Existence of transitions between stationary regimes of the Navier-Stokes equations in the entire space. (Russian, English) Zbl 1299.35228

Zh. Vychisl. Mat. Mat. Fiz. 53, No. 9, 1555-1568 (2013); translation in Comput. Math. Math. Phys. 53, No. 9, 1377-1391 (2013).
Summary: The Navier-Stokes equations are considered in the entire space. It is shown that they have a solution that links two stable steady flows.


35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
Full Text: DOI


[1] R. Finn, “Stationary solutions of the Navier-Stokes equations,” Proc. Symp. Appl. Math. Am. Math. Soc. 17, 121-153 (1965). · Zbl 0148.21602
[2] G. P. Galdi, J. G. Heywood, and Y. Shibata, “On the global existence and convergence to steady state of Navier-Stokes flow past an obstacle that in stated from rest,” Arch. Rational Mech. Anal. 138, 307-318 (1997). · Zbl 0898.35071
[3] P. E. Sobolevskii, “Application of fractional powers of self-adjoint operators to the study of the Navier-Stokes equations,” Dokl. Akad. Nauk SSSR 55(1), 50-53 (1964).
[4] V. I. Yudovich, Linearization Method in Hydrodynamic Stability Theory (Rostov. Gos. Univ., Rostov-on-Don, 1984) [in Russian].
[5] T. Kato, “Strong <Emphasis Type=”Italic“>Lp-solutions of the Navier-Stokes equation in <Emphasis Type=”Italic“>Rm, with applications to weak solutions,” Math. Z. 187, 471-480 (1984). · Zbl 0545.35073
[6] Kobayashi, T.; Shibata, Y., On the Oseen equation in exterior domains (1995)
[7] L. I. Sazonov, “Justification of the linearization method in the flow problem,” Izv. Math. 45, 315-337 (1995). · Zbl 0844.76026
[8] L. I. Sazonov, “On the existence of transitions between steady states in a flow problem,” Vladikavk. Mat. Zh. 13(4), 60-69 (2011). · Zbl 1326.35251
[9] L. I. Sazonov, “Estimates for the perturbed Oseen semigroup,” Vladikavk. Mat. Zh. 11(3), 51-61 (2009). · Zbl 1324.35136
[10] Sazonov, L. I., On the stability of steady solutions of a flow problem, 195-200 (2009)
[11] L. I. Sazonov, “Estimates for the perturbed Oseen semigroup and applications,” Itogi Nauki Yug Ross. Mat. Forum 4, 293-302 (2010).
[12] L. I. Sazonov, “Estimates for the perturbed Oseen semigroup <Emphasis Type=”Italic“>Rn and stability of steady solutions to the Navier-Stokes equations,” Vladikavk. Mat. Zh. 12(3), 71-82 (2010). · Zbl 1221.35286
[13] L. I. Sazonov, “Estimates of perturbed Oseen semigroups and their applications to the Navier-Stokes system in ℝn,” Math. Notes 91, 833-846 (2012). · Zbl 1283.35067
[14] L. I. Sazonov, “The three-dimensional steady flow problem at small Reynolds numbers,” Izv. Math. 75, 1185-1214 (2011). · Zbl 1239.35129
[15] K. I. Babenko, “On stationary solutions of the problem of flow past a body of a viscous incompressible fluid,” Math. USSR-Sb. 20(1), 1-25 (1973). · Zbl 0285.76009
[16] L. I. Sazonov, “On the asymptotics of the solution to the three-dimensional problem of flow far from streamlined bodies,” Izv. Math. 59, 1051-1075 (1995). · Zbl 0922.76102
[17] G. P. Galdi, “Further properties of steady-state solutions to the Navier-Stokes problem past a three-dimensional obstacle,” J. Math. Phys. 48, 1-43 (2007). · Zbl 1144.81345
[18] G. P. Galdi, An Introduction to the Mathematical Theory of Navier-Stokes Equations: Steady-State Problems (Springer, Berlin, 2011). · Zbl 1245.35002
[19] S. M. Nikol’skii, Approximation of Functions of Several Variables and Imbedding Theorems (Springer-Verlag, New York, 1975).
[20] D. Henry, Geometric Theory of Semilinear Parabolic Equations (Springer-Verlag, New York, 1981). · Zbl 0456.35001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.