Mechanisms of Lagrangian Analyticity in Fluids

Abstract

Certain systems of inviscid fluid dynamics have the property that for solutions with just a modest amount of regularity 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, obtaining analyticity for vortex patch solutions in particular. 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 damages the potential analytic properties of the trajectory map. In this instance, we find that a key nonlocal operator does not preserve analytic dependence in time. As a result, we can construct highly regular initial data with corresponding Lagrangian phase space flows that are not analytic as functions of time into the space of C^1 diffeomorphisms.