Path-connectedness of incompressible Euler solutions

We study the incompressible Euler equation and prove that the set of weak solutions is path-connected. More precisely, we construct paths of Hölder regularity $C^{1\mathrm{/}2}$, valued in $C_{t,loc}^0{L}_x^2$ endowed with the strong topology. The main result relies on a convex integration construction adapted from the seminal work of De Lellis and Székelyhidi [14, The Euler equations as a differential inclusion], extending it to a more broader geometric framework, replacing balls with arbitrary convex compact sets.