arXiv subjects

Stephen C. Preston

Publications and source records attributed to Stephen C. Preston.

At least 19 recordsLinked to original sources

arXiv2026

Exact non-stationary solutions of the Euler equations in two and three dimensions

We develop, via Arnold's geometric framework, a mechanism for constructing explicit, smooth, global-in-time, and typically non-stationary solutions of the incompressible Euler equations. The approach introduces a notion of generalized Coriolis force, whose spectrum underlies the construction of these solutions. We recover classical exact solutions such as Kelvin and Rossby-Haurwitz waves, while also producing new explicit examples on curved surfaces and three-dimensional manifolds including the round three-sphere. Furthermore, we obtain a complete classification in two dimensions and a partial classification in three dimensions of the Riemannian manifolds that admit such solutions. The method is in fact formulated in the general Euler-Arnold setting and yields a simple criterion for non-stationarity.

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arXiv2025

Conjugate points on Lie groups with left-invariant metrics

We prove sufficient conditions for the existence of conjugate points along geodesics of a left-invariant metric on a Lie group, using a reformulation of the index form in terms of the adjoint action. In the compact semisimple case, with an arbitrary left-invariant metric, we show that all geodesics must have a conjugate point, and we give upper and lower bounds on conjugate times. In particular this applies to the left-invariant metrics on $SU(n)$ and $SO(n)$ which are of importance in fluid dynamics and rigid body motion, and yields estimates for the diameter and injectivity radius. We also establish criteria in the noncompact case: we show that every closed nonhomogeneous geodesic has a conjugate point, and determine explicit conditions for them in the three-dimensional unimodular case. For homogeneous geodesics, we relate conjugate points to Lagrangian stability, and Eulerian stability of the corresponding steady velocity. Finally, we obtain as by-products criteria for conjugate points in general homogeneous spaces, by lifting the problem to the total Lie group of the quotient and using a result of O'Neill. Through several examples, we show that our theorems apply when well-known criteria relying on positive Ricci curvature or other curvature bounds fail, and in some cases even when Ricci curvature is negative in all directions.

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arXiv2025

Ricci curvature for hydrodynamics on the sphere

The geometric description of incompressible hydrodynamics, as geodesic motion on the infinite-dimensional group of volume-preserving diffeomorphisms, enables notions of curvature in the study of fluids in order to study stability. Formulas for Ricci curvature are often simpler than those for sectional curvature, which typically takes both signs, but the drawback is that Ricci curvature is rarely well-defined in infinite-dimensional spaces. Here we suggest a definition of Ricci curvature in the case of two-dimensional hydrodynamics, based on the finite-dimensional Zeitlin models arising in quantization theory, which gives a natural tool for renormalization. We provide formulae for the finite-dimensional approximations and give strong numerical evidence that these converge in the infinite-dimensional limit, based in part on four new conjectured identities for Wigner $6j$ symbols. The suggested limiting expression for (average) Ricci curvature is surprisingly simple and demonstrates an average instability for high-frequency modes which helps explain long-term numerical observations of spherical hydrodynamics due to mixing.

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arXiv2025

Breakdown of smooth solutions to the subcritical EPDiff equation

We consider the EPDiff equation on $\mathbb{R}^n$ with the integer-order homogeneous Sobolev inertia operator $A=(-Δ)^k$. We prove that for arbitrary radial initial data and a sign condition on the initial momentum, the corresponding radial velocity solution has $C^1$ norm that blows up in finite time whenever $0\le k<n/2+1.$ Our approach is to use Lagrangian coordinates to formulate EPDiff as an ODE on a Banach space, enabling us to use a comparison estimate with the Liouville equation. Along the way we derive the Green function in terms of hypergeometric functions and discuss their properties. This is a step toward proving the general conjecture that the EPDiff equation is globally well-posed for any Sobolev inertia operator of any real order $k$ if and only if $k\ge n/2+1$.

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arXiv2025

Characterization of geodesic completeness for landmark space

We provide a full characterization of geodesic completeness for spaces of configurations of landmarks with smooth Riemannian metrics that satisfy a rotational and translation invariance and which are induced from metrics on subgroups of the diffeomorphism group for the shape domain. These spaces are widely used for applications in shape analysis, for example, for measuring shape changes in medical imaging and morphometrics in biology. For statistics of such data to be well-defined, it is imperative to know if geodesics exist for all times. We extend previously known sufficient conditions for geodesic completeness based on the regularity of the metric to give a full characterization for smooth Riemannian metrics with a rotational and translation invariance by means of an integrability criterion that involves only the behavior of the cometric kernel as landmarks approach collision. We further use the integrability criterion for geodesic completeness and previous work on stochastic completeness to construct a family of Riemannian landmark manifolds that are geodesically complete but stochastically incomplete.

