×

On Leray’s self-similar solutions of the Navier-Stokes equations. (English) Zbl 0884.35115

Leray’s problem of the existence of self-similar solutions of the Navier-Stokes equations \[ \frac{\partial u}{\partial t}+u\cdot\nabla u -\nu\Delta u +\nabla p= 0, \quad \nabla\cdot u=0\;\text{in}\;\mathbb R^3\times(t_1,t_2) \] is considered. These are solutions of the form \[ u(x,t)=\frac 1{\sqrt{2a(T-t)}}U\left(\frac x{\sqrt{2a(T-t)}}\right) \] where \(T\in \mathbb R,\;a>0,\;U:\mathbb R^3\to \mathbb R^3\). \(U\) satisfies the system \[ -\nu\Delta U +aU+a(y\cdot\nabla)U+(U\cdot\nabla)U+ \nabla P=0,\qquad \nabla U=0. \tag{1} \] The main result of the paper is the following
Theorem. Let \(U\) be a weak solution of (1) belonging to \(L^3(\mathbb R^3)\). Then \(U\equiv 0\) in \(\mathbb R^3\).

MSC:

35Q30 Navier-Stokes equations
35C06 Self-similar solutions to PDEs
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] [Br]Browder, F., Regularity of solutions of elliptic equations.Comm. Pure Appl. Math., 9 (1956), 351–361. · Zbl 0070.09601
[2] [Ca]Cattabriga, L., Su un problema al retorno relativo al sistema di equazioni di Stokes.Rend. Sem. Mat. Univ. Padova, 31 (1961), 308–340. · Zbl 0116.18002
[3] [CKN]Caffarelli, L., Kohn, R. &Nirenberg, L., Partial regularity of suitable weak solutions of the Navier-Stokes equations.Comm. Pure Appl. Math., 35 (1982), 771–831. · Zbl 0509.35067
[4] [CZ]Calderón, A. P. &Zygmund, A., On existence of certain singular integrals.Acta Math., 88 (1952), 85–139. · Zbl 0047.10201
[5] [FR1]Frehse, J. &Ružička, M., On the regularity of the stationary Navier-Stokes equations.Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 21 (1994), 63–95.
[6] [FR2] – Regularity for the stationary Navier-Stokes equations in bounded domains.Arch. Rational Mech. Anal., 128 (1994), 361–381. · Zbl 0832.35108
[7] [Ga]Galdi, G. P.,An Introduction to the Mathematical Theory of Navier-Stokes Equations, Vol. 1: Linearized stationary problems. Springer Tracts Nat. Philos. 38. Springer-Verlag, New York, 1994.
[8] [Gi]Giga, Y., Solutions for semilinear parabolic equations inL p and regularity of weak solutions of the Navier-Stokes equations with data inL p ,J. Differential Equations, 62 (1986), 186–212. · Zbl 0577.35058
[9] [GT]Gilbarg, D. &Trudinger, N. S.,Elliptic Partial Differential Equations of Second Order, 2nd edition. Grundlehren Math. Wiss. 224. Springer-Verlag, Berlin-New York, 1983. · Zbl 0562.35001
[10] [GW]Gilbarg, D. &Weinberger, H. F., Asymptotic properties of Leray’s solution of the stationary two-dimensional Navier-Stokes equations.Russian Math. Surveys, 29:2 (1974), 109–123. · Zbl 0304.35071
[11] [La]Ladyzhenskaya, O.,The Mathematical Theory of Viscous Incompressible Flow, 2nd English edition. Math. Appl., 2. Gordon and Breach, New York, 1969. · Zbl 0184.52603
[12] [Le]Leray, J., Sur le mouvement d’un liquide visqueux emplissant l’espace.Acta Math., 63 (1934), 193–248. · JFM 60.0726.05
[13] [LUS]Ladyzhenskaya, O., Ural’ceva, N. &Solonnikov, V.,Linear and Quasi-Linear Equations of Parabolic Type. Transl. Math. Monographs, 23. Amer. Math. Soc., Providence, RI, 1968.
[14] [Mo]Morrey, Ch. B.,Multiple Integrals in the Calculus of Variations. Grundlehren Math. Wiss., 130, Springer-Verlag, New York, 1966.
[15] [Oh]Ohyama, T., Interior regularity of weak solutions of the time-dependent Navier-Stokes equation.Proc. Japan Acad. Ser. A Math. Sci., 36 (1960), 273–277. · Zbl 0100.22404
[16] [Sch]Scheffer, V., Partial regularity of solutions to the Navier-Stokes equations.Pacific J. Math., 66 (1976), 535–552. · Zbl 0325.35064
[17] [Se1]Serrin, J., On the interior regularity of weak solutions of the Navier-Stokes equations.Arch. Rational Mech. Anal., 9 (1962), 187–195. · Zbl 0106.18302
[18] [Se2] –, Mathematical principles of classical fluid mechanics, inHandbuch der Physik, 8:1, pp. 125–263. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1959.
[19] [So]Sohr, H., Zur Regularitätstheorie der instationären Gleichungen von Navier-Stokes.Math. Z., 184 (1983), 359–376. · Zbl 0506.35084
[20] [St]Stein, E.,Singular Integrals and the Differentiability Properties of Functions. Princeton Math. Ser., 30. Princeton Univ. Press, Princeton, NJ, 1970. · Zbl 0207.13501
[21] [Str1]Struwe, M., On partial regularity results for the Navier-Stokes equations.Comm. Pure Appl. Math., 41 (1988), 437–458. · Zbl 0632.76034
[22] [Str2]Struwe, M. Regular solutions of the stationary Navier-Stokes equations onR 5. To appear inMath. Ann.
[23] [Ta]Takahashi, S., On interior regularity criteria for weak solutions of the Navier-Stokes equations.Manuscripta Math., 67 (1990), 237–254. · Zbl 0718.35022
[24] [vW]Wahl, W. von,The Equations of Navier-Stokes and Abstract Parabolic Equations. Aspects of Math., E8. Vieweg, Braunschweig, 1985.
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.