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

TL;DR

This paper constructs weak solutions to the Navier-Stokes inequality with internal singularities, including point and Cantor set blow-ups, offering a simpler approach and extending to Euler inequalities with similar blow-up behavior.

Contribution

It provides a simplified method to construct weak solutions with prescribed singularities and norm inflation, extending previous results to more general sets and inequalities.

Findings

Constructed solutions blow up at points or Cantor sets with specified Hausdorff dimension.

Achieved solutions with approximate equality and arbitrary norm inflation.

Extended approach to Euler inequality with similar blow-up properties.

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.