A Geometric Rigidity Theorem for Hydrodynamical Blowup

Abstract

Suppose there is a smooth solution u of the Euler equation on a 3-dimensional manifold M, with Lagrangian flow η, such that for some Lagrangian path η(t, x) and some time T, we have . Then in particular smoothness breaks down at time T by the Beale-Kato-Majda criterion. We know by the work of Arnold that the Lagrangian solution is a geodesic in the group of volume-preserving diffeomorphisms. We show that either there is a sequence t n ↗ T such that the corresponding geodesic fails to minimize length on each [t n , t n+1], or there is a basis {e 1, e 2, e 3} of T x M with e 3 parallel to the initial vorticity vector ω0(x) such that the components of the stretching matrix Λ(t, x) = (Dη(t, x))T Dη(t, x) satisfy The former possibility can be studied in terms of the two-point minimization approach of Brenier on volume-preserving maps, while the latter gives a precise sense in which the vorticity vector tends to align with the intermediate eigenvector of the stretching matrix Λ.