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

Abstract: 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. In the setting of ideal hydrodynamics, the construction recovers 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. We further 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 also formulated in the general Euler-Arnold setting and yields a simple criterion for non-stationarity.

• The paper develops a spectral-geometric method that combines Killing fields, coadjoint eigenfields, and inertia-operator eigenfields to produce exact, smooth, global-in-time nonlinear Euler solutions without small-amplitude assumptions.
• In two dimensions, the required structure exists exactly on constant-curvature surfaces, recovering Kelvin waves and Rossby–Haurwitz waves while introducing new hyperbolic-ball solutions with discrete Legendre-function spectra.
• In three dimensions, the construction yields wave families on circle bundles and the round three-sphere, but the classification of admissible geometries, twisted torus metrics, and the stability of these flows remain open problems.

Overview

This paper develops a geometric mechanism, built on Arnold's framework for ideal hydrodynamics, for constructing explicit, smooth, global-in-time solutions of the incompressible Euler equations on compact Riemannian manifolds of dimension two and three. The solutions are typically non-stationary and trigonometric in time. The construction recovers classical examples—Kelvin waves on flat domains and Rossby–Haurwitz waves on the round two-sphere—and produces new explicit families on curved surfaces (e.g., a geodesic ball in hyperbolic space) and three-dimensional manifolds including the round three-sphere. Beyond hydrodynamics, the method is formulated for general Euler–Arnold equations on Lie groups with right-invariant metrics, where it admits an interpretation via a generalized Coriolis force. The paper also classifies, completely in two dimensions and partially in three, the manifolds admitting the Killing-field structure required by the construction.

The construction mechanism

The setting is a compact Riemannian manifold $(M,g)$, possibly with boundary, with $X(M)$ the space of smooth divergence-free fields tangent to the boundary. The inertia operator is $A = \Delta$ (Hodge Laplacian) in two dimensions and $A = \curl$ in three; its inverse is compact and self-adjoint on the complement of harmonic fields, so $A$ has discrete spectrum with finite-dimensional eigenspaces (2602.13929). Writing the Euler equations as $\partial_t A u + [u, Au] = 0$, the central result is: if $u_0$ is a steady solution ($[u_0, Au_0]=0$) and there exists a complex field $z = v + iw$ that is simultaneously an eigenfield of the coadjoint operator $K_{u_0} = A^{-1}[v, Au_0]$, of $X(M)$0, and of $X(M)$1, then

$X(M)$2

is a complex-valued exact solution of the fully nonlinear Euler equations. Its real part solves the nonlinear equations while its imaginary part solves the linearized Euler equations along it—a notable feature, since the construction yields paired solutions of both the nonlinear and linearized problems. Crucially, no smallness assumption is imposed: the nonlinearity closes exactly on the finite-dimensional span of $X(M)$3 and $X(M)$4. The real solution is stationary precisely when $X(M)$5.

The abundance question reduces to finding Killing fields $X(M)$6 such that $X(M)$7 is also Killing. For such $X(M)$8, the operators $X(M)$9, $A = \Delta$0, and $A = \Delta$1 commute pairwise and are skew-adjoint, so they admit a discrete simultaneous eigenbasis; each eigenfield generates a two-parameter family of exact solutions. A moving-frame interpretation clarifies when these are genuinely non-trivial: the flow satisfies $A = \Delta$2, and eigenfields with $A = \Delta$3 produce only trivial time dependence (a steady pattern advected by the isometry), whereas $A = \Delta$4 corresponds to genuine vorticity twisting.

