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On the Strong Solvability of the Navier-Stokes Equations

Summary: On the Strong Solvability
of the Navier-Stokes Equations
Herbert Amann
Abstract. In this paper we study the strong solvability of the Navier-Stokes equations for rough
initial data. We prove that there exists essentially only one maximal strong solution and that
various concepts of generalized solutions coincide. We also apply our results to Leray-Hopf weak
solutions to get improvements over some known uniqueness and smoothness theorems. We deal
with rather general domains including, in particular, those having compact boundaries.
Mathematical Subject Classi cation (1991). 35Q30, 76D05, 35K55.
Keywords. Weak, very weak, and strong solutions; existence and uniqueness theorems with rough
initial data.
0. Introduction
Throughout this paper m  2 and
= R m
is a subdomain of R m with
a smooth boundary @

We consider the nonstationary Navier-Stokes equations


Source: Amann, Herbert - Institut für Mathematik, Universität Zürich


Collections: Mathematics