Geometric Hydrodynamics in Open Problems

The uploaded paper states Theorem 6.1: for 2D incompressible Euler on a compact real-analytic surface or a bounded analytic domain, any H^s (s>2) solution with slip boundary condition has particle trajectories that are real-analytic in time; it sketches two proof paradigms, including the Lagrangian/geodesic approach on the real-analytic Banach manifold Ds_μ(M) with an analytic spray, then invoking analytic ODE theory on complex Banach manifolds to conclude analyticity of the flow and, by evaluation, of trajectories. This matches the model’s solution essentially step-for-step (geometric formulation, analytic structure of Ds_μ, analytic spray, complexification, analytic ODE, evaluation). See the theorem and discussion in §6.1 (Analyticity of particle trajectories) , the formulation via an analytic vector field on Ds_μ(M) , and the underlying geodesic framework and exponential map set-up in §§1 and 5.2 .