The Multipoint Initial-Final Value Condition for the Navier - Stokes Linear Model

The Navier - Stokes system models the dynamics of a viscous incompressible fluid. The problem of existence of solutions of the Cauchy - Dirichlet problem for this system is included in the list of the most serious problems of this century. In this paper it is proposed to consider the multipoint initial-final conditions instead of the Cauchy conditions. It should be noted that nowadays the study of solvabilityof initial-final value problems is a new and actively developing direction of the Sobolev type equations theory. The main result of the paper is the proof of unique solvability of the stated problem for the system of Navier - Stokes equations.

Keywords

relatively p-sectorial operators; the multipoint initial-final value condition; the Navier - Stokes linear model.

References

1. Zagrebina S.A. The Initial-Finite Problems for Nonclassical Models of Mathematical Physics. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming & Computer Software, 2013, vol. 6, no. 2, pp. 5-24.
2. Ladyzhenskaya O.A. The Mathematical Theory of Viscous Incompressible Flow. N.-Y., London, Paris, Gordon and Breach, Science Publ., 1969.
3. Temam, R. Navier-Stokes Equations. Theory and Numerical Analysis. Amsterdam, N.-Y., Oxford, North Holland Publ. Co., 1979.
4. Sviridyuk G.A. On a Model of the Dynamics of a Weakly Compressible Viscoelastic Fluid. Russian Mathematics (Izvestiya VUZ. Matematika), 1994, vol. 38, no. 1, pp. 59-68.
5. Sviridyuk G.A., Kuznetsov G.A., Relatively Strongly p- Sectorial Linear Operators. Doklady Mathematics, 1999, vol. 59, no. 2, pp. 298-300.
6. Matveeva O.P., Sukacheva T.G. Matematicheskie modeli vyazkouprugikh neszhimaemykh zhidkostey nenulevogo poryadka [The Mathematical Model of a Viscoelastic Incompressible Fluid of Nonzero Order]. Chelyabinsk, Publishing Center of South Ural State University, 2014. (in Russian)
7. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, Boston, Koln, Tokyo, VSP, 2003. DOI:10.1515/9783110915501
8. Zagrebina S.A. Multipoint Initial-Final Value Problem for the Linear Model of Plane-Parallel Thermal Convection in Viscoelastic Incompressible Fluid. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming & Computer Software, 2014, vol. 7, no. 3, pp. 5-22.