h-Principles for the incompressible Euler equations

TL;DR

This paper extends the construction of weak solutions to the incompressible Euler equations, establishing optimal h-principles in 2D and 3D, and characterizing which subsolutions can be approximated by exact solutions.

Contribution

It identifies all subsolutions approximable by exact solutions in the $H^{-1}$-norm and demonstrates the three-dimensional nature of these flows.

Findings

Characterization of subsolutions approximable by exact solutions.

Extension of h-principles to 2D and 3D Euler equations.

Existence of genuinely three-dimensional solutions not reducible from 2D flows.

Abstract

Recently, De Lellis and Sz\'ekelyhidi constructed H\"older continuous, dissipative (weak) solutions to the incompressible Euler equations in the torus $\mathbb T^3$. The construction consists in adding fast oscillations to the trivial solution. We extend this result by establishing optimal h-principles in two and three space dimensions. Specifically, we identify all subsolutions (defined in a suitable sense) which can be approximated in the $H^{-1}$-norm by exact solutions. Furthermore, we prove that the flows thus constructed on $\mathbb T^3$ are genuinely three-dimensional and are not trivially obtained from solutions on $\mathbb T^2$.