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arXiv2024

Zeitlin's model for axisymmetric 3-D Euler equations

Zeitlin's model is a spatial discretization for the 2-D Euler equations on the flat 2-torus or the 2-sphere. Contrary to other discretizations, it preserves the underlying geometric structure, namely that the Euler equations describe Riemannian geodesics on a Lie group. Here we show how to extend Zeitlin's approach to the axisymmetric Euler equations on the 3-sphere. It is the first discretization of the 3-D Euler equations that fully preserves the geometric structure, albeit restricted to axisymmetric solutions. Thus, this finite-dimensional model admits Riemannian curvature and Jacobi equations, which are discussed.

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arXiv2024

Liouville comparison theory for blowup of Euler-Arnold equations

In this article we introduce a new blowup criterion for (generalized) Euler-Arnold equations on $\mathbb R^n$. Our method is based on treating the equation in Lagrangian coordinates, where it is an ODE on the diffeomorphism group, and comparison with the Liouville equation; in contrast to the usual comparison approach at a single point, we apply comparison in an infinite dimensional function space. We thereby show that the Jacobian of the Lagrangian flow map of the solution reaches zero in finite time, which corresponds to $C^1$-blowup of the velocity field solution. We demonstrate the applicability of our result by proving blowup of smooth solutions to some higher-order versions of the EPDiff equation in all dimensions $n\geq 3$. Previous results on blowup of higher dimensional EPDiff equations were only for versions where the geometric description corresponds to a Sobolev metric of order zero or one. In these situations the behavior does not depend on the dimension and thus already solutions to the one-dimensional version were exhibiting blowup. In the present paper blowup is proved even in situations where the one-dimensional equation has global solutions, such as the EPDiff equation corresponding to a Sobolev metric of order two.

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arXiv2023

Geometric Analysis of the Generalized Surface Quasi-Geostrophic Equations

We investigate the geometry of a family of equations in two dimensions which interpolate between the Euler equations of ideal hydrodynamics and the inviscid surface quasi-geostrophic equation. This family can be realised as geodesic equations on groups of diffeomorphisms. We show precisely when the corresponding Riemannian exponential map is non-linear Fredholm of index 0. We further illustrate this by examining the distribution of conjugate points in these settings via a Morse theoretic approach.

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arXiv2023

Conjugate points along Kolmogorov flows on the torus

The geodesics in the group of volume-preserving diffeomorphisms (volumorphisms) of a manifold $M$, for a Riemannian metric defined by the kinetic energy, can be used to model the movement of ideal fluids in that manifold. The existence of conjugate points along such geodesics reveal that these cease to be infinitesimally length-minimizing between their endpoints. In this work, we focus on the case of the torus $M=\T^2$ and on geodesics corresponding to steady solutions of the Euler equation generated by stream functions $ψ=-\cos(mx)\cos(ny)$ for integers $m$ and $n$, called Kolmogorov flows. We show the existence of conjugate points along these geodesics for all pairs of strictly positive integers $(m,n)$, thereby completing the characterization of all pairs $(m,n)$ such that the associated Kolmogorov flow generates a geodesic with conjugate points.

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arXiv2023

Isometric Immersions and the Waving of Flags

In this article we propose a novel geometric model to study the motion of a physical flag. In our approach a flag is viewed as an isometric immersion from the square with values in $\mathbb R^3$ satisfying certain boundary conditions at the flag pole. Under additional regularity constraints we show that the space of all such flags carries the structure of an infinite dimensional manifold and can be viewed as a submanifold of the space of all immersions. In the second part of the article we equip the space of isometric immersions with its natural kinetic energy and derive the corresponding equations of motion. This approach can be viewed in a similar spirit as Arnold's geometric picture for the motion of an incompressible fluid.

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arXiv2023

3 methods to put a Riemannian metric on Shape Space

In many applications, one is interested in the shape of an object, like the contour of a bone or the trajectory of joints of a tennis player, irrespective of the way these shapes are parameterized. However for analysis of these shape spaces, it is sometimes useful to have a parameterization at hand, in particular if one is interested in deforming shapes. The purpose of the paper is to examine three different methods that one can follow to endow shape spaces with a Riemannian metric that is measuring deformations in a parameterization independent way.

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arXiv2022

Conjugate point criteria on the area-preserving diffeomorphism group

This paper answers some questions about conjugate points along the geodesics corresponding to steady 2D Euler flows, posed by a paper of Drivas-Misiolek-Shi-Yoneda. We present a new sufficient criterion for the existence of conjugate points, which improves on the criterion of Misiolek. It applies in any rotational cell of a steady 2D Euler flow, and in case of rotational symmetry it captures all known conjugate points. We give a general construction of the surfaces that admit a steady fluid with given area form, velocity profile, and vorticity profile, and from this we show how to detect conjugate points in a single rotational cell of a steady flow. When the velocity profile has a local extremum, the criterion becomes particularly simple. Several examples are provided, and in an appendix we use the Misiołek criterion to give some new examples of conjugate points along Kolmogorov flows on the torus.

