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 Holder spaces $C^{1,mu}$. We review the Eulerian-Lagrangian formulation of the equations and prove that given initial data in $H^s$ for $ngeq2$ and $s>frac{n}{2}+1$, a unique local-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 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.
This paper establishes the local-in-time existence and uniqueness of strong solutions in $H^{s}$ for $s > n/2$ to the viscous, non-resistive magnetohydrodynamics (MHD) equations in $mathbb{R}^{n}$, $n=2, 3$, as well as for a related model where the a
dvection terms are removed from the velocity equation. The uniform bounds required for proving existence are established by means of a new estimate, which is a partial generalisation of the commutator estimate of Kato & Ponce (Comm. Pure Appl. Math. 41(7), 891-907, 1988).
