Regularity of solutions to the Navier-Stokes equations (Q1360814)

The regularity of solutions of the stationary and nonstationary Navier-Stokes equations is studied. A distribution \(u(x)\in D'(\Omega)\) is said to be \(H^s\)-regular at a point \((x_0,\xi_0)\in\Omega\times (\mathbb{R}^n\backslash \{0\})\) (in this case we write \(u(x)\in H^s(x_0,\xi_0))\) if one has the representation \(u(x)= u_1(x)+ u_2(x)\), where \(u_1\in H^s(\Omega)\) and \((x_0,\xi_0)\) does not belong to the wave front \(WF(u_2)\) of \(u_2(x)\). First the linear system is considered and the local regularity of its solution is shown. The passage to the nonlinear system is based on the following two facts: (1) for \(s>n/2\) the space \(H^s_{\text{loc}}(\Omega)\) is a ring (the Schauder theorem); (2) for \(n/2< s<r<2s-n/2\) and \(\xi_0\neq 0\) the set \(H^s_{\text{loc}}(\Omega)\cap H^r(x_0,\xi_0)\) is a ring (the Rauch lemma).