Smoothness of the trajectories of ideal fluid particles with Yudovich vorticities in a planar bounded domain

TL;DR

This paper investigates the smoothness properties of fluid particle trajectories in 2D Euler flows with bounded vorticity in bounded domains, establishing regularity results based on boundary smoothness and initial vorticity conditions.

Contribution

It proves that boundary smoothness determines the regularity of particle trajectories and extends Yudovich's results to Gevrey classes and cases with less regular vorticities.

Findings

Particle trajectories are infinitely smooth if the boundary is smooth.

Gevrey regularity of trajectories depends on boundary Gevrey class.

Holder continuity of vorticity propagates along flow lines.

Abstract

We consider the incompressible Euler equations in a (possibly multiply connected) bounded domain of R^2, for flows with bounded vorticity, for which Yudovich proved, in 1963, global existence and uniqueness of the solution. We prove that if the boundary of the domain is C^infty (respectively Gevrey of order M > 1) then the trajectories of the fluid particles are C^infty (resp. Gevrey of order M + 2). Our results also cover the case of "slightly unbounded" vorticities for which Yudovich extended his analysis in 1995. Moreover if in addition the initial vorticity is Holder continuous on a part of the domain then this Holder regularity propagates smoothly along the flow lines. Finally we observe that if the vorticity is constant in a neighborhood of the boundary, the smoothness of the boundary is not necessary for these results to hold.