Global regularity for a model of three-dimensional Navier–Stokes equation

Abstract

The Navier–Stokes equation on the Euclidean space R3 can be expressed in the form ∂tu−Δu=−R×R×[S(u)u], where S(u) is an anti-symmetric matrix defined by S(u)=∇u−(∇u)⊤ and R is a Riesz operator defined by R=|∇|−1∇. In this paper, we propose a model ∂tu+D2u=−R×[S(u)(R×u)], where D=|∇|ln−14⁡(e+λln⁡(e+|∇|)) with λ≥0. We prove that the model is globally well-posed for any initial data in Sobolev space Hs with s≥3. In a very recent work [14], by using a highly non-trivial averaged version of nonlinearity, Tao proposed a Navier–Stokes model and constructed a smooth solution which develops a finite time singularity. Both Tao's model and ours (in the case of λ≡1) obey the fundamental energy identity of the Navier–Stokes equation.Those results demonstrate that finer structures of the nonlinearity in the Navier–Stokes equation are crucial for the study of this equation, beyond the validity of the energy identity and incompressibility which are the most fundamental properties of Navier–Stokes equation. Without further understanding of the structure of nonlinearity, any attempt to positively resolve the Navier–Stokes global regularity problem in three dimensions is impossible due to Tao's example, and any attempt to negatively resolve the same problem for a Navier–Stokes model is also not convincing to yield a negative answer to the global regularity problem of the original Navier–Stokes equation due to the example given in this paper.