Existence of weak solutions for the Navier-Stokes equations with initial data in \(L^ p\). (English) Zbl 0707.35118

The author proves global existence of weak solutions to the Navier-Stokes equations posed on \({\mathbb{R}}^ n\), when the initial data is in \(L^ p({\mathbb{R}}^ n)\) (for \(2<p\leq n\) and \(n=3,4)\). These results are generalized in an addendum to include the case where \(3\leq n\) and \(1<p<\infty\). In this addendum the author also derives a uniqueness result under a condition on the size of the data. This is an attempt to bridge some of the gaps in the theory of the Navier-Stokes equations, i.e., the gaps between the Hopf-Leray theory and those described by E. B. Fabes, B. F. Jones and N. M. Riviere [Arch. Ration. Mech. Anal. 45, 222-240 (1972; Zbl 0254.35097)], Y. Giga and T. Miyakawa [ibid. 89, 267-281 (1985; Zbl 0587.35078)], H. Kozono [J. Differ. Equations 79, No.1, 79-88 (1989)] and R. Racke [ibid. 76, No.2, 312-338 (1988; Zbl 0669.35046)] and the references therein.
The paper is interesting in that it uses methods of classical harmonic analysis to get the results. The author extends the methods and results of Fabes, Jones and Riviere (loc. cit.); the use of a divergence free fundamental solution to the n-dimensional heat equation, to construct solutions to the Navier-Stokes equations.
It is unfortunate that these methods apply only to problems posed on all of space, and cannot be adapted to problems posed on bounded domains. For related results (using different approaches) that apply to both bounded and unbounded domains one may consult the works of Giga and Miyakawa, Kozono and Racke (loc. cit.) and the references therein.
Reviewer: A.J.Meir


35Q30 Navier-Stokes equations
35K55 Nonlinear parabolic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
35K15 Initial value problems for second-order parabolic equations
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