A Geometric Rigidity Theorem for Hydrodynamical Blowup

作者：Preston Stephen C^*

来源：Communications in Partial Differential Equations, 2010, 35(11): 2007-2020.

DOI：10.1080/03605302.2010.490864

• Conjugate points
• Diffeomorphism group
• Finite time blowup
• Global existence
• 3D Euler equations

Suppose there is a smooth solution u of the Euler equation on a 3-dimensional manifold M, with Lagrangian flow , such that for some Lagrangian path (t, x) and some time T, we have [image omitted]. 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 tnT such that the corresponding geodesic fails to minimize length on each [tn, tn+1], or there is a basis {e1, e2, e3} of TxM with e3 parallel to the initial vorticity vector 0(x) such that the components of the stretching matrix (t, x)=(D(t, x))TD(t, x) satisfy [image omitted] 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 .

• 出版日期2010