Regularity of weak solutions for the Navier-Stokes equations in the class \(L^\infty \)(BMO\(^{-1}\)) (Q2905291)

Regularity theory is applied to weak solutions in the class \(L^\infty (-1,0; \mathrm{BMO}^{-1}(\mathbb{R}^{3}))\) of the Navier-Stokes equations. Namely, any \(L^\infty (-1,0; \mathrm{BMO}^{-1}(\mathbb{R}^{3}))\) solution \(u\) is regular in \(\mathbb{R}^{3} \times ]-1,0]\) if one of the following conditions holds: (1) the function \(\nabla\times u/|\nabla\times u|\) verifies a Lipschitz-type continuity in \(\{ x\in\mathbb{R}^{3}: \) \(|\nabla\times u(x,t)|>d\}\times ]-1,0[\), for a given \(d>0\); (2) \(u(t)\in\) VMO\(^{-1}(\mathbb{R}^{3})\) for any \(t\in ]-1,0]\), and \(u\) verifies \(|u(x,t)|\leq c_0 /|x|\) for any \((x,t)\in \mathbb{R}^{3} \times ]-1,0]\), for some \(c_0>0\); (3) \(u\) is suitable and axisymmetric, and the swirl component of velocity \(u_\theta\) verifies \(\sup_{0<R<1}R^{-1}\int_{Q(z_0,R)}|u_\theta|^2dxdt\leq c_0\) for any \(z_0\in \mathbb{R}^{3} \times ]-1,0[\), for some \(c_0>0\).