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

Abstract

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,µ .We review the Eulerian-Lagrangian formulation of the equations and prove that given initial data in H s for n ≥ 2 and s > 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. * BCP is supported by an EPSRC Doctoral Training Award. † B.C.Pooley@warwick.ac.uk

Euler-Lagrangian approach to 3D stochastic Euler equations

“…Analogously to the deterministic case, see Proposition 2 in [19], by simple calculation we have the identities…”

“…We study a Lagragian formulation (following [7], [11] and [19] ) of the incompressible Euler equations on a domain T d . The Euler equations with transport noise model the flow of an incompressible inviscid fluid and are (classically) formulated in terms of a divergence-free vector field u (i.e.…”

On a model for the Navier–Stokes equations using magnetization variables

“…The following propositions show that the systems (1,2) and (3)+(4) are equivalent for classical solutions on the interior of a domain Ω ⊆ R 3 (or on the torus T 3 ). The manipulations in the proofs are similar to the derivation of the Weber formula [27] for the Euler equations as described by Constantin [5], see also [20].…”

“…From the heat equation satisfied by v n , it is easy to check that for any t ≥ 0 and any n (20) v n (t) 2 1/2 + 2 t 0 v n (s) 2 3/2 ds ≤ P n w 0 2 1/2 , hence v n ∈ L ∞ (0, T ; H 1/2 ) ∩ L 2 (0, T ; H 3/2 ) is uniformly bounded (independent of n and t). It therefore suffices to find estimates on z in the same spaces.…”

“…From the heat equation satisfied by v n , it is easy to check that for any t ≥ 0 and any n (20) v n (t) [23]) that z n L ∞ (0,Tn;H 1/2 ) ≤ 1/4 and z n L 2 (0,Tn;H 3/2 ) ≤ 1/2 when (21) Tn 0 δ n (t) dt ≤ 1/8. Now since v n (t) 1 ≤ v(t) 1 for all t ≥ 0, we can choose T > 0 such that (21) holds with T n = T for all n. For this T we have uniform bounds on z n in L ∞ (0, T ; H 1/2 ) and L 2 (0, T ; H 3/2 ).…”