Y spaces and global smooth solution of fractional Navier-Stokes equations with initial value in the critical oscillation spaces (Q683479)

The Cauchy problem to the fractional Navier-Stokes equations is studied in the paper. \[ \frac{\partial u}{\partial t}+(u\cdot\nabla)u+(-\Delta)^\beta u-\nabla p=0,\quad \text{div}\,u=0, \quad x\in\mathbb{R}^n,\;t>0, \] \[ u(x,0)=u_0(x),\;x\in\mathbb{R}^n. \] Here \(n\geq 2\), \(1/2<\beta<1\), \((-\Delta)^\beta\) is the \(\beta\)-order Laplace operator. The problem is investigated in a wide class of spaces including Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces. It is proved that if the initial data \(u_0\) has a small norm then the problem has a unique smooth global solution. The spaces used are described in detail in the article. It should be noted that not all designations are defined, and this makes it difficult to read the work.