On the advection-diffusion equation with rough coefficients: Weak solutions and vanishing viscosity

LDP and CLT for SPDEs with transport noise

“…3 therein. Their results are comparable to ours, although the derivations are quite different; the authors in [7] mostly employ commutator estimates in the style of Di Perna-Lions theory, while we make use of classical tools from parabolic theory like maximal regularity, Lions-Magenes lemma and duality techniques. Propositions 2.8 and 2.9 below have the advantage of providing several stability results, both in strong and weak topologies, which are needed for our purposes; moreover they allow for non-divergence free vector fields.…”

“…Similarly to(3.23),σ k • ∇ξ s H β−2+ε e k ξ s H β−1+ε and thus k |k| −2α σ k • ∇ξ s 2 H β−2+ε k,l |k| −2α |l| −2(1−β−ε) | ξ s , e k−l |Note that the right-hand side can be written as a, b * c 2 , wherea k = |k| −2α , b k = |k| −2(1−β−ε) and c k = | ξ s , e k | 2 , k ∈ Z 2 0 ; c ∈ 1 (Z 2 0 ) with c 1 = ξ s and for any p ∈ (1/α, 1/(β +ε)), one has a ∈ p (Z 2 0 ), b ∈ p (Z 2 0 ), where p is the conjugate in particular, v ∈ L ∞ t C δ x for any ∈ [0, 1 − γ and v ∈ L 2 t C δ x for any δ < 2 − γ . Remark A 7. By time reversal, similar results hold if we considered the backward equation (∂ t + )h = b • ∇h + c h with given terminal condition h T ∈ L ∞ .…”

“…11931004, 12090014), and the Youth Innovation Promotion Association, CAS (Y2021002). Both authors thank Massimo Sorella for pointing out the reference [7].…”

“…While working on the current manuscript, we became aware of the recent work [7], which contains a comprehensive study of such advection-diffusion equations, see Sect. 3 therein.…”

Remarks on anomalous dissipation for passive scalars

“…Their example of anomalous dissipation corresponds to the borderline case α=1−. We also mention the recent preprint [49], where the authors also consider the advection–diffusion equation (2.1) with rough advecting field and study in particular the vanishing viscosity limit. They prove that, if θ0∈L2false(double-struckT2false), as long as u∈L1false(false(0,Tfalse);W1,1false(double-struckT2false)false), then no anomalous dissipation is possible, because the limit solutions are Lagrangian.…”