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

Reinhard Farwig; Hideo Kozono; Hermann Sohr

Banach Center Publications (2008)

- Volume: 81, Issue: 1, page 175-184
- ISSN: 0137-6934

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topReinhard 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 -

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