Abstract
We prove the existence of a unique global strong solution for a stochastic two-dimensional Euler vorticity equation for incompressible flows with noise of transport type. In particular, we show that the initial smoothness of the solution is preserved. The arguments are based on approximating the solution of the Euler equation with a family of viscous solutions which is proved to be relatively compact using a tightness criterion by Kurtz.
Highlights
Consider the two-dimensional Euler equation modelling an incompressible flow perturbed by transport type stochasticity dωt + ut · ∇ωt dt + £i ωt ◦ d Wti = 0 i =1 with initial condition ω0, wherei are time-independent divergence-free vector fields, the operator £i is given by £i ωt = ξi · ∇ωt, and (W i )i∈N is a sequence of independent Brownian motions
The Euler equation is used to model the motion of an incompressible inviscid fluid
A representative aspect in this context is the study of the fluid vortex dynamics modelled by the vorticity equation
Summary
Global existence of smooth solutions for the stochastic Euler equation with multiplicative noise in both 2D and 3D has been obtained in [31]. Existence of a solution for the two-dimensional stochastic Euler equation with noise of transport type has been considered in [12]. The main result of this paper is the following: Theorem: Under certain conditions on the vector fields (ξi )i the two-dimensional stochastic Euler vorticity equation dωt + ut · ∇ωt dt + (ξi · ∇ωt ) ◦ d Wti = 0, ω0 ∈ Wk,2(T2),. 5 we show existence, uniqueness, and continuity for the approximating sequence of solutions constructed in Sect. The paper is concluded with an “Appendix” that incorporates a number of proofs of the technical lemmas and statements of some classical results
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