open access

https://doi.org/10.14943/84236

https://hdl.handle.net/2115/69896

The Navier-Stokes equations with initial values in Besov spaces of type $B_{q,\infty}^{-1+3/q}$

英語

restricted Serrin's uniqueness theorem

2016-06-20

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

1092

1-21

We consider weak solutions of the instationary Navier-Stokes system in a smooth bounded domain R3 with initial value u0 2 L2 ( ). It is known that a weak solution is a local strong solution in the sense of Serrin if u0 satis es the optimal initial value condition u0 2 B􀀀1+3=q q;sq with Serrin exponents sq > 2; q > 3 such that 2 sq + 3 q = 1. This result has recently been generalized by the authors to Rweighted Serrin conditions such that u is contained in the weighted Serrin class T 0 ( ku( )kq)s d < 1 with 2 s + 3 q = 1 􀀀 2 , 0 < < 1 2 . This regularity is guaranteed if and only if u0 is contained in the Besov space B􀀀1+3=q q;s . In this article we consider the limit case of initial values in the Besov space B􀀀1+3=q q;1 and in its subspace B 􀀀1+3=q q;1 based on the continuous interpolation functor. Special emphasis is put on questions of uniqueness within the class of weak solutions.

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