TY - JOUR

T1 - Energy-based regularity criteria for the Navier-Stokes equations

AU - Farwig, Reinhard

AU - Kozono, Hideo

AU - Sohr, Hermann

PY - 2009/10

Y1 - 2009/10

N2 - We present several new regularity criteria for weak solutions u of the instationary Navier-Stokes system which additionally satisfy the strong energy inequality. (i) If the kinetic energy 1/2|u(t)\|22 is Hölder continuous as a function of time t with Hölder exponent α in (1/2, 1), then u is regular. (ii) If for some α in (1/2, 1) the dissipation energy satisfies the left-side condition lim rm inf δrightarrow 01δα int-δt∇ u 22dτ <∞ for all t of the given time interval, then u is regular. The proofs use local regularity results which are based on the theory of very weak solutions, see [1], [4], and on uniqueness arguments for weak solutions. Finally, in the last section we mention a local space-time regularity condition.

AB - We present several new regularity criteria for weak solutions u of the instationary Navier-Stokes system which additionally satisfy the strong energy inequality. (i) If the kinetic energy 1/2|u(t)\|22 is Hölder continuous as a function of time t with Hölder exponent α in (1/2, 1), then u is regular. (ii) If for some α in (1/2, 1) the dissipation energy satisfies the left-side condition lim rm inf δrightarrow 01δα int-δt∇ u 22dτ <∞ for all t of the given time interval, then u is regular. The proofs use local regularity results which are based on the theory of very weak solutions, see [1], [4], and on uniqueness arguments for weak solutions. Finally, in the last section we mention a local space-time regularity condition.

KW - Energy inequality

KW - Instationary Navier-Stokes equations

KW - Local in time regularity

KW - Serrin's condition

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U2 - 10.1007/s00021-008-0267-0

DO - 10.1007/s00021-008-0267-0

M3 - Article

AN - SCOPUS:70350336961

SN - 1422-6928

VL - 11

SP - 428

EP - 442

JO - Journal of Mathematical Fluid Mechanics

JF - Journal of Mathematical Fluid Mechanics

IS - 3

ER -