THE WELL-POSEDNESS OF FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS IN COMPLEX BANACH SPACES

Let X be a complex Banach space and let B and C be two closed linear operators on X satisfying the condition D(B)?D(C),and let d∈L^(1)(R_(+))and 0≤β<α≤2.We characterize the well-posedness of the fractional integro-differential equations D^(α)u(t)+CD^(β)u(t)=Bu(t)+∫_(-∞)td(t-s)Bu(s)ds+f(t),(0≤t≤2π)on periodic Lebesgue-Bochner spaces L^(p)(T;X)and periodic Besov spaces B_(p,q)^(s)(T;X).

ALMOST SURE GLOBAL WELL-POSEDNESS FOR THE FOURTH-ORDER NONLINEAR SCHR?DINGER EQUATION WITH LARGE INITIAL DATA

We consider the fourth-order nonlinear Schr?dinger equation(4NLS)(i?t+εΔ+Δ2)u=c1um+c2(?u)um-1+c3(?u)2um-2,and establish the conditional almost sure global well-posedness for random initial data in Hs(Rd)for s∈(sc-1/2,sc],when d≥3 and m≥5,where sc:=d/2-2/(m-1)is the scaling critical regularity of 4NLS with the second order derivative nonlinearities.Our proof relies on the nonlinear estimates in a new M-norm and the stability theory in the probabilistic setting.Similar supercritical global well-posedness results also hold for d=2,m≥4 and d≥3,3≤m<5.

GLOBAL WELL-POSEDNESS AND OPTIMAL TIME DECAY RATES FOR THE GENERALIZED PHAN-THIEN-TANNER MODEL IN R^(3)

In this paper, we consider the initial value problem for the incompressible generalized Phan-Thien-Tanner(GPTT) model. This model pertains to the dynamic properties of polymeric fluids. Under appropriate assumptions on smooth function f, we find a particular solution to the GPTT model. In dimension three, we establish the global existence and the optimal time decay rates of strong solutions provided that the initial data is close to the particular solution. The results which are presented here are generalizations of the network viscoelastic models.

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.

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.

GLOBAL WELL-POSEDNESS FOR FRACTIONAL NAVIER-STOKES EQUATIONS IN VARIABLE EXPONENT FOURIER-BESOV-MORREY SPACES

In this paper we study the Cauchy problem of the incompressible fractional Navier-Stokes equations in critical variable exponent Fourier-Besov-Morrey space F N^(s(·))p(·),h(·),q(R^(3))with s(·)=4-2α-3/p(·).We prove global well-posedness result with small initial data belonging to FN^(4-2α-3/p(·))p(·),h(·)q(R^(3)).The result of this paper extends some recent work.

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 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 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.

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.

On well-posedness of two-phase nonlocal integral models for higher-order refined shear deformation beams

Due to the conflict between equilibrium and constitutive requirements,Eringen’s strain-driven nonlocal integral model is not applicable to nanostructures of engineering interest.As an alternative,the stress-driven model has been recently developed.In this paper,for higher-order shear deformation beams,the ill-posed issue(i.e.,excessive mandatory boundary conditions(BCs)cannot be met simultaneously)exists not only in strain-driven nonlocal models but also in stress-driven ones.The well-posedness of both the strain-and stress-driven two-phase nonlocal(TPN-Strain D and TPN-Stress D)models is pertinently evidenced by formulating the static bending of curved beams made of functionally graded(FG)materials.The two-phase nonlocal integral constitutive relation is equivalent to a differential law equipped with two restriction conditions.By using the generalized differential quadrature method(GDQM),the coupling governing equations are solved numerically.The results show that the two-phase models can predict consistent scale-effects under different supported and loading conditions.

STOKES COUPLING METHOD FOR THE EXTERIOR FLOW PART Ⅱ: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.

GlobalWell-Posedness for the 2-DMHD Equations with Magnetic Diffusion

In this paper,we consider the 2-D MHD equations with magnetic resistivity but without dissipation on the torus.We prove that if the initial data is small in H4(T2),then the 2-D MHD equations are globally well-posed.To our knowledge,this is the first global well-posedness result for this system.

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.

THE LOCAL WELL-POSEDNESS OF A CHEMOTAXIS-SHALLOW WATER SYSTEM WITH VACUUM

In this paper we prove the local well-posedness of strong solutions to a chemotaxisshallow water system with initial vacuum in a bounded domainΩ■R^(2)without the standard compatibility condition for the initial data.This improves some results obtained in[J.Differential Equations 261(2016),6758-6789].

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.

GLOBAL WELL-POSEDNESSOF A PRANDTL MODEL FROM MHD IN GEVREY FUNCTION SPACES

We consider a Prandtl model derived from MHD in the Prandtl-Hartmann regime that has a damping term due to the effect of the Hartmann boundary layer.A global-in-time well-posedness is obtained in the Gevrey function space with the optimal index 2.The proof is based on a cancellation mechanism through some auxiliary functions from the study of the Prandtl equation and an observation about the structure of the loss of one order tangential derivatives through twice operations of the Prandtl operator.

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 .