A Solution of the Navier-Stokes Problem for an Incompressible Fluid
A solution of the Navier-Stokes problem
Keywords:
Navier-Stokes Equation, Partial Differential Equations (PDE), Incompressible Fluid, Inhomogeneous Linear Equations, Solution UniquenessAbstract
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.
References
Fefferman, C. Existence and smoothness of the Navier- Stokes equation. http: // claymath.org / Millenium Prize Problems / Navier-Stokes Equations. Cambridge, MA: Clay Mathematics Institute: 1-5 (2000).
Caffarelli, L., R. Kohn, & L. Nirenberg. Partial regularity of suitable weak solutions of the Navier–Stokes equations, Communication in Pure & Applied Math. 35:771–831 (1982).
Leray. J. Sur le mouvement d’un liquidevisquexemplissentl’espace, Acta Mathematica Journal, 63: 193–248 (1934).
Lin. F.H. A new proof of the Caffarelli–Kohn–Nirenberg theorem, Communication in Pure and Applied Mathematics, 51: 241–257 (1998).
Scheffer. V. Turbulence and Hausdorff dimension, in Turbulence and the Navier–Stokes Equations, Lecture Notes in Math. 565, Springer Verlag, Berlin: 94–112 (1976).
Landau, L.D, & E.M. Lifshith. Fluid Mechanics. Vol. 6 (2nd ed.). Butterworth-Heinemann. 2nd ed. – Pergamon Press, XIV (1987).
Ladyzhenskya. O.A., The Mathematical Theory of Viscous Incompressible Flows (2nd edition), Gordon and Breach, New York (1969).
Prantdl. L. Gesammelte Abhandlungen zur angewandten Mechanik, Hudro-und Aerodynamic. Springer, Berlin (1961).
Schlichting. H. Boundary-Layer Theory, Nauka, Moscow (1974).
Omurov. T.A. Existence and Uniqueness of a Solution of the nD Navier-Stokes Equation, Advances and Applications in Fluid Mechanics, 19(3): 589-604 (2016).
Sobolev. L.S. Equations of Mathematical Physics, Nauka, Moscow (1966).
Friedman. A. Boundary estimates for second-order parabolic equations and their application. Journal of Mathematics and Mechanics, 7(5) 771-791 (1958).
