Fluid Structure Interaction Problems:the Necessity of a Well Posed,Stable and Accurate Formulation

We investigate problems of fluid structure interaction type and aim for a formulation that leads to a well posed problem and a stable numerical procedure.Our first objective is to investigate if the generally accepted formulations of the fluid structure interaction problem are the only possible ones.Our second objective is to derive a stable numerical coupling.To accomplish that we will use a weak coupling procedure and employ summation-by-parts operators and penalty terms.We compare the weak coupling with other common procedures.We also study the effect of high order accurate schemes.In multiple dimensions this is a formidable task and we start by investigating the simplest possible model problem available.As a flow model we use the linearized Euler equations in one dimension and as the structure model we consider a spring.

ON WELL-POSEDNESS OF THE MULTIOBJECTIVE GENERALIZED GAME

By extending the concept of asymptotic weakly Pareto-Nash equilibrium point to vector-valued case, Tikhonov well-posedness and Hadamard well-posedness results of the multiobjective generalized games are established in this paper.

THE WELL-POSEDNESS AND ASYMPTOTICS OF MULTI-DIMENSIONAL QUANTUM HYDRODYNAMICS

The multi-dimensional quantum hydrodynamic equations for charge transport in ultra-small electronic devices like semiconductors, where quantum effects （like particle tunnelling through potential barriers and built-up in quantum wells） take place, is considered in the present paper, and the recent progress on well-posedness, stability analysis, and small scaling limits are reviewed.

GLOBAL WELL-POSEDNESS OF THE STOCHASTIC 2D BOUSSINESQ EQUATIONS WITH PARTIAL VISCOSITY

This paper deals with the stochastic 2D Boussinesq equations with partial viscosity. This is a coupled system of Navier-Stokes/Euler equations and the transport equation for temperature under additive noise. Global well-posedness result of this system under partial viscosity is proved by using classical energy estimates method.

GLOBAL WELL-POSEDNESS OF THE 2D INCOMPRESSIBLE MICROPOLAR FLUID FLOWS WITH PARTIAL VISCOSITY AND ANGULAR VISCOSITY

This paper is concerned with the two-dimensional equations of incompress- ible micropolar fluid flows with mixed partial viscosity and angular viscosity. The global existence and uniqueness of smooth solution to the Cauchy problem is established.

WELL-POSEDNESS IN CRITICAL SPACES FOR THE FULL COMPRESSIBLE MHD EQUATIONS

In this paper we prove local well-posedness in critical Besov spaces for the full compressible MHD equations in R^N, N≥ 2, under the assumptions that the initialdensity is bounded away from zero. The proof relies on uniform estimates for a mixed hyperbolic/parabolic linear system with a convection term.

LEVITIN-POYAK WELL-POSEDNESS OF GENERALIZED VECTOR EQUILIBRIUM PROBLEMS WITH FUNCTIONAL CONSTRAINTS

In this article, we study Levitin-Polyak type well-posedness for generalized vector equilibrium problems with abstract and functional constraints. Criteria and characterizations for these types of well-posednesses are given.

STOKES COUPLING METHOD FOR THE EXTERIOR FLOW PARTⅡ:WELL-POSEDNESS ANALYSIS

In this paper,we recall the Stokes coupling method for solving the exteriorunsteady Navier-Stokes equations.Moreover, we derive the coupling variational formulation of the Stokes coupling problem by use of the integral representations of the solntion of the Stokes equations at an infinite domain.Finain Finally, we provide some properties of the integral operators over the articleal boundary and the well-posedness of the coupling variational formulation.

ON THE WELL-POSEDNESS OF THE INITIAL VALUE PROBLEM OF NON-STATIC ROTATING FLUID

A systematic study was made on the topological nature of the system of non-static rotating fluid. Several initial (boundary) value problems and their well-posedness were discussed.

GLOBAL WELL-POSEDNESS FOR A FIFTH-ORDER SHALLOW WATER EQUATION ON THE CIRCLE

The periodic initial value problem of a fifth-order shallow water equation t u 2 x t u ＋ 3 x u 5 x u ＋ 3u x u 2 x u 2 x u u 3 x u = 0 is shown to be globally well-posed in Sobolev spaces˙ H s （T） for s 〉 2/3 by I-method. For this equation lacks scaling invariance, we first reconsider the local result and pay special attention to the relationship between the lifespan of the local solution and the initial data, and then prove the almost conservation law, and finally obtain the global well-posedness by an iteration process.

On the Well-Posedness for Optimization Problems: A Theoretical Investigation

In this paper, some theoretical notions of well-posedness and of well-posedness in the generalized sense for scalar optimization problems are presented and some important results are analysed. Similar notions of well-posedness, respectively for a vector optimization problem and for a variational inequality of differential type, are discussed subsequently and, among the various vector well-posedness notions known in the literature, the attention is focused on the concept of pointwise well-posedness. Moreover, after a review of well-posedness properties, the study is further extended to a scalarizing procedure that preserves well-posedness of the notions listed, namely to a result, obtained with a special scalarizing function, which links the notion of pontwise well-posedness to the well-posedness of a suitable scalar variational inequality of differential type.

