  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.