Two-dimensional classification and examples

In two dimensions the requirement that both $A = \Delta$5 and $A = \Delta$6 be Killing forces constant sectional curvature, proved via the Weitzenböck formula $A = \Delta$7 and the identity $A = \Delta$8 vanishing identically. This is a complete classification: only constant-curvature surfaces admit the structure. Three canonical examples illustrate how curvature controls the dynamics:

Manifold	Killing field	$A = \Delta$9	Stationary iff
Flat torus / disk	translation / rotation	$A = \curl$0	never (nontrivial modes)
Round $A = \curl$1	equatorial rotation	$A = \curl$2	$A = \curl$3 or $A = \curl$4
Hyperbolic ball	rotation	$A = \curl$5	$A = \curl$6 or $A = \curl$7

On the flat torus and disk, $A = \curl$8, so all admissible eigenvalues satisfy $A = \curl$9; nevertheless the solutions are genuinely non-stationary because $A$0 (with $A$1). These recover the classical Kelvin waves, expressed through Fourier modes and Bessel functions respectively. On the round sphere, positive curvature makes $A$2, producing nonzero coadjoint eigenvalues and recovering Rossby–Haurwitz waves with their standard phase speed. On a geodesic ball in hyperbolic space, negative curvature reverses the sign ($A$3), yielding a new family of Kelvin-type waves built from associated Legendre functions $A$4 with discrete spectrum determined by Dirichlet conditions at the boundary. In every case the imaginary part supplies an explicit oscillatory solution of the linearized Euler equation.

Three-dimensional geometry

Here $A$5, and the analysis splits according to whether $A$6 and $A$7 are parallel. A rigidity lemma shows that if both are Killing, then $A$8 is constant; consequently, in the parallel case $A$9 with $\partial_t A u + [u, Au] = 0$0 constant and $\partial_t A u + [u, Au] = 0$1 constant.

Circle bundles. When $\partial_t A u + [u, Au] = 0$2 with $\partial_t A u + [u, Au] = 0$3 unit-length and regular, the dual one-form is a K-contact form, making $\partial_t A u + [u, Au] = 0$4 Sasakian; the Boothby–Wang theorem yields a circle fibration over a surface $\partial_t A u + [u, Au] = 0$5 classified by genus and Euler class $\partial_t A u + [u, Au] = 0$6, with $\partial_t A u + [u, Au] = 0$7. Conversely, any Riemannian surface and nonzero integer arise this way. Nonzero Euler class is essential for nontrivial dynamics—the trivial bundle gives $\partial_t A u + [u, Au] = 0$8 and hence only moving-frame-trivial solutions. A generalized Chandrasekhar–Kendall lemma reduces curl eigenfields to Laplacian eigenfunctions: given $\partial_t A u + [u, Au] = 0$9 with $u_0$0 and $u_0$1, explicit combinations of $u_0$2, $u_0$3, and $u_0$4 diagonalize curl on a three-dimensional invariant subspace, with eigenvalues $u_0$5.

Round three-sphere. Applying this to the Hopf fibration produces what the authors describe as higher-dimensional analogues of Rossby–Haurwitz waves: periodic-in-time solutions occurring in families of multiplicity greater than one, parametrized by integers $u_0$6 and degree $u_0$7, with Jacobi-polynomial radial profiles. The simplest case ($u_0$8, $u_0$9) is written explicitly as polynomial vector fields on $[u_0, Au_0]=0$0 restricted to $[u_0, Au_0]=0$1, rotating at frequency $[u_0, Au_0]=0$2. Arbitrary linear combinations of eigenfields sharing fixed $[u_0, Au_0]=0$3 and $[u_0, Au_0]=0$4 remain exact solutions.

Torus bundles. When $[u_0, Au_0]=0$5 and $[u_0, Au_0]=0$6 are independent, a local normal-form theorem gives coordinates $[u_0, Au_0]=0$7 in which

$[u_0, Au_0]=0$8

with $[u_0, Au_0]=0$9 and $z = v + iw$0. The parameter $z = v + iw$1 genuinely changes the geometry (it enters the scalar curvature and, for $z = v + iw$2, defines a contact form); notably, the authors state they do not know whether any closed compact 3-manifold admits such a metric with $z = v + iw$3, since the scalar curvature typically blows up where $z = v + iw$4 vanishes. On solid-torus domains $z = v + iw$5, curl eigenfields reduce to a coupled ODE system for radial profiles with Bessel-type boundary conditions. The untwisted case $z = v + iw$6, $z = v + iw$7 recovers Kelvin's 1880 cylindrical modes—derived originally as solutions of the linearized equation, later observed by Dritschel to solve the nonlinear equation for arbitrary amplitude. A concrete twisted example with $z = v + iw$8 yields elementary closed-form solutions involving half-integer Bessel functions.

