Mathematical Analysis of the Navier-Stokes Equations by Matthias Hieber & James C. Robinson & Yoshihiro Shibata

Author:Matthias Hieber & James C. Robinson & Yoshihiro Shibata
Language: eng
Format: epub
ISBN: 9783030362263
Publisher: Springer International Publishing

Since we have

since the intervals (t i − r∕3, t i) are disjoint we have

and the result follows. □

2.3 Serrin’s Local Regularity Result for u ∈ L 5+(Q ∗)

In this section we prove a local conditional regularity result, due to Serrin [37]. We take a solution that solves the Navier–Stokes equations only on some region U of space-time, and show that if u ∈ L α(U) for some α > 5 then in fact u is smooth in the spatial variables within U. We then give a sketch of the proof of the same result when u ∈ L 5(U).

[Two comments are in order here. First, Serrin’s result is more general, allowing for different integrability in space and time: if u ∈ L r(a, b;L s( Ω)) with

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