TIME GLOBAL MILD SOLUTIONS OF NAVIER-STOKES-OSEEN EQUATIONS

In this paper we prove the existence and uniqueness of time global mild solutions to the Navier-Stokes-Oseen equations,which describes dynamics of incompressible viscous fluid flows passing a translating and rotating obstacle,in the solenoidal Lorentz space L_(σ,w)^(3)·Besides,boundedness and polynomial stability of these solutions are also shown.

Existence and Continuous Dependence of Mild Solutions for Some Fractional Neutral Differential Equations with Nonlocal Initial Conditions

The existence,uniqueness,and continuous dependence to the mild solutions of the nonlocal Cauchy problem were proved for a class of semilinear fractional neutral differential equations.The results are obtained by using the Krasnoselskii's fixed point theorem and the theory of resolvent operators for integral equations.

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].

EXISTENCE OF MILD SOLUTIONS TO SEMILINEAR FRACTIONAL ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS

In this paper,we use the analytic semigroup theory of linear operators and fixed point method to prove the existence of mild solutions to a semilinear fractional order functional differential equations in a Banach space.

Controllability of Singular Distributed Parameter Systems in the Sense of Mild Solution

Necessary and sufficient conditions for the exact controllability and approximate controllability of a singular distributed parameter system are obtained.These general results are used to examine the exact controllability and approximate controllability of the Dzektser equation in the theory of seepage.

Mild Solution of Stochastic Equations with Levy Jumps： Existence, Uniqueness, Regularity and Stability

The existence and uniqueness of mild solution to stochastic equations with jumps are established, a stochastic Fubini theorem and a type of Burkholder-Davis- Gundy inequality are proved, and the two formulas are used to study the regularity property of the mild solution of a general stochastic evolution equation perturbed by Levy process. Then the authors prove the moment exponential stability, almost sure exponential stability and comparison principles of the mild solution. As applications, the stability and comparison principles of stochastic heat equation with Levy jump are given.

Weighted Pseudo Asymptotically Periodic Mild Solutions of Evolution Equations

In this paper, we propose a new class of functions called weighted pseudo S-asymptotically periodic function in the Stepanov sense and explore its properties in Banach space including composition theorems. Furthermore, the existence, uniqueness of the weighted pseudo S-asymptotically periodic mild solutions to partial evolution equations and nonautonomous semilinear evolution equations are investigated. Some interesting examples are presented to illustrate the main findings.

Existence of Mild Solutions for a Class of Fractional Non-autonomous Evolution Equations with Delay

In this paper,we consider the existence and uniqueness of the mild solutions for a class of fractional non-autonomous evolution equations with delay and Caputo's fractional derivatives.By using the measure of noncompactness,β-resolvent family,fixed point theorems and Banach contraction mapping principle,we improve and generalizes some related results on this topic.At last,we give an example to illustrate the application of the main results of this paper.

Analyticity of Mild Solution for the 3D Incompressible Magneto-hydrodynamics Equations in Critical Spaces

The Cauchy problem for the 3D incompressible magneto-hydrodynamics equations in crit- cal spaces is considered. We first prove the global well-posedness of mild solution for the system in some time dependent spaces. Furthermore, we obtain analyticity of the solution.

EXISTENCE OF MILD SOLUTION TO NONLINEAR NEUTRAL STOCHASTIC DIFFERENTIAL SYSTEM WITH DELAY

This paper is concerned with the existence of solution to a nonlinear neutral stochastic diferential system with delay in a Hilbert Space. Sufcient conditions for the existence are obtained using the Schaefer fxed point theorem.

STABILITY OF THE MILD SOLUTION OF STOCHASTIC NONLINEAR EVOLUTION DIFFERENTIAL EQUATIONS

In this paper, we consider the stability problem associated with the mild solutions of stochastic nonlinear evolution differential equations in Hilbert space under hypothesis which is weaker than Lipschitz condition. And the result is established by employing the Ito-type inequality and the extension of the Bihari＇s inequality.

EXISTENCE OF MILD SOLUTIONS TO NEUTRAL FRACTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

In this paper, we consider the existence of mild solution to a class of neutral fractional differential equations with infinite delay. By means of fixed points methods, we obtain some sufficient conditions for the existence and uniqueness of mild solutions, which extend some known results.

