Steady 3d Euler flows via a topology-preserving convex integration scheme

Abstract

Given any smooth solenoidal vector field $v_0$ on $\mathbf T^3$, we show the existence of infinitely many H\"older-continuous steady Euler flows $v$ with the same topology as $v_0$, in certain weak sense. In particular, we show that $v$ possesses a unique flow of the highest H\"older regularity, which is conjugate to the flow of $v_0$ via a volume-preserving H\"older homeomorphism of $\mathbf T^3$. This result extends to the case of Euler equations on toroidal domains, which has applications to the study of plasmas. The proof relies on a novel convex integration scheme incorporating the key idea that the velocity field of the subsolutions must remain diffeomorphic to $v_0$ at each iteration step.