Convergence properties of additive and multiplicative Schwarz iterations for solving linear systems of equations with a symmetric positive semidefinite matrix are analyzed. The analysis presented applies to matrices whose principal submatrices are nonsingular, i.e., positive definite. These matrices appear in discretizations of some elliptic partial differential equations, e.g., those with Neum...

In this paper, we deal with controllability properties of linear and nonlinear Korteweg–de Vries equations in a bounded interval. The main part of this paper is a result of uniform controllability of a linear KdV equation in the limit of zero-dispersion. Moreover, we establish a result of null controllability for the linear equation via the left Dirichlet boundary condition, and of exact contro...

In this paper we consider the three–dimensional Navier–Stokes equations subject to periodic boundary conditions or in the whole space. We provide sufficient conditions, in terms of one direction derivative of the velocity field, namely, uz , for the regularity of strong solutions to the three-dimensional Navier–Stokes equations.

In this paper we show the existence of regular solutions of the Rothe–approximation of the unsteady Navier–Stokes equations with periodic boundary condition in arbitrary dimension. The result relies on techniques developed by the authors in the study of the higher–dimensional steady Navier–Stokes equations.

In this paper, we present an efficient method for determining the solution of the stochastic second kind Volterra integral equations (SVIE) by using the Taylor expansion method. This method transforms the SVIE to a linear stochastic ordinary differential equation which needs specified boundary conditions. For determining boundary conditions, we use the integration technique. This technique give...

In this paper we deal with the three-dimensional Navier-Stokes system, posed in a cube. In this context, we prove a result concerning its global approximate controllability by means of boundary controls which act in some part of the boundary.

In 1999, J.C. Mattingly and Ya. G. Sinai used elementary methods to prove the existence and uniqueness of smooth solutions to the 2D Navier-Stokes equations with periodic boundary conditions. And they were almost successful in proving the existence and uniqueness of smooth solutions to the 3D Navier-Stokes equations with periodic boundary conditions using the same strategy. In this paper, we mo...

Abstract. We consider the incompressible, two dimensional Navier Stokes equation with periodic boundary conditions under the effect of an additive, white in time, stochastic forcing. Under mild restrictions on the geometry of the scales forced, we show that any finite dimensional projection of the solution possesses a smooth density with respect to Lebesgue measure. We also show that under natu...