Existence of Solutions for a Non-Autonomous Evolution Equations with Nonlocal Conditions

The existence of mild solutions for non-autonomous evolution equations with nonlocal conditions in Banach space is studied in this article. We obtained the existence of at least one mild solution to the evolution equations by using Krasnoselskii’s fixed point theorem as well as the theory of the evolution family. The interest of this paper is that any assumptions are not imposed on the nonlocal terms and Green’s functions and a new alternative method is applied to prove the existence of mild solutions. The results obtained in this paper may improve some related conclusions on this topic. An example is given as an application of the results.

Existence of Solutions for Semilinear Impulsive Delay Differential Equations with Nonlocal Conditions

In this paper,by using Schaefer fixed-point theorem,the existence of mild solutions of semilinear impulsive delay differential equations with nonlocal conditions is studied.The results obtained are a generalization and continuation of the recent results on this issue.In the end,an example is given to show the application of the results.

Stability of Solution Sets of Evolution Inclusions

In this paper we consider the stability of fixed points of the multivalued Nemitsky operator and, further, obtain stability of solution sets of evolution inclusions. Our work extends the stability results of Lim and Constantin.

Existence of almost periodic solutions to a class of non- autonomous functional integro-differential stochastic equations

In this paper, a class of non-autonomous functional integro-differential stochastic equations in a real separable Hilbert space is studied. When the operators A（t） satisfy Acquistapace-Terreni conditions, and with some suitable assumptions, the existence and uniqueness of a square-mean almost periodic mild solution to the equations are obtained.

A Study on Stochastic Differential Equation Using Fractional Power of Operator in the Semigroup Theory

Stochastic differential equation (SDE) is an ordinary differential equation with a stochastic process that can model the unpredictable real-life behavior of any continuous systems. It is the combination of differential equations, probability theory, and stochastic processes. Stochastic differential equations arise in modeling a variety of random dynamic phenomena in physical, biological and social process. The SDE theory is traditionally used in physical science and financial mathematics. Recently, more researchers have been conducted in the application of SDE theory to various areas of engineering. This dissertation is mainly concerned with the existence of mild solutions for impulsive neutral stochastic differential equations with nonlocal conditions in Hilbert spaces. The results are obtained by using fractional powers of operator in the semigroup theory and Sadovskii fixed point theorem.

Entropy Formulation for Triply Nonlinear Degenerate Elliptic-Parabolic-Hyperbolic Equation with Zero-Flux Boundary Condition

In this note, we investigated existence and uniqueness of entropy solution for triply nonlinear degenerate parabolic problem with zero-flux boundary condition. Accordingly to the case of doubly nonlinear degenerate parabolic hyperbolic equation, we propose a generalization of entropy formulation and prove existence and uniqueness result without any structure condition.

Existence results for a class of parabolic evolution equations in Banach spaces

We discuss the existence results of the parabolic evolution equation d(x(t)+g(t,x(t)))/dt+A(t)x(t)=f(t,x(t)) in Banach spaces, where A(t) generates an evolution system and functions f,g are continuous. We get the theorem of existence of a mild solution, the theorem of existence and uniqueness of a mild solution and the theorem of existence and uniqueness of an S-classical (semi-classical) solution. We extend the cases when g(t)=0 or A(t)=A.

Nonlocal controllability for semilinear problems in Banach spaces

If A： D（A） X→X is a densely defined and closed linear operator, which generates a linear semigroup S （t） in Banach space X. The nonlocal control/ability for the following nonlocal semilinear problems： u＇ （t） = Au （t） ＋ Bx（ t） ＋ f（ t, u（t） ）, 0≤t ≤ T with nonlocal initial condition u（0） = u0 ＋ g（u） is discussed in Banach space X. The results show that if semigroup S（t） is strongly continuous, the functionsf and g are compact and the control B is bounded, then it is nonlocally controllable. The nonlocal controllability for the above nonlocal problem is also studied when B and W are unbounded and the semigroup S（t） is compact or strongly continuous. For illustration, a partial differential equation is worked out.