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

Sazonov, L. I.

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.

MSC:

35Q30	Navier-Stokes equations
76D05	Navier-Stokes equations for incompressible viscous fluids

Keywords:

Navier-Stokes equations; Ozeen semigroups; perturbations

References:

[1]	R. Finn, “Stationary solutions of the Navier-Stokes equations,” Proc. Symp. Appl. Math. Am. Math. Soc. 17, 121-153 (1965). · Zbl 0148.21602 · doi:10.1090/psapm/017/9933
[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 · doi:10.1007/s002050050043
[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). · Zbl 0201.18204
[4]	V. I. Yudovich, Linearization Method in Hydrodynamic Stability Theory (Rostov. Gos. Univ., Rostov-on-Don, 1984) [in Russian]. · Zbl 0553.76038
[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 · doi:10.1007/BF01174182
[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 · doi:10.1070/IM1995v045n02ABEH001661
[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 · doi:10.1134/S0001434612050306
[14]	L. I. Sazonov, “The three-dimensional steady flow problem at small Reynolds numbers,” Izv. Math. 75, 1185-1214 (2011). · Zbl 1239.35129 · doi:10.1070/IM2011v075n06ABEH002569
[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 · doi:10.1070/SM1973v020n01ABEH001823
[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 · doi:10.1070/IM1995v059n05ABEH000047
[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 · doi:10.1063/1.2425099
[18]	G. P. Galdi, An Introduction to the Mathematical Theory of Navier-Stokes Equations: Steady-State Problems (Springer, Berlin, 2011). · Zbl 1245.35002 · doi:10.1007/978-0-387-09620-9
[19]	S. M. Nikol’skii, Approximation of Functions of Several Variables and Imbedding Theorems (Springer-Verlag, New York, 1975). · Zbl 0307.46024 · doi:10.1007/978-3-642-65711-5
[20]	D. Henry, Geometric Theory of Semilinear Parabolic Equations (Springer-Verlag, New York, 1981). · Zbl 0456.35001