Global existence, uniqueness and estimates of the solution to the Navier-Stokes equations

The Navier-Stokes (NS) problem consists of finding a vector-function v from the Navier Stokes equations. The solution v to NS problem is defined in this paper as the solution to an integral equation. The kernel G of this equation solves a linear problem which is obtained from the NS problem by dropping the nonlinear term (v . del)v. The kernel G is found in closed form. Uniqueness of the solution to the integral equation is proved in a class of solutions v with finite norm N-1(v) = sup(xi is an element of R3),(t is an element of[0,T]) (1 + vertical bar xi vertical bar)(vertical bar v vertical bar+ vertical bar del v vertical bar) <= c(*), where T > 0 and C > 0 are arbitrary large fixed constants. In the same class of solutions existence of the solution is proved under some assumption. Estimate of the energy of the solution is given. (C) 2017 Elsevier Ltd. All rights reserved.