On weak solutions to the Navier-Stokes inequality with internal singularities

Abstract

We construct weak solutions to the Navier-Stokes inequality, $$u\cdot \left({\partial }_{t}u-\nu \Delta u+(u\cdot \nabla )u+\nabla p\right)\le 0$$ in ${\mathbb{R}}^3$, which blow up at a single point $(x_0,T_0)$ or on a set $S\times \{T_0\}$, where $S\subset {\mathbb{R}}^3$ is a Cantor set whose Hausdorff dimension is at least $\xi$ for any preassigned $\xi \in (0,1)$. Such solutions were constructed by Scheffer, Comm. Math. Phys., 1985 & 1987. Here we offer a simpler perspective on these constructions. We sharpen the approach to construct smooth solutions to the Navier-Stokes inequality on the time interval $[0,1]$ satisfying the "approximate equality" $${\Vert u\cdot \left({\partial }_{t}u-\nu \Delta u+(u\cdot \nabla )u+\nabla p\right)\Vert }_{L^{\infty }}\le \vartheta ,$$ and the "norm inflation" $\Vert u(1){\Vert }_{L^{\infty }}\ge \mathcal{N}\Vert u(0){\Vert }_{L^{\infty }}$ for any preassigned $\mathcal{N}>0$, $\vartheta >0$. Furthermore we extend the approach to construct a weak solution to the Euler inequality $$u\cdot \left({\partial }_{t}u+(u\cdot \nabla )u+\nabla p\right)\le 0,$$ which satisfies the approximate equality with $\nu =0$ and blows up on the Cantor set $S\times \{T_0\}$ as above.

86 pages, 23 figures

References in corpus (2)

• Leray's fundamental work on the Navier-Stokes equations: a modern review of "Sur le mouvement d'un liquide visqueux emplissant l'espace"
• Partial regularity for a surface growth model