Well-Posedness of an N-Unit Series System with Finite Number of Vacations

We investigate the solution of an N-unit series system with finite number of vacations. By using C0-semigroup theory of linear operators, we prove well-posedness and the existence of the unique positive dynamic solution of the system.

GLOBAL WELL-POSEDNESS AND SCATTERING FOR THE MASS-CRITICAL HARTREE EQUATION IN HIGH DIMENSIONS

We obtain global well-posedness and scattering, and global L2（d＋2）/d t,x spacetime bounds for solutions to the defocusing mass-critical Hartree equation in Rt×Rx^d,d≥5.

LOCAL WELL-POSEDNESS IN SOBOLEV SPACES WITH NEGATIVE INDICES FOR A SEVENTH ORDER DISPERSIVE EQUATION

This paper is concerned with the Cauchy problem of a seventh order dispersive equation. We prove local well-posedness with initial data in Sobolev spaces Hs（R） for negative indices of s〉-114 .

WELL-POSEDNESS OF INITIAL VALUE PROBLEM FOR EULER EQUATIONS OF INVISCID COMPRESSIBLE ADIABATIC FLUID

The well-posedness of the initial value problem of the Euler equations was mainly discussed based on the stratification theory, and the necessary and sufficient conditions of well-posedness are presented for some representative initial or boundary value problem, thus the structure of solution space for local （exact） solution of the Euler equations is determined. Moreover the computation formulas of the analytical solution of the well-posed problem are also given.

SHARP WELL-POSEDNESS OF THE CAUCHY PROBLEM FOR THE HIGHER-ORDER DISPERSIVE EQUATION

This current paper is devoted to the Cauchy problem for higher order dispersive equation ut＋δx^2n＋1u=δx（uδx^nu）＋δx^n-1（ux^2）, n ≥ 2, n ∈ N^＋. Ut By using Besov-type spaces, we prove that the associated problem is locally well-posed in H（-n/2＋3/4,-1/2n）. The new ingredient is that we establish some new dyadic bilinear estimates. When n is even, we also prove that the associated equation is ill-posed in H^（s,a）（R） with s〈-n/2＋3/4 and all a∈R.

GLOBAL WELLPOSEDNESS OF MAGNETOHYDRODYNAMICS SYSTEM WITH TEMPERATURE-DEPENDENT VISCOSITY

The initial boundary value problem of the one-dimensional magneto-hydrodynamics system, when the viscosity, thermal conductivity, and magnetic diffusion coefficients are general smooth functions of temperature, is considered in this article. A unique global classical solution is shown to exist uniquely and converge to the constant state as the time tends to infinity under certain assumptions on the initial data and the adiabatic exponent γ. The initial data can be large if γ is sufficiently close to 1.

On the well-posedness for stochastic fourth-order Schrdinger equations

The influence of the random perturbations on the fourth-order nonlinear SchrSdinger equations,iut＋△^2u＋ε△u＋λ｜u｜^p-1u=ξ,（t,x）∈R^＋×R^n,n≥1,ε∈{-1,0,＋1},is investigated in this paper. The local well-posedness in the energy space H^2（R^n） are proved for p 〉n＋4/n＋2,and p≤2^#-1 if n≥5.Global existence is also derived for either defocusing or focusing L^2-subcritical nonlinearities.

Well-posedness of the time-varying linear electromagnetic initial-boundary value proble

The well-posedness of the initial-boundary value problem of the time-varying linear electromagnetic field in a multi-medium region is investigated. Function spaces are defined, with Faraday＇s law of electromagnetic induction and the initial-boundary conditions considered as constraints. Gauss＇s formula applied to a multi-medium region is used to derive the energy-estimating inequality. After converting the initial-boundary conditions into homogeneous ones and analysing the characteristics of an operator introduced according to the total current law, the existence, uniqueness and stability of the weak solution to the initial-boundary value problem of the time-varying linear electromagnetic field are proved.

GLOBAL WELL-POSEDNESS IN ENERGY SPACE OF SMALL AMPLITUDE SOLUTIONS FOR KLEIN-GORDON-ZAKHAROV EQUATION IN THREE SPACE DIMENSION

The Cauchy problem of the Klein-Gordon-Zakharov equation in three dimensional space is considered. It is shown that it is globally well-posed in energy space H^1 × L^2 × L^2 × H^-1 if small initial data （u0 （x）, u1 （x）, n0 （x）, n1 （x）） ∈ （H^1 ×L^2× L^2 × H^-1）. It answers an open problem： Is it globally well-posed in energy space H^1 × L^2 × L^2 × H^-1 for 3D Klein-Gordon- Zakharov equation with small initial data [1, 2]？ The method in this article combines the linear property of the equation （ dispersive property） with nonlinear property of the equation （energy inequalities）. We mainly extend the spaces F^s and N^3 in one dimension [3] to higher dimension.