Mechanisms of Lagrangian Analyticity in Fluids

Certain systems of inviscid fluid dynamics have the property that for solutions that are only slightly better than differentiable in Eulerian variables, the corresponding Lagrangian trajectories are analytic in time. We elucidate the mechanisms in fluid dynamics systems that give rise to this automatic Lagrangian analyticity, as well as mechanisms in some particular fluids systems which prevent it from occurring. We give a conceptual argument for a general fluids model which shows that the fulfillment of a basic set of criteria results in the analyticity of the trajectory maps in time. We then apply this to the incompressible Euler equations to prove analyticity of trajectories for vortex patch solutions. We also use the method to prove that the Lagrangian trajectories are analytic for solutions to the pressureless Euler–Poisson equations, for initial data with moderate regularity. We then examine the compressible Euler equations, and find that the finite speed of propagation in the system is incompatible with the Lagrangian analyticity property. By taking advantage of this finite speed we construct smooth initial data with the property that some corresponding Lagrangian trajectory is not analytic in time. We also study the Vlasov–Poisson system, uncovering another mechanism that deters the analyticity of trajectories. In this case, a key nonlocal operator does not preserve analytic dependence in time. For this system we also construct smooth initial data for which the corresponding solution has some non-analytic Lagrangian trajectory, providing a counterexample to Lagrangian analyticity for a system in which there is an infinite speed of propagation