The energy conservation and regularity for the Navier-Stokes equations

In this paper, we consider the energy conservation and regularity of the weak solution u to the Navier-Stokes equations in the endpoint case. We first construct a divergence-free field u(t, x) which satisfies limt→T √ T − t||u(t)||BMO <∞ and limt→T √ T − t||u(t)||L∞ = ∞ to demonstrate that the Type II singularity is admissible in the endpoint case u ∈ L(BMO). Secondly, we prove that if a suitable weak solution u(t, x) satisfying ||u||L2,∞([0,T ];BMO(Ω)) <∞ for arbitrary Ω ⊆ R then the local energy equality is valid on [0, T ]×Ω. As a corollary, we also prove ||u||L2,∞([0,T ];BMO(R)) <∞ implies the global energy equality on [0, T ]. Thirdly, we show that as the solution u approaches a finite blowup time T , the norm ||u(t)||BMO must blow up at a rate faster than c T−t with some absolute constant c > 0. Furthermore, we prove that if ||u3||L2,∞([0,T ];BMO(R)) = M < ∞ then there exists a small constant cM depended on M such that if ||uh||L2,∞([0,T ];BMO(R)) ≤ cM then u is regular on (0, T ]× R.

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