On NAVIER-STOKES and KELVIN-VOIGT Equations in Three Dimensions in Interpolation Spaces

We show for the three-dimensional initial-boundary value problem of the NAVIER -STOKES and KELVIN -VOIGT equation over bounded domains and for the corresponding stationary problem the existence of weak solutions in intermediate spaces between some of the usual spaces for weak solutions of the NAVIER -STOKES equations and spaces for the corresponding strong solutions. The proofs yield estimates of the solutions in terms of the data which are supposed to be in appropriate intermediate spaces. The basic ingredients of the proof are well-known results for weak and strong solutions and some nonlinear interpolation arguments.     

热方程的一些端点估计及其在Navier-Stokes方程中的应用——献给陆善镇教授75华诞

本文首先讨论热方程初值问题的解在Hardy、BMO（bounded mean oscillation）和Besov型空间中的估计.然后本文结合Coifmann-Lions-Meyer-Semmes在Hardy空间中的补偿紧性结果,给出Navier-Stokes方程整体弱解的二阶导数的一些端点估计.     

Sobolev type embedding and weak solutions with a prescribed singular set

New Sobolev type embeddings for some weighted Banach spaces are established. Using such embeddings and the singular positive radial entire solutions, we construct singular positive weak solutions with a prescribed singular set for a weighted elliptic equation. Our main results in this paper also provide positive weak solutions with a prescribed singular set to an equation with Hardy potential.     

Some nonlinear elliptic systems with right-hand side integrable data with respect to the distance to the boundary

In this paper, we study the very weak solutions to some nonlinear elliptic systems with right-hand side integrable data with respect to the distance to the boundary. Firstly, we study the existence of the approximate solutions. Secondly, a priori estimates are given in the framework of weighted spaces. Finally, we prove the existence, uniqueness and regularity of the very weak solutions.     

Remarks on regularity criteria for the weak solutions of liquid crystals

In this paper, we consider the regularity criteria for weak solutions of liquid crystals. It is proved that the solution is in fact smooth if the velocity or the velocity gradient belongs to some critical multiplier spaces or Tribel–Lizorkin spaces. As a corollary, we obtain the Beal–Kato–Majda criteria for liquid crystals.     

On a theorem by Sohr for the Navier-Stokes equations

In this paper we study some criteria for the full (space-time) regularity of weak solutions to the Navier-Stokes equations. In particular, we generalize some classical and very recent criteria involving the velocity, or its derivatives. More specifically, we show with elementary tools that if a weak solution, or its vorticity, is small in appropriate Marcinkiewicz spaces, then it is regular.     

Regularity for P-Harmonic Type Systems with the Gradients below the Controllable Growth

In this paper, we shall establish that each weak solution of p-harmonic type systems with the gradients below the controllable growth belongs to, Holder continuity spaces with any HSlder exponent α∈ [0, 1）. Furthermore, we can obtain that the gradients of the corresponding weak solutions also belong to locally Hoelder continuity spaces with some Hoelder exponent. Keywords controllable growth, p-harmonic systems, full regularity MR（2000） Subject Classification 35J60, 35B65     

On Generalized Vector Equilibrium Problems

In this paper we prove the existence of solutions of the generalized vector equilibrium problem in the setting of Hausdorff topological vector spaces. As applications, we present some relevant particular cases: a generalized vector variational-like inequality in Hausdorff topological vector spaces, and equilibrium problem in the case of pseudomonotone real functions, and a generalized weak Pareto optima problem.     

Spaces of Quasi-Periodic Functions and Oscillations in Differential Equations

We build spaces of q.p. (quasi-periodic) functions and we establish some of their properties. They are motivated by the Percival approach to q.p. solutions of Hamiltonian systems. The periodic solutions of an adequatez partial differential equation are related to the q.p. solutions of an ordinary differential equation. We use this approach to obtain some regularization theorems of weak q.p. solutions of differential equations. For a large class of differential equations, the first theorem gives a result of density: a particular form of perturbated equations have strong solutions. The second theorem gives a condition which ensures that any essentially bounded weak solution is a strong one.     

On the Existence of Weak Solutions of Nonlinear Integral Equations in Banach Spaces

In this paper, we use the techniques of measure of weak noncompactness and Henstock–Kurzweil–Pettis integrals to discuss the existence theorem of weak solutions for a class of the nonlinear integral equations and obtain a new result, which improves and extends some relevant results for differential and integral equations in Banach spaces.     

