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On the singular set and the uniqueness of weak solutions of the Navier- Stokes equations. (English) Zbl 0567.35069
This paper establishes three results for weak solutions to the Navier- Stokes equations. Firstly it proves a uniqueness theorem by showing that if u, v are weak solutions with the same data, if \(u\in L^{\infty}((0,T),L^ n(\Omega))\) with \(\Omega\) bounded and if v satisfies the energy inequality everywhere then \(u=v\). Secondly, let u be any weak solution in the class \(L^{\infty}((0,T),L^ n(\Omega)).\) Then there are at most countably many points in [0,T] where \(\| u(t)\|_{L^ n(\Omega)}\) is discontinuous from the left while u is continuous from the right at each t in [0,T], in the \(L^ n(\Omega)\)- topology. Finally it is shown that for \(n=4\) any weak solution for which the energy inequality holds in regular with the exception of a bounded set of measure 0.
Reviewer: B.Straughan

35Q30 Navier-Stokes equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35D10 Regularity of generalized solutions of PDE (MSC2000)
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[1] Fujita, H., and Kato, T.: On the Navier-Stokes initial value problem I. Arch. Rat. Mech. Anal. 16, 269-315 (1964) · Zbl 0126.42301
[2] Fujiwara, D., and Morimoto, H.: An Lr-theorem of the Helmholtz decomposition of vector fields. J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 24, 685-700 (1977) · Zbl 0386.35038
[3] Giga, Y.: Analyticity of the Semigroup Generated by the Stokes Operator in Lr Spaces. Math. Z. 178, 297-329(1981) · Zbl 0461.47019
[4] Giga, Y.: Domains in Lr spaces of fractional powers of the Stokes operator, to appear in Archive Rat. Mech. Anal.
[5] Giga, Y.: Regularity criteria for weak solutions of the Navier-Stokes system. Preprint · Zbl 0598.35094
[6] Giga, Y., and Miyakawa, T.: Solutions in Lr to the Navier-Stokes initial value problem, to appear in Archive Rat. Mech. Anal. · Zbl 0587.35078
[7] Kiselev, A.A., and Lady?enskaja, O.A.: On the Existence and Uniqueness of Solutions of the Nonstationary Problems for Flows of Non-Compressible Fluids. Am. Math. Soc. Translations, Ser. 2, 24, 79-106(1963) · Zbl 0131.41201
[8] Kaniel, S., and Shinbrot, M.: The Initial Value Problem for the Navier-Stokes Equations. Arch. Rat. Mech. Anal. 21, 279-285(1966) · Zbl 0148.45504
[9] Kato, T.: Strong Lp-solutions of the Navier-Stokes equations in ?m. Math. Z., to appear.
[10] Lions, J.L.: Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires. Paris: Dunod, Gauthier-Villars 1969
[11] Masuda, K.: On the stability of incompressible viscous fluid motions past objects.J. Math. Soc. Japan 27, 294-327(1975) · Zbl 0303.76011
[12] Sobolevskii, P.E.: Study of the Navier-Stokes Equations by the Methods of the Theory of Parabolic Equations in Banach Spaces. Soviet Math. Dokl. 5, 720-723(1964)
[13] Serrin, J.: The initial value problem for the Navier-Stokes equations. Nonlinear Problems (R. Langer ed.), 69-98, Madison: The University of Wisconsin press 1963 · Zbl 0115.08502
[14] Sohr, H.: Zur Regularitätstheorie der instationären Gleichungen von Navier-Stokes. Math. Z. 184, 359-376(1983) · Zbl 0506.35084
[15] Sohr, H.: Optimale lokale Existenzsätze für die Gleichungen von Navier-Stokes. Math. Ann. 267, 107-123 (1984) · Zbl 0552.35059
[16] Solonnikov, V.A.: Estimates for solutions of nonstationary Navier-Stokes equations. J. Soviet Math. 8, 467-529(1977) · Zbl 0404.35081
[17] Solonnikov, V.A.: Estimates of the solutions of a nonstationary linearized system of Navier-Stokes. Am. Math. Soc. Translations 75, 1-116 (1968) · Zbl 0187.03402
[18] Témam, R.: Navier-Stokes Equations. Amsterdam-New York-Oxford: North Holland 1977
[19] Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. Amsterdam-New York-Oxford: North Holland 1978 · Zbl 0387.46032
[20] Wahl, W. von: Über das Verhalten für t?O der Lösungen nichtlinearer parabolischer Gleichungen, insbesondere der Gleichungen von Navier-Stokes. Sonder-forschungsbereich 72 ?Approximation und Optimierung?, Universität Bonn. Preprint no. 602 (1983)
[21] Wahl, W. von: Berichtigung zur Arbeit: ?Über das Verhalten für t?O der Lösungen nichtlinearer parabolischer Gleichungen, insbesondere der Gleichungen von Navier-Stokes? (preprint no. 602). Sonderforschungs-bereich 72 ?Approximation und Optimierung?, Universität Bonn. Preprint-Reihe (1983)
[22] Wahl, W. von: Regularity Questions for the Navier-Stokes Equations. Approximation Methods for Navier-Stokes Problems. Proceedings, Paderborn, Germany 1979. Lecture Notes in Mathematics 771 (1980) · Zbl 0451.35051
[23] Wahl, W. von: Regularity of Weak Solutions of the Navier-Stokes Equations. To appear in the Proceedings of the 1983 AMS Summer Institute on Nonlinear Functional Analysis and Applications. Proceedings of Symposia in Pure Mathematics. Am. Math. Soc.: Providence, Rhode Island
[24] Wahl, W. von: Regularitätsfragen für die instationären Navier-Stokesschen Gleichungen in höheren Dimensionen. J. Math. Soc. Japan 32, 263-283 (1980) · Zbl 0456.35073
[25] Weissler, F.B.: The Navier-Stokes Initial Value Problem in Lp, Arch. Rat. Mech. Anal. 74, 219-230 (1981) · Zbl 0454.35072
[26] Yosida, K.: Functional Analysis. Grundlehren d. Math. Wiss. 123. Berlin-Heidelberg-New York: Springer 1965 · Zbl 0126.11504
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