Title: Mechanisms of Lagrangian analyticity in fluids,
Abstract: 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 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 are able to
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 instance, we find that a key nonlocal operator does not
preserve analytic dependence in time. For this system we can also construct
smooth initial data for which the corresponding solution has some non-analytic
Lagrangian trajectory. This provides a counterexample to Lagrangian analyticity
for a system in which there is an infinite speed of propagation, unlike the
compressible Euler equations.