Euler–Arnold framework and generalized Coriolis force

The hydrodynamic results are subsumed into a general Lie-theoretic correspondence. For a Lie group $z = v + iw$9 with right-invariant metric and an element $K_{u_0} = A^{-1}[v, Au_0]$0 satisfying $K_{u_0} = A^{-1}[v, Au_0]$1 (the abstract analogue of being Killing), the curve $K_{u_0} = A^{-1}[v, Au_0]$2 solves the Euler–Arnold equation if and only if $K_{u_0} = A^{-1}[v, Au_0]$3 solves

$K_{u_0} = A^{-1}[v, Au_0]$4

The second term is interpreted as a generalized Coriolis force; on $K_{u_0} = A^{-1}[v, Au_0]$5 with $K_{u_0} = A^{-1}[v, Au_0]$6 it reduces to the classical Coriolis operator $K_{u_0} = A^{-1}[v, Au_0]$7. Unlike related "magnetic" constructions on central extensions of diffeomorphism groups, this works directly on the group and arises purely from a change to a frame rotating under the isometric flow. A spectral criterion follows: any eigenvector of $K_{u_0} = A^{-1}[v, Au_0]$8 with $K_{u_0} = A^{-1}[v, Au_0]$9 generates exact solutions, stationary exactly when $X(M)$00. The construction is further motivated perturbatively: expanding geodesics in powers of a perturbation parameter yields a hierarchy of ODEs, and truncation after first order—available precisely because $X(M)$01 for Killing $X(M)$02—recovers the same structure without any smallness assumption on the resulting solution.

Limitations and open questions

Several restrictions are acknowledged explicitly. All explicit examples require the steady base flow $X(M)$03 and $X(M)$04 to be simultaneously Killing; the authors do not know whether this is essential or merely sufficient, i.e., whether the hypotheses of the main theorem can be satisfied by other means. The three-dimensional classification is partial: a complete characterization of manifolds admitting a Killing field whose curl is also Killing remains open, as does the existence of any closed compact 3-manifold supporting the twisted ($X(M)$05) torus-bundle metric. The inertia-operator formulation is specific to dimensions two and three, so nothing is known about extensions to higher dimensions. Although each solution comes with an oscillatory solution of the linearized Euler equation, no other linearized solutions are known and Eulerian stability is unresolved—even for classical Rossby–Haurwitz waves this is difficult. Lagrangian stability, conjugate points, and the possibility of closed particle trajectories or simultaneously closing trajectories (closed geodesics in the diffeomorphism group) are raised but not settled. Extensions to ideal magnetohydrodynamics and to Navier–Stokes on Einstein 3-manifolds are proposed as tractable directions rather than established results.

Conclusion

The paper provides a systematic, dimension-specific recipe for converting simultaneous spectral data of the inertia operator, the coadjoint operator, and the adjoint action of a Killing field into exact, globally defined, time-periodic solutions of the nonlinear Euler equations, together with matched solutions of the linearized problem. It unifies Kelvin waves, Rossby–Haurwitz waves, Chandrasekhar–Kendall modes, and their hyperbolic and three-dimensional analogues under a single generalized-Coriolis mechanism, and it delineates precisely which geometries—in two dimensions exhaustively, in three partially—support the construction. The main open issues concern the necessity of the double-Killing hypothesis, completion of the three-dimensional classification, and stability of the resulting flows.