Weak solutions to the Navier-Stokes inequality with arbitrary energy profiles

TL;DR

This paper constructs localized weak solutions to the Navier-Stokes inequality with arbitrary nonincreasing energy profiles, extending previous methods and providing insights into partial regularity and blow-up phenomena.

Contribution

It introduces a new method to build weak solutions to the Navier-Stokes inequality with prescribed energy profiles, including blow-up scenarios, without requiring full Navier-Stokes equations.

Findings

Constructed weak solutions with arbitrary nonincreasing energy profiles.

Demonstrated solutions satisfy partial regularity theory of Caffarelli, Kohn & Nirenberg.

Extended Scheffer's results to include prescribed energy profiles and blow-up solutions.

Abstract

In a recent paper, Buckmaster & Vicol (arXiv:1709.10033) used the method of convex integration to construct weak solutions $u$ to the 3D incompressible Navier-Stokes equations such that $\| u(t) \|_{L^2} =e(t)$ for a given non-negative and smooth energy profile $e: [0,T]\to \mathbb{R}$. However, it is not known whether it is possible to extend this method to construct nonunique suitable weak solutions (that is weak solutions satisfying the strong energy inequality (SEI) and the local energy inequality (LEI)), Leray-Hopf weak solutions (that is weak solutions satisfying the SEI), or at least to exclude energy profiles that are not nonincreasing. In this paper we are concerned with weak solutions to the Navier-Stokes inequality on $\mathbb{R}^3$, that is vector fields that satisfy both the SEI and the LEI (but not necessarily solve the Navier-Stokes equations). Given $T>0$ and a nonincreasing energy profile $e\colon [0,T] \to [0,\infty )$ we construct weak solution to the Navier-Stokes inequality that are localised in space and whose energy profile $\| u(t)\|_{L^2 (\mathbb{R}^3 )}$ stays arbitrarily close to $e(t)$ for all $t\in [0,T]$. Our method applies only to nonincreasing energy profiles. The relevance of such solutions is that, despite not satisfying the Navier-Stokes equations, they satisfy the partial regularity theory of Caffarelli, Kohn & Nirenberg (Comm. Pure Appl. Math., 1982). In fact, Scheffer's constructions of weak solutions to the Navier-Stokes inequality with blow-ups (Comm. Math. Phys., 1985 & 1987) show that the Caffarelli, Kohn & Nirenberg's theory is sharp for such solutions. Our approach gives an indication of a number of ideas used by Scheffer. Moreover, it can be used to obtain a stronger result than Scheffer's. Namely, we obtain weak solutions to the Navier-Stokes inequality with both blow-up and a prescribed energy profile.