The methods of integral transformations in the theory of partial differential equations are well recognised for allowing the solution of numerous problems and the clarification of the physical significance of some important laws and phenomena in fluid mechanics. In this sense, this chapter looks at the Navier-Stokes system, which explains the flow of a viscous incompressible fluid. Furthermore, the developed method is used to convert the original problem to a system of second-order integral equations, and the theory of these systems is used to prove the existence and uniqueness of the solution of the non-stationary Navier-Stokes problem in the special space, which was introduced in this chapter. The discovered pressure distribution law, which is explained by a Poisson type equation and plays a crucial role in the theory of Navier-Stokes systems in generating analytic smooth (conditionally smooth) solutions, was also obtained in an analytical form.

**Author(S) Details**

**Taalaibek D. Omurov
**Kyrgyz National University named after Jusup Balasagyn, Kyrgyzstan.

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