h-Principles for the incompressible Euler equations

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 $\mathbb T^2$.