A Grothendieck compactness principle for the Mackey dual topology

Grothendieck [6] proved that every norm compact subset of a Banach space is contained in the closed convex hull of a norm null sequence. In a recent paper [3], an analogous result for weak compactness in a Banach space is shown to be equivalent to the Schur property. In this article, we obtain a similar type result in the Mackey dual of a Banach space. A related result for weak^? compactness is also obtained.     

Random attractors for stochastic reaction‐diffusion equations with multiplicative noise in

In this paper, we study the random dynamical system generated by a stochastic reaction‐diffusion equation with multiplicative noise and prove the existence of an ‐random attractor for such a random dynamical system. The nonlinearity f is supposed to satisfy some growth of arbitrary order .     

The defocusing energy-supercritical Hartree equation

In this paper,we study the global well-posedness and scattering problem for the energysupercritical Hartree equation iut+Δu.(|x|.γ.|u|2)u=0 with γ4 in dimension d γ.We prove that if the solution u is apriorily bounded in the critical Sobolev space,that is,u ∈Lt∞(I;Hxsc(Rd)) with sc:= γ/2.11,then u is global and scatters.The impetus to consider this problem stems from a series of recent works for the energy-supercritical nonlinear wave equation(NLW) and nonlinear Schrdinger equation(NLS).We utilize the strategy derived from concentration compactness ideas to show that the proof of the global well-posedness and scattering is reduced to disprove the existence of three scenarios:finite time blowup;soliton-like solution and low to high frequency cascade.Making use of the No-waste Duhamel formula,we deduce that the energy of the finite time blow-up solution is zero and so get a contradiction.Finally,we adopt the double Duhamel trick,the interaction Morawetz estimate and interpolation to kill the last two scenarios.     

Fock-Sobolev空间上有界、紧与S_p-类算子

本文研究Fock-Sobolev空间上稠密定义算子,将这些算子统一表示成积分算子,利用积分算子的方法得到了它们的一个充分条件,并构造反例说此充分条件是非必要的,还得到这些算子为紧算子的两个充分条件.最后构造符号函数在复平面上每一点处本性无界的紧和Sp-类(0p∞)Toeplitz算子.     

ON THE BOUNDEDNESS OF SPHERES IN F-SPACES

In this paper, some results concerning the relationship between the bounded-ness of some spheres and the local boundedness of the F*-space are presented. Moreover, some results about the compactness are also given.     

Weak Continuity and Compactness for Nonlinear Partial Differential Equations

This paper presents several examples of fundamental problems involving weak continuity and compactness for nonlinear partial differential equations, in which compensated compactness and related ideas have played a significant role. The compactness and convergence of vanishing viscosity solutions for nonlinear hyperbolic conservation laws are first analyzed, including the inviscid limit from the Navier-Stokes equations to the Euler equations for homentropic flow, the vanishing viscosity method to construct the global spherically symmetric solutions to the multidimensional compressible Euler equations, and the sonic-subsonic limit of solutions of the full Euler equations for multi-dimensional steady compressible fluids. Then the weak continuity and rigidity of the Gauss-Codazzi-Ricci system and corresponding isometric embeddings in differential geometry are revealed. Further references are also provided for some recent developments on the weak continuity and compactness for nonlinear partial differential equations.     

Combined effects of singular and critical nonlinearities in elliptic problems

In this paper we consider the existence and multiplicity of positive weak solutions for a class of elliptic problems with the nonlinearity containing both singular and critical terms. By means of the concentration compactness principle due to Lions and Ekeland’s variational principle, two positive weak solutions are obtained.     

Relative Compactness for Capacities, Measures, Upper Semicontinuous Functions and Closed Sets

We prove an Ascoli theorem for capacities. This theorem which characterizes relatively compact sets of capacities is widely applicable and many Ascoli theorems for particular classes of capacities can immediately be deduced as corollaries. Indeed it is usually necessary only to demonstrate that these classes are closed and then to simplify the characterization when possible. In particular, we show that the proof of the classical Prohorov theorem can be naturally factored into the shorter proof of the Ascoli theorem for capacities and into the somewhat longer proof that the class of probability measures is closed in the class of capacities. We also deduce new and known Ascoli theorems for sup measures, upper semi-continuous functions, the Vietoris hyperspace topology, and various classes of measures.     

带有临界Sobolev指数的奇异双调和问题

LetΩR~N be a smooth bounded domain such that 0∈Ω,N≥5,2~*:=(2N)/(N-4) is the critical Sobolev exponent,and f(x) is a given function.By using the variational methods, the paper proves the existence of solutions for the singular critical in the homogeneous problemΔ~u-μu/(|x|~4)=|u|~(2~*-2)u f(x) with Dirichlet boundary condition on Ωunder some assumptions on f(x) andμ.