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 × (0, T), one can choose an admissible weak solution on R3 × (0, T) of class Cβ for any β locβ 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 T3 × (0, T) and any open set U ⊂ T3, 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 × (0, T) of full Hausdorff dimension. These solutions are of class Cβ outside their singular set. © The Author(s), 2025. Published by Cambridge University Press.
Enciso et al. (Wed,) studied this question.