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Criteria of local in time regularity of the Navier-Stokes equations beyond Serrin's condition

Reinhard FarwigHideo KozonoHermann Sohr — 2008

Banach Center Publications

Let u be a weak solution of the Navier-Stokes equations in a smooth bounded domain Ω ⊆ ℝ³ and a time interval [0,T), 0 < T ≤ ∞, with initial value u₀, external force f = div F, and viscosity ν > 0. As is well known, global regularity of u for general u₀ and f is an unsolved problem unless we pose additional assumptions on u₀ or on the solution u itself such as Serrin’s condition | | u | | L s ( 0 , T ; L q ( Ω ) ) < where 2/s + 3/q = 1. In the present paper we prove several local and global regularity properties by using assumptions...

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