Euler-Lagrangian approach to 3D stochastic Euler equations

Abstract

3D stochastic Euler equations with a special form of multiplicative noise are considered. A Constantin-Iyer type representation in Euler-Lagrangian form is given, based on stochastic characteristics. Local existence and uniqueness of solutions in suitable Hölder spaces is proved from the Euler-Lagrangian formulation.

Circulation and Energy Theorem Preserving Stochastic Fluids

“…See Appendix A for more explanation of the notation ♯. For a proof of the representation (20), see [18]. This result can be expressed also at the level of vorticity ω t = curl(u t ), where one has an exact Cauchy formula of the form…”

“…Formula (63) in the proof below exhibits the terms that are omitted in (18) relative to the full double-Lie diffusion appearing in equation (16). Consequently, this alternative pathwise Kelvin theorem provides a characterization of smooth solutions u t to (7) with the same noise σ…”

“…Direction 1: Stochastic Euler-Poincaré solutions have a pathwise Kelvin Theorem. In view of Proposition 1, one direction is simple: by using equation (7) with f t and {σ (k) t } k∈N defined by (10) and (11) in Theorem 1, and applying Proposition 1 to the unique smooth solution u t for given initial conditions u 0 (which always exists provided, at least, that T is taken sufficiently small [8,18], see Remark 3), one has that realization-by-realization of the Brownian processes {W For the other direction, assume that the circulation is conserved along all material loops Γ. Since u t and {ξ (k) } k∈N are assumed smooth, the map x → X t (x) is a F t -adapted diffeomorphism [17,37].…”

“…In § 3.3 of [14], it is shown there exists (for data in the appropriate Sobolev space) a maximal stopping time τ max : Ξ → [0, ∞) and a unique solution u ∈ C(0, τ; W 3,2 (T 3 ; R 3 )) for all τ τ max . Subsequently, [25] established local existence of (1.5)-(1.6) in Hölder spaces C(0, T ; C n+1,α (Ω)) for some n ∈ N and some α ∈ (0, 1), by using the Weber formula (2.17) and following the Eulerian-Lagrangian scheme of Constantin [7]. Thus, in view of remark 2.3, regularity in the appropriate Hölder spaces can be taken as the precise meaning of 'smooth' in theorem 2.1 as well as in proposition 2.5 and theorem 2.12 appearing below.…”

On the Well-Posedness of Stochastic Boussinesq Equations with Transport Noise

“…Several results have also been established regarding the two and three-dimensional Euler equation [Bes99,BF99,Kim02,Kim09,CFM07,GHV14]. Recently, solution properties of a newly derived stochastic model of the Euler equation were investigated in [CHF17,FL18]. This model was proposed by D. Holm in [Hol15] and presents an innovative geometric approach for including stochastic processes as cylindrical transport noise in PDE systems via a stochastic variational principle.…”