Stephen C. Preston

The motion of whips and chains

Abstract We study the motion of an inextensible string (a whip) fixed at one point in the absence of gravity, satisfying the equations η t t = ∂ s ( σ η s ) , σ s s − | η s s | 2 σ = − | η s t | 2 , | η s | 2 ≡ 1 with boundary conditions η ( t , 1 ) = 0 and σ ( t , 0 ) = 0 . We prove local existence and uniqueness in the space defined by the weighted Sobolev energy ∑ l = 0 m ∫ 0 1 s l | ∂ s l η t | 2 d s + ∫ 0 1 s l + 1 | ∂ s l + 1 η | 2 d s , when m ⩾ 3 . In addition we show persistence of smooth solutions as long as the energy for m = 3 remains bounded. We do this via the method of lines, approximating with a discrete system of coupled pendula (a chain) for which the same estimates hold.

Geometric investigations of a vorticity model equation

Abstract This article consists of a detailed geometric study of the one-dimensional vorticity model equation ω t + u ω x + 2 ω u x = 0 , ω = H u x , t ∈ R , x ∈ S 1 , which is a particular case of the generalized Constantin–Lax–Majda equation. Wunsch showed that this equation is the Euler–Arnold equation on Diff ( S 1 ) when the latter is endowed with the right-invariant homogeneous H ˙ 1 / 2 -metric. In this article we prove that the exponential map of this Riemannian metric is not Fredholm and that the sectional curvature is locally unbounded. Furthermore, we prove a Beale–Kato–Majda-type blow-up criterion, which we then use to demonstrate a link to our non-Fredholmness result. Finally, we extend a blow-up result of Castro–Cordoba to the periodic case and to a much wider class of initial conditions, using a new generalization of an inequality for Hilbert transforms due to Cordoba–Cordoba.

A Geometric Rigidity Theorem for Hydrodynamical Blowup

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.

The Geometry of Barotropic Flow

In this paper we construct a new noninvariant Riemannian metric on the semidirect product of the diffeomorphism group of a manifold and the space of positive functions on that manifold, which has the property that certain geodesics give the equations of barotropic fluid mechanics. We compute a formula for its curvature, analyze the sign of the curvature, and determine directly the growth of Jacobi fields in a few special cases.