Typicality results for weak solutions of the incompressible Navier--Stokes equations

TL;DR

This paper demonstrates that within the class of weak solutions to the incompressible Navier-Stokes equations, smooth solutions are rare and Leray solutions form a nowhere dense set, highlighting the typicality of irregular solutions.

Contribution

It establishes that smooth solutions are meagre and Leray solutions are nowhere dense in the space of weak solutions, providing new insights into the structure of solution sets.

Findings

Smooth solutions are meagre in the solution space.

Leray solutions form a nowhere dense set.

Irregular solutions are typical in the considered class.

Abstract

In this work we show that, in the class of $L^\infty((0,T);L^2(\mathbb{T}^3))$ distributional solutions of the incompressible Navier-Stokes system, the ones which are smooth in some open interval of times are meagre in the sense of Baire category, and the Leray ones are a nowhere dense set.