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

The Navier-Stokes equation on the Euclidean space R-3 can be expressed in the form partial derivative(t)u - Delta u = -R x R x [S(u)u], where S(u) is an anti-symmetric matrix defined by S(u) = del u - (del u)(T) and R is a Riesz operator defined by R = vertical bar del vertical bar(-1)del. In this paper, we propose a model partial derivative(t)u + D(2)u = -R x [S(u)(R x u)], where D = vertical bar del vertical bar ln(-1/4) (e + lambda ln(e + vertical bar del vertical bar)) with lambda >= 0. We prove that the model is globally well-posed for any initial data in Sobolev space H-s 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 lambda equivalent to 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. 2015 Elsevier Inc. All rights reserved.