Global regularity for a model of three-dimensional Navier-Stokes equation (Q2350016)

In this paper, the author proposes a model equation \(\partial u_t +D^2 u =-{\mathcal R}\times[{\mathcal S} (u)({\mathcal R}\times u)],\,\) where \(D=|\nabla|\ln^{-1/4}(e+ \lambda \ln(e+|\nabla|))\), with \(\lambda\geq 0\), and \({\mathcal R}\) is a Riese operator defined by \({\mathcal R}=|\nabla|^{-1}\nabla\). The author proves that the model is globally well-posed for any initial data in Sobolev space \({\mathcal H}^s\) with \(s\geq 3\). The model for the 3D incompressible Navier-Stokes equation is globally well-posed if \(u_0\in {\mathcal H}^s\) with \(s\geq 3\).