New Developments on Existence and Uniqueness of Solutions to Some Models in Atmospheric Dynamics

This survey is concerned with the new developments on existence and uniqueness of solutions of some basic models in atmospheric dynamics, such as two-and three-dimensional quasi-geostrophic models and three-dimensional balanced model. The main aim of this paper is to introduce some results about the global and local (with respect to time) existence of solutions given by the authors in recent years, but others' important contributions and the literature on this subject are also quoted. We discuss briefly the relationships among the existence and uniqueness, physical instability and computational instability. In the appendixes, some key mathematical techniques in obtaining our results are presented, which are of vital importance to other problems in geophysical fluid dynamics as well.

UNIQUENESS FOR THE SOLUTIONS OF ELASTIC THIN PLATES AND SHALLOW SHELLS AS WELL AS THEIR CONTACT PROBLEM WITH HALF SPACE

The uniqueness for the solutions mentioned in the subject is proved by using the uniqueness of the solution for the internal boundary problem of Laplace and bi-Laplace equations of the first kind as well as of the second.

UNIQUENESS OF SOLUTIONS OF SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS FOR THIRD QRDER NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS

By making use of the differential inequalities, in this paper we study the uniqueness of solutions of the two kinds of the singularly perturbed boundary value problems for the nonlinear third order ordinary differential equation with a small parameter ε>0: where i=1, 2; a(?)(ε), β(ε) and γ(ε) are functions defined on (0, ε_o], while ε_o>0 is a constant.This paper is the continuation of our works [4, 6].

UNIQUENESS OF THE MILD SOLUTION OF SEMILINEAR STOCHASTIC EVOLUTION EQUATION IN HILBERT SPACE

In this paper, we will consider following initial value problem of semilinear stochastic evolution equation in Hilbert Space: [GRAPHICS] where W(t) is a wiener process in H, H and Y are two real separable Hilbert Spaces, A is an infinitesimal generator of a strongly continuous semigroup s(t) on Y, f(t, y): [0, T] x Y --> Y, and G(t, y): [0, T] X Y --> L(H, Y), y0: OMEGA --> Y is a ramdom variable of square integrable. We apply theory of the semigroup and obtain two conclusions of uniqueness of the mild solution of (1) which include the corresponding results in [4].

Pathwise Uniqueness of the Solutions toVolterra Type Stochastic DifferentialEquations in the Plane

In this paper we prove the pathwise uniqueness of a kind of two-parameter Volterra type stochastic differential equations under the coefficients satisfy the non-Lipschitz conditions. We use a martingale formula in stead of Ito formula, which leads to simplicity the process of proof and extends the result to unbounded coefficients case.

EXISTENCE,UNIQUENESS AND APPROXIMATION OF THE SOLUTION OF AN ODE WITH (INFINITELY MANY) STATE-DEPENDENT IMPULSES VIA FIXED MESH GALERKIN FORMULATION

A fixed mesh variational formulation is used to establish existence and uniqueness of the solution of ordinary differential equations with (in finitely many) state-dependent in pulses on the right-hand side. This approach gives a natural numerical scheme to approximate the solution.The convergence of the approximation is proved and its asymptatic order obtained.

Semi-analytic solution of Eringen's two-phase local/nonlocal model for Euler-Bernoulli beam with axial force

Eringen’s two-phase local/nonlocal model is applied to an Euler-Bernoulli nanobeam considering the bending-induced axial force, where the contribution of the axial force to bending moment is calculated on the deformed state. Basic equations for the corresponding one-dimensional beam problem are obtained by degenerating from the three-dimensional nonlocal elastic equations. Semi-analytic solutions are then presented for a clamped-clamped beam subject to a concentrated force and a uniformly distributed load, respectively. Except for the traditional essential boundary conditions and those required to be satisfied by transferring an integral equation to its equivalent differential form, additional boundary conditions are needed and should be chosen with great caution, since numerical results reveal that non-unique solutions might exist for a nonlinear problem if inappropriate boundary conditions are used. The validity of the solutions is examined by plotting both sides of the original integro-differential governing equation of deflection and studying the error between both sides. Besides, an increase in the internal characteristic length would cause an increase in the deflection and axial force of the beam.

Linear mathematical model for the unique solution of 3D ptychographic iterative engine

Diffraction intensities of the 3D ptychographic iterative engine(3PIE)were written as a set of linear equations of the selfcorrelations of Fourier components of all sample slices,and an effective computing method was developed to solve these linear equations for the transmission functions of all sample slices analytically.With both theoretical analysis and numerical simulations,this study revealed the underlying physics and mathematics of 3PIE and demonstrated for the first time,to our knowledge,that 3PIE can generate mathematically unique reconstruction even with noisy data.

ON THE SOLUTION OF NONLINEARTWOPOINT BOUNDARY VALUE PROBLEM──u" + g(t.u) = f(t),u(0) = u(2π) = 0

In this paper, a non-variational version of a max-min principle is proposed, andan existence and uniqueness result is obtained for the nonlinear two-point boundaryvalue problenl u' + g(t.u) = f(t),u(0) = u(2π) = 0

Uniqueness for the Linear Theory of Generalized Thermo-Elasticity

In this paper, the uniqueness for the linear theory of a new generalized thermo-elastic model of the continuum with the centre symmetry is shown under less assumptions, whose constitutive equations contain deformation, temperature and its rate as well as its gradient, electric field and its gradient. So the phase variation is pemitted when the deformation proceeds as long as the constitutive equations preserve their Original forms.

