Energy-Based Regularity Criteria for the Navier–Stokes Equations

Abstract

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 \(\frac{1}{2}\|u(t)\|_2^2\) is Holder continuous as a function of time t with Holder exponent \(\alpha \in (\frac{1}{2}, 1),\) then u is regular. (ii) If for some \(\alpha \in (\frac{1}{2}, 1)\) the dissipation energy satisfies the left-side condition lim \({\rm inf}_{\delta\rightarrow 0} \frac{1}{{\delta}^{\alpha}} \int_{t-\delta}^t\|\nabla u\|_2^2 d\tau < \infty\) 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.