Well-posedness of higher-order Camassa-Holm equations

We consider higher-order Camassa-Holm equations describing exponential curves of the manifold of smooth orientation-preserving diffeomorphisms of the unit circle in the plane. We establish the existence of global weak solutions. We also present some invariant spaces under the action of the equation. Moreover, we prove a “weak equals strong” uniqueness result.     

The initial boundary value problems for the semilinear diffusion equations with data in L p spaces

In this paper, I have established conditions under which the existence and uniqueness of weak solutions of some Semilinear diffusion equations with initial and boundary data in fractional LL ^p spaces can be established in a bounded domain with smooth boundary The interest of this method relies on the fact that it is by successive approximations and hence amenable to numerical treatment. The paper also considers the semigroups theory on the existence of weak classical solutions.     

Clifford Valued Weighted Variable Exponent Spaces with an Application to Obstacle Problems

In this paper, we introduce the weighted variable exponent spaces in the context of Clifford algebras. After discussing the properties of these spaces, we obtain the existence of weak solutions for obstacle problems for nondegenerate A-Dirac equations with variable growth in the setting of these spaces. Furthermore, we also obtain the existence and uniqueness of weak solutions to the scalar parts of nondegenerate A-Dirac equations in Dirac Sobolev spaces.     

ON HYPERSTABILITY OF THE BIADDITIVE FUNCTIONAL EQUATION

We present results on approximate solutions to the biadditive equation f(x+y,z-w) + f(x-y,z+w) = 2f(x,z)-2f(y,w)on a restricted domain. The proof is based on a quite recent fixed point theorem in some function spaces. Our main results state that,under some weak natural assumptions,functions satisfying the equation approximately(in some sense) must be actually solutions to it. In this way we obtain inequalities characterizing biadditive mappings and inner product spaces.Our outcomes are connected with the well known issues of Ulam stability and hyperstability.     

弱拓扑下的非线性随机积分和微分方程组的解*

在本文中,我们首先对具有随机定义域的弱连续随机算子组证明了一个Darbo型随机不动点定理.利用这一定理,我们对Banach空间中关于弱拓扑的非线性随机Volterra积分方程组给出了随机解的存在性准则.作为应用,我们得到了非线性随机微分方程组的Canchy问题弱随机解的存在定理.也得到了这些随机方程组在Banach空间中关于弱拓扑的极值随机解的存在性和随机比较结果.我们的定理改进和推广了Szep,Mitchell－Smith,Cramer－Lakshmikantham,Lakshmikantham-Leela和丁的相应结果.     

Degenerate non-stationary differential equations with delay in Banach spaces

A generalization of the operator method by Grisvard is used to ensure weak and strict solutions to some degenerate differential equations with delay in Banach spaces, whose operator coefficients are time depending. Some applications to ordinary and partial differential equations with delay are described.     

Sewing method for weak solutions of second-order hyperbolic equation with variable domains of discontinuous unbounded operators

We prove the existence and uniqueness of global weak solutions on the entire interval for the Cauchy problem for hyperbolic differential-operator equations with time-discontinuous operators that have variable domains and satisfy certain matching conditions at the points of discontinuity. To this end, we develop a method of successive sewing of existing local weak solutions of Cauchy problems on the smoothness intervals of the operators. The sewing method is based on special energy inequalities, which imply the time continuity of local weak solutions in the main space and of their first derivatives in some negative spaces and hence the existence of the corresponding limit values at the points of discontinuity. These values, with regard for the matching conditions, are taken for the initial data on each successive interval.     

Iterative Algorithms for Mixed Equilibrium Problems,Strict Pseudocontractions and Monotone Mappings

In this paper, we introduce some iterative algorithms for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a strict pseudocontraction and the set of solutions of a variational inequality for a monotone, Lipschitz continuous mapping. We obtain both weak and strong convergence theorems for the sequences generated by these processes in Hilbert spaces.     

EXISTENCE OF WEAK SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS IN BANACH SPACES

0 IntroductionRecently, the study of ordinary differential equations in a Banach spacehas been developed extensively by using the strong topology, see [2],13],[4]. Atpresent, however, there is no reference or paper which systematically studiesweak solutions of the Cauchy problem of ordinary differential equation in theweak topology. In references [l],15],[7] and [8], several results of weak solutionsare proved. In this paper, we shall give the other results of weak solutions,some of which are…     

次临界增长P-调和组的处处内部正则性

对于低阶梯度项满足次临界增长的p-调和型方程组,本文建立了其弱解梯度具有处处内部H■lder连续性的正则性结果,本文结论就低阶项的增长指标来说已经达到最佳.