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Uniqueness and regularity of weak solutions to the Navier-Stokes equations. (English) Zbl 0941.35065
Kozono, Hideo (ed.) et al., Recent topics on mathematical theory of viscous incompressible fluid. Proceedings of the regional workshop, Tsukuba University, Japan, December 2-12, 1996. Tokyo: Kinokuniya. Lect. Notes Numer. Appl. Anal. 16, 161-208 (1998).
The first purpose of this article is to prove the existence of weak solutions of the Navier-Stokes equation using an extension of the Masuda space test functions, functions whose local singularities with respect to the weak $$L^n$$ norm are uniformly small in the time interval $$(0,T)$$. The second purpose is to extend Servin-Masuda’s criterion on uniqueness of weak solutions, i.e. to prove that if the measure of a spatial singular set of weak solutions is sufficient small with respect to the local weak $$L^n$$ norm on the time interval $$(0,T)$$ in a certain sense, then uniqueness holds. The third purpose of this paper is to investigate local behaviour of weak solutions in 3-dimensional domains. The author proves interior regularity of weak solutions and characterizes removable singularities.
For the entire collection see [Zbl 0892.00038].

##### MSC:
 35Q30 Navier-Stokes equations 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 35D05 Existence of generalized solutions of PDE (MSC2000) 35D10 Regularity of generalized solutions of PDE (MSC2000)