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

Abstract

We construct weak solutions to the Navier-Stokes inequality, $$ u\cdot \left(\partial_t u -\nu \Delta u + (u\cdot \nabla) u +\nabla p \right) \leq 0 $$ 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" $$ \left\| u\cdot \left(\partial_t u-\nu \Delta u + (u\cdot \nabla) u +\nabla p \right) \right\|_{L^\infty}\leq \vartheta, $$ and the "norm inflation" $\| u(1) \|_{L^\infty} \geq \mathcal{N} \| u(0) \|_{L^\infty}$ for any preassigned $\mathcal{N}>0$, $\vartheta >0$. Furthermore we extend the approach to construct a weak solution to the Euler inequality $$u\cdot \left(\partial_t u+ (u\cdot \nabla) u +\nabla p \right) \leq 0, $$ which satisfies the approximate equality with $\nu =0$ and blows up on the Cantor set $S\times \{T_0 \}$ as above.