Convergence of transport noise to Ornstein–Uhlenbeck for 2D Euler equations under the enstrophy measure

Abstract

We consider the vorticity form of the 2D Euler equations which is perturbed by a suitable transport type noise and has white noise initial condition. It is shown that, under certain conditions, this equation converges to the 2D Navier-Stokes equation driven by the space-time white noise.

Eddy heat exchange at the boundary under white noise turbulence

“…One of the main ideas used in this work goes back to [10,11], see also [12,13], but several other aspects are new: first of all the way to overcome the difficulties due to the boundary, but also the more quantitative presentation of the results, which required new proofs.…”

Coagulation dynamics under environmental noise: scaling limit to SPDE

“…From Stratonovich-to-Itô correction, we obtain an extra second-order divergence-form operator, the "eddy diffusion". This opens the door to the investigation of diffusion and coagulation enhancement, along the lines of [10,11,12,7,18,16,15] and references therein (also for other independent works on mixing or diffusion enhancement). We stress that the philosophy underlying the emergence of enhanced diffusion starting from a transporttype noise and through a suitable scaling limit was first discovered by Galeati [18].…”

The infinitesimal generator of the stochastic Burgers equation

“…Now we take again δ = 1/2, bring the integral term from the right hand side to the left, and send t → ∞ to deduce (23). The extension to general ϕ m 0 ∈ (1 + N ) −α L 2 follows from Fatou's lemma, as in step 3.…”

“…For singular SPDEs the situation is somewhat unsatisfactory because while the pathwise approach applies to a wide range of equations, it seems completely unclear how to set up a general probabilistic solution theory. There are some exceptions, for example martingale techniques tend to work in the "not-so-singular" case when the equation is singular but can be handled via a simple change of variables and does not require regularity structures (sometimes this is called the Da Prato-Debussche regime [12,13]); see [52,53] and also [22,23] for a an example where the change of variable trick does not work but still the equation is not too singular. For truly singular equations there exist only very few probabilistic results.…”