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It is known that the methods of integral transformations in the theory of partial differential equations made it possible to find solutions to many problems and clarify the physical meaning of some basic laws and phenomena in fluid mechanics. In this regard, in the present work, we study the Navier-Stokes system, which describe the flow of a viscous incompressible fluid. Moreover, on the basis of the developed method, the original problem is transformed to the system of Volterra and Volterra-Abel integral equations of the second kind, and taking into account the theory of these systems, the existence and uniqueness of the solution of the non-stationary Navier-Stokes problem in the special space, which was introduced in the paper, are proved. The solution was obtained for velocity and pressure in an analytical form, in addition, the found pressure distribution law, which is described by a Poisson type equation and plays a fundamental role in the theory of Navier-Stokes systems in constructing analytic smooth (conditionally smooth) solutions.
Taalaibek D. Omurov. (2019) A Solution of the Navier-Stokes Problem for an Incompressible Fluid, , Proc. of the PAS: A; 56, issue 4.
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