Abstract
Two fluid configurations along a flow are conjugate if there is a one parameter family of geodesics (fluid flows) joining them to infinitesimal order. Geometrically, they can be seen as a consequence of the (infinite dimensional) group of volume preserving diffeomorphisms having sufficiently strong positive curvatures which ‘pull’ nearby flows together. Physically, they indicate a form of (transient) stability in the configuration space of particle positions: a family of flows starting with the same configuration deviate initially and subsequently re-converge (resonate) with each other at some later moment in time. Here, we first establish existence of conjugate points in an infinite family of Kolmogorov flows—a class of stationary solutions of the Euler equations—on the rectangular flat torus of any aspect ratio. The analysis is facilitated by a general criterion for identifying conjugate points in the group of volume preserving diffeomorphisms. Next, we show non-existence of conjugate points along Arnold stable steady states on the annulus, disk and channel. Finally, we discuss cut points, their relation to non-injectivity of the exponential map (impossibility of determining a flow from a particle configuration at a given instant) and show that the closest cut point to the identity is either a conjugate point or the midpoint of a time periodic Lagrangian fluid flow.
Résumé
Deux configurations de fluides le long d’un écoulement sont conjuguées s’il existe une famille à un paramètre de géodésiques (écoulements fluides) les reliant à un ordre infinitésimal. D’un point de vue géométrique, ces géodésiques peuvent être considérées comme une conséquence du groupe (de dimension infinie) de difféomorphismes préservant le volume ayant des courbures positives suffisamment fortes pour “rapprocher” les écoulements voisins. Physiquement, ils indiquent une forme de stabilité (transitoire) dans l’espace de configuration des positions des particules: une famille d’écoulements commençant avec la même configuration dévie initialement et les écoulements reconvergent les uns avec les autres ensuite (résonnent) à un moment ultérieur du temps. Ici, nous établissons d’abord l’existence de points conjugués dans une famille infinie d’écoulements de Kolmogorov - une classe de solutions stationnaires des équations d’Euler - sur le tore rectangulaire plat de tout rapport de forme. L’analyse est facilitée par un critère général pour l’identification des points conjugués dans le groupe des difféomorphismes préservant le volume. Ensuite, nous montrons la non-existence de points conjugués le long des états stables d’Arnold sur l’anneau, le disque et le canal. Enfin, nous discutons des points de coupure, de leur relation avec la non-injectivité de la carte exponentielle (impossibilité de déterminer un écoulement à partir d’une configuration de particules à un instant donné) et nous montrons que le point de coupure le plus proche de l’identité est soit un point conjugué, soit le point milieu d’un écoulement fluide Lagrangien périodique en temps.
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Notes
See for example his 1982 Crafoord Prize lecture.
The only exception is when the eigenfunction corresponds to the smallest eigenvalue on the domain. In this case, the laminar solution is the unique stationary solution at all Reynolds number which, moreover, attracts all orbits at long times.
In fact, (a minor modification of) the work of Chepyzhov, Vishik and Zelik [10] shows that the sequence of global attractors \(\{\mathcal {A}^\nu \}_{\nu >0}\) for 2D Navier–Stokes with forcing strength commensurate with the viscosity \(\nu \) converges to a so-called trajectory attractor for the unforced Euler equations. See also the work of Glatt-Holtz et al. [16] for an analogous statement in a stochastically forced setting. This attractor is known to be supported on \(H^1\) velocity fields whose corresponding vorticity is bounded (the Yudovich space). It is not expected to be supported on velocity fields with Sobolev regularity \(s>n/2+1\), which is the setting in which the Euler equations can currently be understood geometrically. Thus, the precise connection with \(\mathscr {D}_\mu ^s(M)\) is not yet clear and building a geometric theory for Yudovich solutions may be of crucial importance for advancing our understanding of high-Reynolds number 2D turbulence.
A. Shnirelman, private communication.
The multiplicity of a conjugate point is the dimension of the kernel of the differential of \(\mathrm {exp}_e\) at that point.
In 3D hydrodynamics both cut and conjugate points are plentiful. On the one hand (local) cut points exist on any sufficiently long geodesic [40]. On the other hand conjugate points cluster or even densely fill out finite geodesic segments [13, 35]. This is in sharp contrast with 2D hydrodynamics and it is tempting to relate this phenomenon to the failure of Arnold’s stability criterion due to indefiniteness of the quadratic form [36]. In fact, it can be show that (with \(v \in T_e\mathscr {D}_\mu ^s(M)\) and \( \Vert v\Vert _{L^2}^2=1\)) we have
$$\begin{aligned} m_c^{u_0,v}&= -2E''_{u_0}(w) + \big \langle [\mathrm {ad}_v, \mathrm {ad}_v^*] u_0, u_0 \big \rangle _{L^2} \end{aligned}$$where \(w = \mathrm {ad}_v^*u_0\), which may prove useful in investigating this issue.
In fact, the curvature is identically zero. This follows from the fact that the solid body rotation on the disk has velocity which is an isometry of the Euclidean space, so that the sectional curvatures must be non-negative [26]. Combining these two facts gives the claim.
There are examples of compact Riemannian manifolds whose conjugate and cut loci are disjoint, cf. [46].
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Acknowledgements
We thank the anonymous referee for helpful comments and Alexandra Haedrich for help with translating the abstract into French. This work was done while TY was an associate professor at the University of Tokyo, Japan.
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To Sasha, with admiration and respect.
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Drivas, T.D., Misiołek, G., Shi, B. et al. Conjugate and cut points in ideal fluid motion. Ann. Math. Québec 46, 207–225 (2022). https://doi.org/10.1007/s40316-021-00176-4
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DOI: https://doi.org/10.1007/s40316-021-00176-4