# Three Dimensional Flux Problem for the Navier - Stokes Equations

V.V. PukhnachevSolvability of the flux problem for the Navier - Stokes equations has been proven by J. Leray (1933) under an additional condition of zero flux through each connected component of the flow domain boundary. He used arguments from contradiction was by contradiction and did not give a priory estimate of solution. This estimate was obtained by E. Hopf (1941) under the same condition concerning fluxes. The following problem is open up to now: if exists a solution of the flux problem, when only the necessary condition of total zero flux is satisfied? For small fluxes values, solvability of three dimensional problem was established independently by H. Fujita and R. Finn (1961). H. Fujita and H. Morimoto (1995) proved existence theorem for flows, which are close to potential ones). M.V. Korobkov, K. Pileckas and R. Russo (2011 - 2015) gave the positive solution of the flux problem for planar and axially symmetric flows without restrictions on the flux values. The present paper is devoted to consideration of flux problem in the domain of a spherical layer type. We obtained an a priori estimate of solution under following additional conditions: the flow has a symmetry plane; the flux through the inner domain boundary is positive. This estimate implies solvability of the problem.Full text

- Keywords
- flux problem; symmetric solutions; virtual drain.
- References
- 1. Leray J. Etude de diverses equations integrales non lineaires problemes que pose l'hydrodynamique. J. Math. Pures Appl., 1933, vol. 12, no. 9, pp. 1-82.

2. Hopf E. Ein allgemeiner Endlichkeitssatz der Hydrodynamik. Math. Ann., 1941, vol. 117, no. 1, pp. 764-775.

3. Ladyzhenskaya O.A. The Mathematical Theory of Viscous Incompressible Flow. Math. Appl., vol. 2. N.Y., London, Paris, Gordon and Breach Science Publ, 1969. 224 p.

4. Fujita H. On the Existence and Regularity of the Steady-State Solutions of the Navier - Stokes Equations. J. Fac. Sci. Univ. Tokyo Sect. 1, 1961, vol. 9, pp. 59-102.

5. Finn R. On the Steady-State Solutions of the Navier - Stokes Equations. III. Acta Math., 1961, vol. 105, no. 3-4, pp. 197-244. DOI: 10.1007/BF02559590

6. Fujita H., Morimoto H. A Remark on the Existence of the Navier - Stokes Flow with Non-Vanishing Outflow Condition. Nonlinear Waves (Sapporo, 1995), GAKUTO Intern. Ser. Math. Sci. Appl., Gakkotosho, Tokyo, 1997, vol. 10, pp. 53-61.

7. Korobkov M.V., Pileckas K., Pukhnachev V.V., Russo R. The Flux Problem for the Navier - Stokes Equations. Russian Math. Surveys, 2015, vol. 69, no. 6, pp. 1065-1122. DOI: 10.1070/RM2014v069n06ABEH004928

8. Takeshita A. A Remark on Leray's Inequality. Pacific J. Math., 1993, vol. 151, no. 1, pp. 151-158. DOI: 10.2140/pjm.1993.157.151

9. Sazonov L.I. On the Existence of a Stationary Symmetric Solution of the Two-Dimensional Fluid Flow Problem. Math. Notes, 1993, vol. 54, no. 6, pp. 1280-1283. DOI: 10.1007/BF01209092

10. Fujita H. On Stationary Solutions to Navier - Stokes Equations in Symmetric Plane Domains under General Outflow Condition. Navier - Stokes equations: theory and numerical methods (Varenna, 1997), Pitman Res. Notes Math. Ser. Longman, Harlow, 1998, vol. 388, pp. 16-30.

11. Pukhnachev V.V. Viscous Flows in Domains with a Multiply Connected Boundary. New Directions in Mathematical Fluid Mechanics, Adv. Math. Fluid Mech., Basel, Birkhauser Verlag, 2010, pp. 333-348.

12. Pukhnachev V.V. [Symmetries in the Navier - Stokes equations]. Uspekhi mekhaniki, 2006, vol. 4, no. 1, pp. 6-76 (in Russian)

13. Andreev V.K., Kaptsov O.V., Pukhnachov V.V., Rodionov A.A. Application of Group-Theoretical Methods in Hydrodynamics. Dordrecht, Kluwer Acad. Publ., 1998. DOI: 10.1007/978-94-017-0745-9

14. Amick Ch.J. Existence of Solutions to the Nonhomogeneous Steady Navier - Stokes Equations. Indiana Univ. Math. J., 1984, vol. 33, no. 6, pp. 817-830. DOI: 10.1512/iumj.1984.33.33043

15. Fujita H., Morimoto H., Okamoto H. Stability Analysis of Navier - Stokes Flows in Annuli. Math. Methods Appl. Sci., 1997, vol. 20, no. 11, pp. 659-678. DOI: 10.1002/(SICI)1099-1476(19970725)20:11<959::AID-MMA895>3.0.CO;2-D

16. Kochin N.E., Kibel' I.A., Rose N.V. Theoretical Hydromechanics, part 2. N.Y., Interscience Publ., 1964. 569 p.

17. Pukhnachev V.V. Singular Solutions of Navier - Stokes Equations. Proceedings of the St. Petersburg Mathematical Society, vol. XV: Advances in Mathematical Analysis of Partial Differential Equations, Amer. Math. Soc. Transl. Ser. 2, Amer. Math. Soc., 2014, vol. 232, pp. 193-218.