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arXiv2022

Solar models and McKean's breakdown theorem for the $μ$CH and $μ$DP equations

We study the breakdown for $μ$CH and $μ$DP equations on the circle, given by $$m_t + u m_θ + λu_θ m = 0,$$ for $m = μ(u) - u_{θθ}$, where $μ$ is the mean and $λ=2$ or $λ=3$ respectively. It is already known that if the initial momentum $m_0$ never changes sign, then smooth solutions exist globally. We prove the converse: if the initial momentum changes sign, then $C^2$ solutions $u$ must break down in finite time. The technique is similar to that of McKean, who proved the same for the Camassa-Holm equation, but we introduce a new perspective involving a change of variables to treat the equation as a family of planar systems with central force for which the conserved angular momentum is precisely the conserved vorticity. We also demonstrate how this perspective can apply to give some insights for other PDEs of continuum mechanics, such as the Okamoto-Sakajo-Wunsch equation (and in particular the De Gregorio equation).

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arXiv2020

A diffeomorphism-invariant metric on the space of vector-valued one-forms

In this article we introduce a diffeomorphism-invariant Riemannian metric on the space of vector valued one-forms. The particular choice of metric is motivated by potential future applications in the field of functional data and shape analysis and by connections to the Ebin metric on the space of all Riemannian metrics. In the present work we calculate the geodesic equations and obtain an explicit formula for the solutions to the corresponding initial value problem. Using this we show that it is a geodesically and metrically incomplete space and study the existence of totally geodesic subspaces. Furthermore, we calculate the sectional curvature and observe that, depending on the dimension of the base manifold and the target space, it either has a semidefinite sign or admits both signs.

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arXiv2019

Axisymmetric diffeomorphisms and ideal fluids on Riemannian 3-manifolds

We study the Riemannian geometry of 3D axisymmetric ideal fluids. We prove that the $L^2$ exponential map on the group of volume-preserving diffeomorphisms of a $3$-manifold is Fredholm along axisymmetric flows with sufficiently small swirl. Along the way, we define the notions of axisymmetric and swirl-free diffeomorphisms of any manifold with suitable symmetries and show that such diffeomorphisms form a totally geodesic submanifold of infinite $L^2$ diameter inside the space of volume-preserving diffeomorphisms whose diameter is known to be finite. As examples we derive the axisymmetric Euler equations on $3$-manifolds equipped with each of Thurston's eight model geometries.

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arXiv2019

Shape Analysis of Surfaces Using General Elastic Metrics

In this article we introduce a family of elastic metrics on the space of parametrized surfaces in 3D space using a corresponding family of metrics on the space of vector valued one-forms. We provide a numerical framework for the computation of geodesics with respect to these metrics. The family of metrics is invariant under rigid motions and reparametrizations; hence it induces a metric on the "shape space" of surfaces. This new class of metrics generalizes a previously studied family of elastic metrics and includes in particular the Square Root Normal Function (SRNF) metric, which has been proven successful in various applications. We demonstrate our framework by showing several examples of geodesics and compare our results with earlier results obtained from the SRNF framework.

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arXiv2019

Nonpositive Curvature of the quantomorphism group and quasigeostrophic motion

In this paper, we compute the sectional curvature of the quantomorphism group $\mathcal{D}_q(M)$ whose geodesic equation is the quasi-geostrophic (QG) equation in geophysics and oceanography, for flows with a stream function depending on only one variable. Using this explicit formula, we will also derive a criterion for the curvature operator to be nonpositive and discuss the role of the Froude number and the Rossby number on curvature. The main technique to obtain a usable formula is a simplification of Arnold's general formula in the case where a vector field is close to a Killing field, and then use the Green's function explicitly. We show that nonzero Froude number and Rossby numbers both tend to stabilize flows in the Lagrangian sense.

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arXiv2019

Vanishing distance phenomena and the geometric approach to SQG

In this article we study the induced geodesic distance of fractional order Sobolev metrics on the groups of (volume preserving) diffeomorphisms and symplectomorphisms. The interest in these geometries is fueled by the observation that they allow for a geometric interpretation for prominent partial differential equations in the field of fluid dynamics. These include in particular the modified Constantin-Lax-Majda and surface quasi-geostrophic equations. The main result of this article shows that both of these equations stem from a Riemannian metric with vanishing geodesic distance.

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