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

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Farwig R., Kozono H., Sohr H.

Banach Center Publications

2008

tom 81

nr 1

175-184

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 beyond Serrin's condition e.g. as follows: If the norm $||u||_{L^r(0,T;L^q(Ω))}$ and a certain norm of F satisfy a ν-dependent smallness condition, where Serrin's number 2/r + 3/q > 1, or if u satisfies a local leftward $L^{s} - L^{q}$-condition for every t ∈ (0,T), then u is regular in (0,T).

Ograniczanie wyników

1 Banach Center Publications

1 Farwig R.

1 Kozono H.

1 Sohr H.

1 2008

Wyniki wyszukiwania

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