A Solution of the n-Dimensional Navier-Stokes System for an Incompressible Fluid with Cauchy Condition

The main object of study of this work is the system of n-dimensional non-stationary Navier-Stokes equations (NSE) for an incompressible fluid with viscosity in an unbounded domain. In this case, it is necessary to establish the existence, uniqueness and smoothness (conditional smoothness) of the solution of this system with the Cauchy condition in the introduced vector space. It is known that in the theory of NSE for a viscous incompressible fluid, the motion is non-potential, and as soon as the Reynolds number becomes large enough, non-linear convective terms (inertial forces) begin to play a significant role, for example, the NSE with medium or low viscosity (turbulent fluid motion). This means that the difficulty of solving such a NS system is due to its nonlinearity, as well as the need to find the speed and pressure depending on any values ​​of the viscosity parameter. More precisely, there exists no general method capable of transforming the given nonlinear system of NSE together with the continuity equation into an integral form while preserving all convective terms and all terms accounting for viscosity. This is precisely why the NSЕ for an incompressible viscous fluid in both bounded and unbounded domains gave rise to the Millennium Prize Problem.