An extension theorem for weak solutions of the 3d incompressible Euler equations and applications to singular flows

Abstract

We prove an extension theorem for local solutions of the 3d incompressible Euler equations. More precisely, we show that if a smooth vector field satisfies the Euler equations in a spacetime region $\Omega\times(0,T)$, one can choose an admissible weak solution on $\mathbf R^3\times (0,T)$ of class $C^\beta$ for any $\beta<1/3$ such that both fields coincide on $\Omega\times (0,T)$. Moreover, one controls the spatial support of the global solution. Our proof makes use of a new extension theorem for local subsolutions of the incompressible Euler equations and a $C^{1/3}$ convex integration scheme implemented in the context of weak solutions with compact support in space. We present two nontrivial applications of these ideas. First, we construct infinitely many admissible weak solutions of class $C^\beta_{\text{loc}}$ with the same vortex sheet initial data, which coincide with it at each time $t$ outside a turbulent region of width $O(t)$. Second, given any smooth solution $v$ of the Euler equation on $\mathbf T^3\times(0,T)$ and any open set $U \subset \mathbf T^3$, we construct admissible weak solutions which coincide with $v$ outside $U$ and are uniformly close to it everywhere at time 0, yet blow up dramatically on a subset of $U\times (0,T)$ of full Hausdorff dimension. These solutions are of class $C^\beta$ outside their singular set.