An Eulerian–Lagrangian Form for the Euler Equations in Sobolev Spaces

In 2000 Constantin showed that the incompressible Euler equations can be written in an “Eulerian–Lagrangian” form which involves the back-to-labels map (the inverse of the trajectory map for each fixed time). In the same paper a local existence result is proved in certain Hölder spaces $${C^{1,\mu}}$$C1,μ. We review the Eulerian–Lagrangian formulation of the equations and prove that given initial data in Hs for $${n \geq 2}$$n≥2 and $${s > \frac{n}{2}+1}$$s>n2+1, a unique local-in-time solution exists on the n-torus that is continuous into Hs and C1 into Hs-1. These solutions automatically have C1 trajectories. The proof here is direct and does not appeal to results already known about the classical formulation. Moreover, these solutions are regular enough that the classical and Eulerian–Lagrangian formulations are equivalent, therefore what we present amounts to an alternative approach to some of the standard theory.