Local well-posedness of the flag geodesic equation: geodesics on the manifold of isometric immersions of a square into $\mathbb{R}^3$

Bauer, Møller-Andersen, and Preston (Arch. Ration. Mech. Anal. 248(3), 2024) model a waving flag as a geodesic in the space of isometric immersions of the flat unit square into $\mathbb{R}^3$ subject to flagpole boundary conditions, and derive the geodesic equation as two coupled second-order-in-time nonlinear equations for the generating functions $(\kappa,\alpha)$, whose coefficients are determined at each fixed time by an auxiliary first-order boundary-value ODE system of six equations for tension-like fields. They chart the space of flags as an infinite-dimensional manifold but explicitly do not attempt any existence, uniqueness, or stability theory for the derived evolution. The open problem is to establish local-in-time well-posedness of this flag geodesic initial-value problem in function spaces natural to the formulation — including solvability and regularity of the auxiliary tension system and treatment of the degenerate configurations of the ruling geometry — or to rigorously demonstrate that the system as stated is ill-posed.

Background

A flag, in the sense of the source paper, is an isometric immersion of the flat unit square into $\mathbb{R}^3$ — a map $\mathbf{r}:[0,1]^2\to\mathbb{R}^3$ preserving the lengths of all curves on the square, which away from planar points forces the surface to be developable (Gaussian curvature zero, ruled by straight asymptotic lines) — together with boundary conditions attaching one edge to a straight flagpole: $\mathbf{r}(0,v)=v\hat{\jmath}$ and $\mathbf{r}_u(0,v)=\hat{\imath}$. A classical rigidity property of developable surfaces is that, away from flat points, such an immersion is determined by two functions of one variable. Bauer–Møller-Andersen–Preston globalized this: for regular upturned flags (a nondegenerate ruling pattern in which the asymptotic lines emanating from the bottom edge reach every point of the sheet, encoded by a $C^1$ function $\alpha$ with $\alpha(0)=0$, $\alpha(1)>0$, and the noncrossing condition $\lambda(x)>0$ on $[0,1]$, where $\lambda(x)=1+\alpha'(x)\gamma(x)$ with the geometric factor $\gamma(x)=1$ for $x\le x^*$ and $\gamma(x)=(1-x)/\alpha(x)$ for $x>x^*$, $x^*$ denoting the switch point with $\alpha(x^*)=1-x^*$ at which the ruling through the bottom edge bends around the top-right corner of the square; positivity of $\lambda$ is equivalent to positivity of the Jacobian $J(x,y)=1+y\alpha'(x)$ of the change of variables on its domain $0\le y\le\gamma(x)$, and to condition (13) of Definition 7 of the paper), the whole flag is reconstructed uniquely from two generating functions: $\kappa(x)$, the signed curvature of the bottom edge, and $\alpha(x)$, the ruling-slope parameter (the bottom-edge torsion is $\tau=-\alpha\kappa$). Under these nondegeneracy assumptions the space of flags carries the structure of an infinite-dimensional manifold charted globally by $(\kappa,\alpha)$; for smooth flags the chart construction loses derivatives and requires the Nash–Moser implicit function theorem, a variant of the inverse function theorem that tolerates a loss of derivatives at each iteration.

