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Published in:
2013 | OriginalPaper | Chapter
Selected Topics of Local Regularity Theory for Navier–Stokes Equations
Author : Gregory Seregin
Published in: Topics in Mathematical Fluid Mechanics
Publisher: Springer Berlin Heidelberg
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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;