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

摘要

In 2000 Constantin showed that the incompressible Euler equations can bewritten 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 alocal existence result is proved in certain H\"older spaces $C^{1,\mu}$. We review the Eulerian-Lagrangian formulation of the equations and prove thatgiven initial data in $H^s$ for $n\geq2$ and $s>\frac{n}{2}+1$, a uniquelocal-in-time solution exists on the $n$-torus that is continuous into $H^s$and $C^1$ into $H^{s-1}$. These solutions automatically have $C^1$trajectories. The proof here is direct and does not appeal to results already known aboutthe classical formulation. Moreover, these solutions are regular enough thatthe classical and Eulerian-Lagrangian formulations are equivalent, thereforewhat we present amounts to an alternative approach to some of the standardtheory.