EXISTENCE AND UNIQUENESS FOR SECOND-ORDER VECTOR BOUNDARY VALUE PROBLEM OF NONLINEAR SYSTEMS

This paper is concerned with the following second-order vector boundary value problem ：x^R=f（t,Sx,x,x＇）,0〈t〈1,x（0）=A,g（x（1）,x＇（1））=B,where x,f,g,A and B are n-vectors. Under appropriate assumptions,existence and uniqueness of solutions are obtained by using upper and lower solutions method.

The Long-Term Dynamic Behavior of Solutions to a Class of Generalized Higher-Order Kirchhoff-Type Coupled Wave Equations

In this paper, we study the long-term dynamic behavior of a class of generalized high-order Kirchhoff-type coupled wave equations. Firstly, the existence of uniqueness global solution of this kind of equations in Ek space is proved by prior estimation and Galerkin method;Then, through using Rellich-Kondrachov compact embedding theorem, it is proved that the solution semigroup S(t) has the family of the global attractors Ak in space Ek;Finally, through linearization method, proves that the operator semigroup S(t) Frechet differentiable and the attenuation of linearization problem volume element. Furthermore, we can obtain the finite Hausdorff dimension and Fractal dimension of the family of the global attractors Ak.

On Security of Quantum Cryptography by Probabilistic Analysis

The existing quantum cryptography is a classical cryptography in nature and basically insecure because of its classical (conventional) bits, classical encryption algorithm and classical (public) channel. A novel topic about successful communication between the legitimate users, Alice and Bob, is discussed with probability of solution uniqueness of Bob’s decryption equation. We find, by probabilistic analysis, that success of communication between Alice and Bob is probabilistic with a probability bigger than 1/2. It is also novel to define insecurity of the quantum cryptography by probability of solution uniqueness of the search equation of Eve, the eavesdropper. The probability of Eve’s success to find the plain-text of Alice (and Bob) is greater than 1/2, and so the quantum cryptography is seriously insecure.

Uniqueness and stability of solution for Cauchy problem of degenerate quasilinear parabolic equations

The uniqueness and existence of BV-solutions for Cauchy problem of the form (e)u/(e)t=△A(u)+N∑i=1(e)bi(u)/(e)xi, A'u≥0,u(x,0)=uo are proved.

The Dynamic Behavior of a Class of Kirchhoff Equations with High Order Strong Damping

In this paper, we study the long time behavior of a class of Kirchhoff equations with high order strong dissipative terms. On the basis of the proper hypothesis of rigid term and nonlinear term of Kirchhoff equation, firstly, we evaluate the equation via prior estimate in the space E0 and Ek, and verify the existence and uniqueness of the solution of the equation by using Galerkin’s method. Then, we obtain the bounded absorptive set B0k on the basis of the prior estimate. Moreover, by using the Rellich-Kondrachov Compact Embedding theorem, we prove that the solution semigroup S(t) of the equation has the family of the global attractor Ak in space Ek. Finally, we prove that the solution semigroup S(t) is Frechet differentiable on Ek via linearizing the equation. Furthermore, we can obtain the finite Hausdorff dimension and Fractal dimension of the family of the global attractor Ak.

Global Attractors and Their Dimension Estimates for a Class of Generalized Kirchhoff Equations

In this paper, we studied the long-time properties of solutions of generalized Kirchhoff-type equation with strongly damped terms. Firstly, appropriate assumptions are made for the nonlinear source term g (u) and Kirchhoff stress term M (s) in the equation, and the existence and uniqueness of the solution are proved by using uniform prior estimates of time and Galerkin’s finite element method. Then, abounded absorption set B0k is obtained by prior estimation, and the Rellich-kondrachov’s compact embedding theorem is used to prove that the solution semigroup S (t) generated by the equation has a family of the global attractor Ak in the phase space . Finally, linearize the equation and verify that the semigroups are Frechet diifferentiable on Ek. Then, the upper boundary estimation of the Hausdorff dimension and Fractal dimension of a family of the global attractor Ak was obtained.

BIANALYTIC FUNCTIONS, BIHARMONIC FUNCTIONS AND ELASTIC PROBLEMS IN THE PLANE

Let the elastic body only be acted by gravity. By investigating the relations of bianalytic functions and biharmonic functions, the uniqueness and existence of the stress functions (Airy functions) are established in planar simple connected region. Moreover, the integral representation formula of the stress functions in the unit disk of the plane is obtained.

PERIODIC BOUNDARY VALUE PROBLEM AND CAUCHY PROBLEM OF THE GENERALIZED CUBIC DOUBLE DISPERSION EQUATION

In this article, the existence, uniqueness and regularities of the global generalized solution and global classical solution for the periodic boundary value problem and the Cauchy problem of the general cubic double dispersion equationutt - uxx - auxxtt ＋ bux4 - duxxt = f（u）xxare proved, and the sufficient conditions of blow-up of the solutions for the Cauchy problems in finite time are given.

UNIQUE EXISTENCE OF PERIODIC SOLUTIONS OF NEUTRAL VOLTERRA INTEGRODIFFERENTIAL EQUATIONS

This paper deals with the uniqueness and existence of periodic solutions of neutral Volterraintegrodifferential equations (1) and (2). Some new unique existence criteria are obtained.

On the Existence,Uniqueness and Stability of Periodic Solutions of Large Scale Systems with Unbounded Delay

In this paper,we consider the periodic solution problems for the systems with unbounded delay,and the existence,uniqueness and stability of the periodic solution are dealt with unitedly.First we establish the suitable delay-differential inequality,then study seperately the problems of periodic solution for the systems with bounded delay,with unbounded delay and the Volterra integral-dlfferentlal systems with infinite delay by using the character of matrix measure and the asymptotic fixed point theorem of poincaré’s periodic operator in the different phase spaces.A series of simple criteria for the existence,uniqueness and stability of these systems are obtained.