An extension theorem for weak solutions of the 3d incompressible Euler equations and applications to singular flows
Alberto Enciso, Javier Peñafiel-Tomás, Daniel Peralta-Salas
TL;DR
The paper proves an extension theorem for local smooth solutions of the 3D incompressible Euler equations, showing that any local solution on a region Ω×(0,T) can be extended to a global admissible weak solution in C^β with β<1/3, while controlling spatial support and allowing prescribed energy profiles. The approach splits into two stages: first extend the local data to a global smooth subsolution with compact support using a divergence- and Hodge-theoretic extension, then apply a Nash-type convex integration scheme with careful space-time gluing to produce a weak solution whose Reynolds stress vanishes in the limit. The authors also present two applications: constructing infinitely many weak solutions from a vortex-sheet initial data and generating solutions that blow up on subsets of positive measure with full Hausdorff dimension, while remaining Hölder outside the singular set. The combination of an extension theorem for subsolutions and a space-time convex integration framework provides new flexibility in constructing anomalous weak solutions with controlled spatial structure and energy behavior, advancing understanding of Onsager-type phenomena in nontrivial geometries. The results illuminate the nonuniqueness and rich tapestry of weak solutions to the Euler equations and offer new tools for exploring turbulence-like behavior in mathematical fluid mechanics.
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 $Ω\times(0,T)$, one can choose an admissible weak solution on $\mathbf R^3\times (0,T)$ of class $C^β$ for any $β<1/3$ such that both fields coincide on $Ω\times (0,T)$. 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 $C^β_{\text{loc}}$ 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 $C^β$ outside their singular set.