A constructive approach to regularity of Lagrangian trajectories for incompressible Euler flow in a bounded domain

We give a constructive proof of smoothness (in ultradifferentiable classes) of Lagrangian trajectories for 3D incompressible Euler flows in an impermeable bounded domain whose boundary is ultradifferentiable, i.e. may be either analytic or have a regularity between indefinite differentiability and analyticity. Based on a little-known Cauchy Lagrangian formulation of the 3D incompressible Euler equations, we establish novel explicit recursion relations that include contributions from the boundary.

This leads to a constructive proof of time-analyticity of the Lagrangian trajectories with analytic boundaries, which can then be used subsequently for the design of a very high-order Cauchy-Lagrangian method to study numerically, among other, the open issue of finite time blow up of classical solutions.