Energy-based regularity criteria for the Navier-Stokes equations

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-δ^t∇ 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.