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⋅∇)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 N1(v)=supξ∈R3,t∈[0,T](1+|ξ|)(|v|+|∇v|)≤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.