of the general results.     

EXISTENCE OF POSITIVE RADIAL SOLUTIONS FOR SOME SEMILINEAR ELLIPTIC EQUATIONS IN ANNULUS

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｓｔｕｄｉｅｓｏｆｐｏｓｉｔｉｖｅｒａｄｉａｌｓｏｌｕｔｉｏｎｓｆｏｒｆｏｌｌｏｗｉｎｇｓｅｍｉｌｉｎｅａｒｅｌｌｉｐｔｉｃｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍ(Ｐ)　Δｕ(Ｘ) +ｇ( ｜Ｘ｜)ｆ(ｕ(Ｘ) ) =0 ,　　Ｒ1<｜Ｘ｜<Ｒ2 ,ｕ(Ｘ) ｜｜Ｘ｜ =Ｒ1 =ｕ(Ｘ) ｜｜Ｘ｜ =Ｒ2 =0(ｗｈｅｒｅＲ1>0 ,Ｘ ∈Ｒｎ,ｎ ≥ 2 )ａｒｅｂｅｉｎｇｃｏｎｔｉｎｕｅｄｆｏｒｒｅｃｅｎｔ 2 0ｙｅａｒｓｗｉｔｈｏｕｔｉｎｔｅｒｒｕｐｔｉｏｎ[1- 11],ｂｅｃａｕｓｅｔｈｅｐｒｏｂｌｅｍ (Ｐ)ｈａｓｗｉ…     

A stability study of Navier-Stokes equations (III)

In this paper, the necessary conditions of the existence of C² solutions in some initial problems of Navier-Stokes equations are given, and examples of instability of initial value (at t=0) problems are also given. The initial value problem of Navier-Stokes equation is one of the most fundamental problem for this equation various authors studies this problem and contributed a number of results. J. Lerav, a French professor, proved the existence of Navier-Stokes equation under certain defined initial and boundary value conditions. In this paper, with certain rigorously defined key concepts, based upon the basic theory of J. Hadamard partial differential equations¹, gives a fundamental theory of instability of Navier-Stokes equations. Finally, many examples are given, proofs referring to Ref. [4].     

2<Emphasis Type="Italic">T</Emphasis>-periodic solution for<Emphasis Type="Italic">m</Emphasis> order neutral type differential equations with time delays

Periodic solution of m order linear neutral equations with constant coefficient and time delays was studied. Existence and uniqueness of 2 T-periodic solutions for the equation were discussed by using the method of Fourier series. Some new necessary and sufficient conditions of existence and uniqueness of 2T-periodic solutions for the equation are obtained. The main result is used widely. It contains results in some correlation paper for its special case, improves and extends the main results in them. Existence of periodic solution for the equation in larger number of particular case can be checked by using the result, but cannot be checked in another paper. In other words, the main result in this paper is most generalized for (1), the better result cannot be found by using the same method. Foundation item: the Natural Science Foundation of Yunnan Education Committee (990002) Biography: ZHANG Bao-sheng (1962-)     

A class of exact solutions for <Emphasis Type="Italic">N</Emphasis>-dimensional incompressible magnetohydrodynamic equations

In this paper, a sufficient and necessary condition is presented for existence of a class of exact solutions to N-dimensional incompressible magnetohydrodynamic (MHD) equations. Such solutions can be explicitly expressed by appropriate formulae. Once the required matrices are chosen, solutions to the MHD equations are directly constructed.     

An Optimal Control Approach to Minimum-Weight Vibrating Beam Design

ABSTRACT

A gradient projection algorithm is applied to a class of vibrating cantilever beam optimization problems which are formulated as optimal control problems. The cross-section area is distributed along the beam for minimum total weight subject to fixed natural frequency constraints and a minimum allowed area limit. Three topics receive major emphasis: the effects of shear deformation and rotary inertia, higher-mode frequency constraints, and multiple frequency constraints. Computational results are presented for the cases of fixed fundamental frequency, fixed second-mode frequency, and fixed fundamental and second-mode frequencies under a variety of conditions.     

Basic Geometrical Properties of Optimal Flexural Force Transmission Fields

Abstract

On the basis of theories of optimal design by Masur, Prager and Shield, kinematic optimality conditions for elastic and elastic-plastic plane flexural systems of maximum strength and maximum stiffness were previously derived. In the present paper, general geometrical properties of moment and displacement fields in various types of optimal regions are outlined with a view to constructing optimal solutions for any arbitrary set of boundary conditions.     

A note on strochastic optimal control of reflected diffusions with jumps

Stochastic optimal control problems for a class of reflected diffusion with Poisson jumps in a half-space are considered. The nonlinear Nisio’s semigroup for such optimal control problems was constructed. A Hamilton-Jacobi-Bellman equation with the Neumann boundary condition associated with this semigroup was obtained. Then, viscosity solutions of this equation were defined and discussed, and various uniqueness of this equation was also considered. Finally, the value function was such optimal control problems is shown to be a viscosity solution of this equation.