Гомоклінічний хаос та рівняння Нав’є–Стокса

Abstract

In this paper we consider a perturbed system of Navier–Stokes equations, which is rewritten in the form of an operator-differential equation. Using the obtained a priori estimates for the corresponding operator, the property of the exponential dichotomy for a generating homogeneous equation is established. The necessary and sufficient conditions for the existence of solutions of a generating linear homogeneous equation bounded on the entire axis are obtained. The corresponding set of solutions is represented by the constructed Green operator. For a nonlinear Navier–Stokes equation, we introduce the operator equation for generating elements. Using the operator equation for the generating elements, we obtain the necessary condition for the bifurcation of the solutions of the Navier–Stokes equation. It is necessary to find such a solution of the system of equations that is bounded on the entire axis, which transforms into a generating bounded solution of the corresponding homogeneous equation when . In this paper we obtain a sufficient condition for the existence of a solution of the Navier–Stokes equation bounded on the entire axis. An iterative Newton-Kantorovich type algorithm for its finding was constructed. With the help of representation, estimates of the corresponding solutions in the spaces of integrable functions are established