An optimal control problem with feedback for a mathematical model of the motion of weakly concentrated water polymer solutions with objective derivative

We study a problem with feedback for a mathematical model of the motion of weakly concentrated water polymer solutions with smoothed Jaumann objective derivative. We prove the existence of an optimal solution yielding the minimum of a specified bounded lower semicontinuous quality functional. To establish the existence of an optimal solution, we use the topological approximation method for studying problems of hydrodynamics.     

Pullback attractors for a two-dimensional Navier-Stokes model in an infinite delay case

We prove the existence of solutions for a Navier-Stokes model in two dimensions with an external force containing infinite delay effects in the weighted space C_γ(H). Then, under additional suitable assumptions, we prove the existence and uniqueness of a stationary solution and the exponential decay of the solutions of the evolutionary problem to this stationary solution. Finally, we study the existence of pullback attractors for the dynamical system associated to the problem under more general assumptions.     

Optimal control problem for a stationary model of low concentrated aqueous polymer solutions

We study the existence of an optimal control for lower bounded upper semicontinuous functionals and weak solutions of a system of equations describing the stationary motions of low concentrated aqueous polymer solutions in a bounded domain with locally Lipschitz boundary.     

Pullback attractors for a class of non-autonomous nonclassical diffusion equations

We consider the first initial boundary value problem for the non-autonomous nonclassical diffusion equation u_t−εΔu_t−Δu+f(u)=g(t), ε∈[0,1], in a bounded domain in R^N. Under a Sobolev growth rate of the nonlinearity f and a suitable exponential growth of the external force g, using the asymptotic a priori estimate method, we prove the existence of pullback D-attractors in the space and the upper semicontinuity of at ε=0.     

Solvability of a stationary model of motion of weak aqueous polymer solutions

In this paper we study the solvability in a weak sense of the boundary-value problem for a system of equations that describes the stationary motion of weak aqueous polymer solutions in a bounded domain with a locally Lipschitz boundary.     

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.     

Asymptotic autonomous attractors for a stochastic lattice model with random viscosity

We study forward asymptotic autonomy of a pullback random attractor for a non-autonomous random lattice system and establish the criteria in terms of convergence, recurrence, forward-pullback absorption and asymptotic smallness of the discrete random dynamical system. By applying the abstract result to both non-autonomous and autonomous stochastic lattice equations with random viscosity, we show the existence of both pullback and global random attractors such that the time-component of the pullback attractor semi-converges to the global attractor as the time-parameter tends to infinity.     

On the uniqueness of the solution of a boundary-value problem for weakly nonlinear integro-differential equations with parameters

The conditions of existence of the unique solution of a boundary-value problem for weakly nonlinear integro-differential equations with parameters and the equivalence of such problem and the appropriate integral equation are established.     

Approximation of attractors for evolution equations with the help of attractors for finite systems

The problem of approximation of attractors for semidynamical systems (SDS) in a metric space is considered. Let some (exact) SDS possessing an attractor M be inaccurately defined, i.e., another SDS, which is close in some sense to the exact one, be given. The problem is to find a set M that is close to M in the Hausdorff metric. A finite procedure for construction of M is suggested. The results obtained are suitable for numerical construction of attractors for a rather large class of systems, including the ones generated by the Lorenz equations. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 197, pp. 71–86, 1992. Translated by I. N. Kostin.     

Existence of smooth attractors for the Navier-Stokes-omega model of turbulence

We consider the Navier-Stokes- model, given by

     

The exact solution of a linear integral equation with weakly singular kernel

A space , which is proved to be a reproducing kernel space with simple reproducing kernel, is defined. The expression of its reproducing kernel function is given. Subsequently, a class of linear Volterra integral equation (VIE) with weakly singular kernel is discussed in the new reproducing kernel space. The reproducing kernel method of linear operator equation Au=f, which request the image space of operator A is and operator A is bounded, is improved. Namely, the request for the image space is weakened to be L²[a,b], and the boundedness of operator A is also not required. As a result, the exact solution of the equation is obtained. The numerical experiments show the efficiency of our method.     

A common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type

In this paper, we proved a common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type and a property (E.A) introduced in [M. Aamri, D. El Moutawakil, Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl. 270 (2002) 181-188]. Our theorem generalizes Theorem 2.2 of [M. Aamri, D. El Moutawakil, Common fixed points under contractive conditions in symmetric spaces, Appl. Math. E-Notes 3 (2003) 156-162] and Theorem 2 of [M. Aamri, D. El Moutawakil, Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl. 270 (2002) 181-188].     

Regularity of the solution of Hammerstein equations with weakly singular kernel

Some regularity properties of the solution to a class of weakly singular Hammerstein equations are derived. The results obtained in this paper extend the results of C. Schneider [4], where he obtains similar properties for the solution to weakly singular Fredholm equations of the second kind.