Study on a Solution of the Navier-Stokes Problem for an Incompressible Fluid with Viscosity

It is well known that the methods of integral transformations in the theory of partial differential equations enabled the solution of many problems and the clarification of the physical meaning of some fundamental laws and phenomena in fluid mechanics. In this regard, the Navier-Stokes system, which describes the flow of a viscous incompressible fluid, is examined in this chapter. Furthermore, the original problem is transformed to a system of second-order integral equations using the developed method, and the existence and uniqueness of the solution of the non-stationary Navier-Stokes problem in the special space, which was introduced in this chapter, are proved using the theory of these systems. The solution was acquired for velocity and pressure in an analytical form, as well as the discovered pressure distribution law, which is explained by a Poisson type equation and plays a fundamental role in the theory of Navier-Stokes systems in constructing analytic smooth (conditionally smooth) solutions.