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

Reinhard Farwig, Hideo Kozono, and Hermann Sohr. "Criteria of local in time regularity of the Navier-Stokes equations beyond Serrin's condition." Banach Center Publications 81.1 (2008): 175-184. <http://eudml.org/doc/282090>.

@article{ReinhardFarwig2008,
abstract = {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).},
author = {Reinhard Farwig, Hideo Kozono, Hermann Sohr},
journal = {Banach Center Publications},
keywords = {nonstationary Navier-Stokes equations; local in time regularity; Serrin's condition},
language = {eng},
number = {1},
pages = {175-184},
title = {Criteria of local in time regularity of the Navier-Stokes equations beyond Serrin's condition},
url = {http://eudml.org/doc/282090},
volume = {81},
year = {2008},
}

TY - JOUR
AU - Reinhard Farwig
AU - Hideo Kozono
AU - Hermann Sohr
TI - Criteria of local in time regularity of the Navier-Stokes equations beyond Serrin's condition
JO - Banach Center Publications
PY - 2008
VL - 81
IS - 1
SP - 175
EP - 184
AB - 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).
LA - eng
KW - nonstationary Navier-Stokes equations; local in time regularity; Serrin's condition
UR - http://eudml.org/doc/282090
ER -