Invariant measures for passive scalars in the small noise inviscid limit

Abstract

We consider a class of invariant measures for a passive scalar $f$ driven by an incompressible velocity field $\mathbf{u}$, on a $d$-dimensional periodic domain, satisfying $${\partial }_{t}f+\mathbf{u}\cdot \nabla f=0,f(0)=f_0.$$ The measures are obtained as limits of stochastic viscous perturbations. We prove that the span of the $H^1$ eigenfunctions of the operator $\mathbf{u}\cdot \nabla$ contains the support of these measures. We also analyze several explicit examples: when $\mathbf{u}$ is a shear flow or a relaxation enhancing flow (a generalization of weakly mixing), we can characterize the limiting measure uniquely and compute its covariance structure. We also consider the case of two-dimensional cellular flows, for which further regularity properties of the functions in the support of the measure can be deduced. The main results are proved with the use of spectral theory results, in particular the RAGE theorem, which are used to characterize large classes of orbits of the inviscid problem that are growing in $H^1$.

24 pages, 1 figure. V2: fixed the proof of Corollary 2.4

References in corpus (3)

• Enhanced dissipation and inviscid damping in the inviscid limit of the Navier-Stokes equations near the 2D Couette flow
• Anomalous diffusion in fast cellular flows at intermediate time scales
• From averaging to homogenization in cellular flows - an exact description of the transition

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• Suppression of blow-up in Patlak-Keller-Segel via shear flows
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