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Weak and strong solutions of the Navier-Stokes initial value problem. (English) Zbl 0547.35101
The existence, uniqueness and regularity of weak and strong solutions of the initial boundary value problem for the Navier Stokes equations on bounded domains with smooth boundary is reviewed with an emphasize on semigroup theory and the properties of the Stokes operator. One of the aims of the presentation is to clarify the differences between the two and higher dimensional cases. Suitable a priori estimates are derived from the energy estimate. The uniqueness of global weak solution (Leray- Hopf solutions) is proved. Utilizing analytic semigroup theory strong solutions are constructed so that uniqueness holds. Finally, the singularities with respect to time of the Leray-Hopf solutions are studied.
Reviewer: H.Jeggle

MSC:
35Q30 Navier-Stokes equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35D10 Regularity of generalized solutions of PDE (MSC2000)
35D05 Existence of generalized solutions of PDE (MSC2000)
47D03 Groups and semigroups of linear operators
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
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