Pullback attractors for a class of extremal solutions of the 3D Navier-Stokes system

In this paper we construct a dynamical process (in general, multivalued) generated by the set of solutions of an optimal control problem for the three-dimensional Navier-Stokes system. We prove the existence of a pullback attractor for such multivalued process. Also, we establish the existence of a uniform global attractor containing the pullback attractor. Moreover, under the unproved assumption that strong globally defined solutions of the three-dimensional Navier-Stokes system exist, which guaranties the existence of a global attractor for the corresponding multivalued semiflow, we show that the pullback attractor of the process coincides with the global attractor of the semiflow.     

The Khavinson-Shapiro conjecture and polynomial decompositions

The main result of the paper states the following: Let ψ be a polynomial in n variables. Suppose that there exists a constant C>0 such that any polynomial f has a polynomial decomposition f=ψqf+hf with Δkhf=0 and . Then . Here Δk is the kth iterate of the Laplace operator Δ. As an application, new classes of domains in Rn are identified for which the Khavinson-Shapiro conjecture holds.     

Radial symmetry and regularity of solutions for poly-harmonic Dirichlet problems

Let B=B₁(0) be the unit ball in Rn and r=|x|. We study the poly-harmonic Dirichlet problem

     

A sufficient condition of viability for fractional differential equations with the Caputo derivative

In this paper viability results for nonlinear fractional differential equations with the Caputo derivative are proved. We give the sufficient condition that guarantees fractional viability of a locally closed set with respect to nonlinear function. As an example we discuss positivity of solutions, particularly in linear case.     

Zero distribution of solutions of complex linear differential equations determines growth of coefficients

It is shown that the exponent of convergence λ(f) of any solution f of with entire coefficients A₀(z), …, A_k?2(z), satisfies λ(f) ? λ ∈ [1, ∞) if and only if the coefficients A₀(z), …, A_k?2(z) are polynomials such that for j = 0, …, k ? 2. In the unit disc analogue of this result certain intersections of weighted Bergman spaces take the role of polynomials. The key idea in the proofs is W. J. Kim’s 1969 representation of coefficients in terms of ratios of linearly independent solutions. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Global decaying solution to dissipative nonlinear evolution equations with ellipticity

We establish the global existence and decaying results for the Cauchy problem of nonlinear evolution equations:

(E)     

An efficient numerical method for preconditioned saddle point problems

In this paper, we consider the solution of linear systems of saddle point type by a preconditioned numerical method. We first transform the original linear system into two sub-systems with small size by a preconditioning strategy, then employ the conjugate gradient (CG) method to solve the linear system with a SPD coefficient matrix, and a splitting iteration method to solve the other sub-system, respectively. Numerical experiments show that the new method can achieve faster convergence than several effective preconditioners published in the recent literature in terms of total runtime and iteration steps.     

Nonmonotone algorithm for minimax optimization problems

Many real life problems can be stated as a minimax optimization problem, such as the problems in economics, finance, management, engineering and other fields. In this paper, we present an algorithm with nonmonotone strategy and second-order correction technique for minimax optimization problems. Using this scheme, the new algorithm can overcome the difficulties of the Maratos effect occurred in the nonsmooth optimization, and the global and superlinear convergence of the algorithm can be achieved accordingly. Numerical experiments indicate some advantages of this scheme.     

Convergence analysis for finite family of relatively quasi nonexpansive mappings and systems of equilibrium problems

In this paper, a new iterative scheme by hybrid method is constructed. Strong convergence of the scheme to a common element of the set of common fixed points of finite family of relatively quasi-nonexpansive mappings and set of common solutions of a system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space is proved using the properties of generalized f-projection operator. Our results extend important recent results.