The dynamical model follows Arnold's geometric picture of ideal-fluid motion, in which the Euler equations are the geodesic equation of a right-invariant metric on the group of volume-preserving diffeomorphisms, and the program of Ebin and Marsden that converts such PDE into ODE on Banach manifolds and thereby proves local well-posedness. Equipping the space of flags with the natural kinetic-energy ($L^2$-type) metric, the source paper derives (Theorem 29) the geodesic equation on the space of odd ($\kappa(0)=0$), regular, upturned flags: two coupled second-order-in-time nonlinear equations for $(\kappa,\alpha)$ in which the acceleration $\ddot\kappa$ is a second spatial derivative of $\xi\kappa$ plus first spatial derivatives of quadratic combinations of the state and the tension fields, while $\ddot\alpha$ carries an overt division by $\kappa$ (the term $-2\dot\kappa\dot\alpha\alpha'/\kappa$) and spatial derivatives of $\xi\kappa$ and of a combination of the tension fields, each again divided by $\kappa$. Here $\omega,\psi,\chi$ are auxiliary fields defined by the linear ODE system (37)–(39) of the paper from the generating functions and their velocities $(\kappa,\alpha,\dot\kappa,\dot\alpha)$, and the remaining coefficient fields $(\sigma,\rho,g,z,\varphi,q)$ — the analogues of the tension in the inextensible-string equation — are determined, at each fixed time, by an auxiliary linear system of six first-order ODEs (equations (63)–(68) of the paper) with mixed homogeneous boundary conditions at $x=1$ (for $\sigma,\rho,g$) and $x=0$ (for $z,\varphi,q$), together with two algebraic definitions (69)–(70) for $\mu$ and $\xi$. The system is therefore nonlocal and quasilinear: perturbing the initial data anywhere changes the coefficients everywhere through the auxiliary boundary-value problem.

The one-dimensional precursor is the motion of inextensible curves (whips and strings), where the geodesic equation is a single wave equation for the tangent angle with a tension $\sigma$ determined at each time by a second-order spatial ODE; there local well-posedness is known (Preston 2011), and weak, generalized, and a priori-estimate theories have been developed (Preston–Saxton 2013; Şengül–Vorotnikov 2017; Iguchi–Takayama 2024). The flag case is strictly harder: two coupled wave equations instead of one, six tension equations instead of one, derivative loss in the chart map, singular coefficients where $\kappa=0$ (note the division by $\kappa$) and where the geometric factor $\gamma$ vanishes at $x=1$ in the upturned case (Proposition 30 of the paper shows, unconditionally for upturned flags, that the rescaled fields $\sigma/(1-x)^2$, $\rho/(1-x)^2$, and $g/(1-x)^3$ are nonsingular on $[0,1]$, so that (63)–(70) can be rewritten without singular coefficients — a rewriting any theory must then handle), a kink in $\lambda$ at the switch point $x^*$ where $\gamma$ changes formula, and documented shortcut behavior of the weak metric near degenerate configurations. The authors derive the equations formally, prove the manifold structure, and present preliminary numerics for the geodesic boundary-value problem, but explicitly leave the local existence theory open; the zbMATH review of the published version (2024) likewise states that a local existence theorem is currently out of reach.

Problem Statement

Establish — or refute — the local well-posedness of the flag geodesic initial-value problem derived by Bauer–Møller-Andersen–Preston. Concretely: given initial data $(\kappa(0),\alpha(0))$ defining an odd ($\kappa(0)=0$), regular, upturned flag — that is, $\alpha(0)=0$, $\alpha(1)>0$, the noncrossing condition $\lambda(x)>0$ on $[0,1]$, where $\lambda(x)=1+\alpha'(x)\gamma(x)$ with the geometric factor $\gamma(x)=1$ for $x\le x^*$ and $\gamma(x)=(1-x)/\alpha(x)$ for $x>x^*$, $x^*$ the switch point with $\alpha(x^*)=1-x^*$ (equivalently, condition (13) of Definition 7 of the paper), together with the remaining admissibility conditions of Definition 7 — and initial velocities $(\dot\kappa(0),\dot\alpha(0))$ in function spaces consistent with the $C^2$-immersion character of the flag, prove: (a) for each time in a short interval, the auxiliary first-order boundary-value system (63)–(70) — six ODEs for $(\sigma,\rho,g,z,\varphi,q)$ with homogeneous boundary conditions $\sigma(1)=\rho(1)=g(1)=0$ and $z(0)=\varphi(0)=q(0)=0$, together with the algebraic definitions of $\mu,\xi$ — admits a unique solution with the regularity that the evolution equations require; (b) existence and uniqueness of a solution $(\kappa,\alpha)$ of the coupled second-order system of Theorem 29 on a time interval $[0,T)$ depending on the data; and (c) continuous dependence of the solution on the initial data. The theory must either cover or sharply delimit the degenerate configurations intrinsic to the model: the endpoint $x=1$ where the geometric factor $\gamma$ vanishes in the upturned case, points where $\kappa=0$ (where the equations contain divisions by $\kappa$), and the possible loss of smoothness of $\lambda$ at the switch point $x^*$ where $\gamma$ changes formula; it must also propagate, not assume, the admissibility constraint $\lambda>0$. A rigorous demonstration that no such theory exists in any natural class — for example, explicit nonuniqueness within the admissible class, or Hadamard instability of the system as stated — would resolve the question negatively.

