Two examples on the property of the noise in the systems of equations of fluid mechanics

In this manuscript, we discuss two examples of the results on the systems of partial differential equations in fluid mechanics obtained via stochastic analysis, for which deterministic case has no counterpart. Firstly, it is well-known that for a deterministic system of equations in fluid mechanics, a lack of diffusion creates a significant obstacle in proving its global well-posedness for a sufficiently small initial data. Nevertheless, it may be proven that for the Hall-magnetohydrodynamics system forced by Lévy noise, such a result may be obtained even with zero viscous diffusion. Secondly, we may extend the classical Kelvin's conservation of circulation flow, as well as the Cauchy's vorticity transport formula, on the Euler equations to show that the solution to the damped Euler equations enjoys the exact same identities except with an exponential decay in time. Such identities are known to completely fail once we add diffusion and consider the damped Navier–Stokes equations. Nevertheless, discovering a stochastic Lagrangian formulation, we are able to show that such identities indeed do continue to hold informally on average, i.e. via a mathematical expectation, over the random ensemble at earlier time. The stochastic Lagrangian formulation and these identities have also been obtained for the solution to the Boussinesq system. These results are rigorously proven in Mohan and Yamazaki (0000) and Yamazaki (0000). In this manuscript we will not pursue complete proofs, which were in particular mostly inspired by Kim (2011) and Iyer (2006), but rather elaborate on motivations, difficulty and remaining open problems for future works.