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

• Autorzy: Reinhard Farwig , Hideo Kozono , Hermann Sohr
• Języki publikacji: EN

Abstrakty

EN

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).

Kategorie tematyczne

• 76D05: Navier-Stokes equations
• 35Q30: Navier-Stokes equations
• 35B65: Smoothness and regularity of solutions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 2008
• Tom: 81
• Numer: 1
• Strony: 175-184

Daty

wydano

2008

Twórcy

autor

Reinhard Farwig

• Fachbereich Mathematik, Technische Universität Darmstadt, 64289 Darmstadt, Germany

autor

Hideo Kozono

• Mathematical Institute, Tôhoku University, Sendai, 980-8578 Japan

autor

Hermann Sohr

• Fakultät für Elektrotechnik, Informatik und Mathematik, Universität Paderborn, 33098 Paderborn, Germany

Identyfikatory

DOI

10.4064/bc81-0-11