Selected Topics of Local Regularity Theory for Navier–Stokes Equations

Abstract

$\mathbb{R}_{+} =\{ t \in \mathbb{R} :\,\, t > 0\}$, $\mathbb{R}_{-} =\{ t \in \mathbb{R} :\,\, t < 0\}$; $\mathbb{R}_{+}^{d} =\{ x = (x^\prime ,x_{d}) :\,\, x^\prime = (x_{i}),\,\,i = 1,2,\ldots ,d - 1,\,\,x_{d} > 0\}$; $Q_{-} = {\mathbb{R}}^{d} \times R_{-}$, $Q_{+} = {\mathbb{R}}^{d} \times R_{+}$; $Q_{\delta ,T} = \Omega \times ]\delta ,T[$, $Q_{T} = \Omega \times ]0,T[$, $\Omega \subset {\mathbb{R}}^{d}$; B(x, r) is the ball in ${\mathbb{R}}^{d}$ of radius r centered at the point $x \in {\mathbb{R}}^{d}$, B(r) = B(0, r), B = B(1); $B_{+}(x,r) =\{ y = (y^\prime ,y_{d}) \in B(x,r) :\,\, y_{d} > x_{d}\}$ is a half ball, $B_{+}(r) = B_{+}(0,r)$, $B_{+} = B_{+}(1)$; Q(z, r) = B(x, r) ×]t − r ², t[ is the parabolic ball in ${\mathbb{R}}^{d} \times \mathbb{R}$ of radius r centered at the point $z = (x,t) \in {\mathbb{R}}^{d} \times \mathbb{R}$, Q(r) = Q(0, r), Q = Q(1); $Q_{+}(r) = Q_{+}(0,r) = B_{+}(r)\times ] - {r}^{2},0[$; $L_{s}(\Omega )$ and $W_{s}^{1}(\Omega )$ are the usual Lebesgue and Sobolev spaces, respectively; $L_{s,l}(Q_{T}) = L_{l}(0,T;L_{s}(\Omega ))$, $L_{s}(Q_{T}) = L_{s,s}(Q_{T})$; $W_{s,l}^{1,0}(Q_{T}) =\{ \vert v\vert + \vert \nabla v\vert \in L_{s,l}(Q_{T})\}$ and $W_{s,l}^{1,0}(Q_{T}) =\{ \vert v\vert + \vert \nabla v\vert + \vert {\nabla }^{2}v\vert + \vert \partial v\vert \in L_{s,l}(Q_{T})\}$ are parabolic Sobolev spaces; $C_{0,0}^{\infty }(\Omega ) =\{ v \in C_{0}^{\infty }(\Omega ) :\,\, \mathrm{div}\,v = 0\}$; ${ \circ \atop J} (\Omega )$ is the closure of the set $C_{0,0}^{\infty }(\Omega )$ in the space $L_{2}(\Omega )$, ${ \circ \atop J} _{2}^{1}(\Omega )$ is the closure of the same set with respect to the Dirichlet integral; BMO is the space of functions having bounded mean oscillation;