References

1. Martin Bauer, Jakob Møller-Andersen, and Stephen C. Preston. Isometric Immersions and the Waving of Flags. Archive for Rational Mechanics and Analysis 248(3), Paper No. 38, 49 p., 2024. DOI: 10.1007/s00205-024-01978-w. https://doi.org/10.1007/s00205-024-01978-w (preprint arXiv:1905.06378, https://arxiv.org/abs/1905.06378)
2. V. I. Arnold. Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits. Annales de l'Institut Fourier 16, 319–361, 1966. DOI: 10.5802/aif.233. https://doi.org/10.5802/aif.233
3. David G. Ebin and Jerrold E. Marsden. Groups of Diffeomorphisms and the Motion of an Incompressible Fluid. Annals of Mathematics 92, 102–163, 1970. DOI: 10.2307/1970699. https://doi.org/10.2307/1970699
4. Stephen C. Preston. The motion of whips and chains. Journal of Differential Equations 251(3), 504–550, 2011. DOI: 10.1016/j.jde.2011.05.005. https://doi.org/10.1016/j.jde.2011.05.005
5. Stephen C. Preston and Ralph Saxton. An $H^1$ model for inextensible strings. Discrete and Continuous Dynamical Systems 33(5), 2065–2083, 2013. DOI: 10.3934/dcds.2013.33.2065. https://doi.org/10.3934/dcds.2013.33.2065
6. Yasemin Şengül and Dmitry Vorotnikov. Generalized solutions for inextensible string equations. Journal of Differential Equations 262(6), 3610–3641, 2017. DOI: 10.1016/j.jde.2016.11.040. https://doi.org/10.1016/j.jde.2016.11.040
7. Tatsuo Iguchi and Masahiro Takayama. A priori estimates for solutions to equations of motion of an inextensible hanging string. Mathematische Annalen 390(2), 1919–1971, 2024. DOI: 10.1007/s00208-023-02786-5. https://doi.org/10.1007/s00208-023-02786-5
8. Maria Deliyianni and Justin T. Webster. Theory of Solutions for an Inextensible Cantilever. Applied Mathematics & Optimization 84 (Suppl. 2), 1345–1399, 2021. DOI: 10.1007/s00245-021-09798-0. https://doi.org/10.1007/s00245-021-09798-0 (preprint arXiv:2005.11836, https://arxiv.org/abs/2005.11836)
9. Maria Deliyianni, Kevin McHugh, Justin T. Webster, and Earl Dowell. Dynamic Equations of Motion for Inextensible Beams and Plates. Archive of Applied Mechanics 92(6), 1929–1952, 2022. DOI: 10.1007/s00419-022-02157-7. https://doi.org/10.1007/s00419-022-02157-7 (preprint arXiv:2107.07454, https://arxiv.org/abs/2107.07454)
10. Po-Chun Kuo, Ming-Chih Lai, Yoichiro Mori, and Analise Rodenberg. The tension determination problem for an inextensible interface in 2D Stokes flow. Research in the Mathematical Sciences 10(4), Paper No. 46, 2023. DOI: 10.1007/s40687-023-00406-x. https://doi.org/10.1007/s40687-023-00406-x
11. Eduardo García-Juárez, Po-Chun Kuo, and Yoichiro Mori. The Immersed Inextensible Interface Problem in 2D Stokes Flow. SIAM Journal on Mathematical Analysis 57(4), 3454–3487, 2025. DOI: 10.1137/24m1658504. https://doi.org/10.1137/24m1658504