Enhanced Dissipation, Hypoellipticity, and Anomalous Small Noise Inviscid Limits in Shear Flows

By Jacob Bedrossian, Michele Coti Zelati. 2015

157 citations recorded by the source corpus.

We analyze the decay and instant regularization properties of the evolution semigroups generated by two-dimensional drift-diffusion equations in which the scalar is advected by a shear flow and dissipated by full or partial diffusion. We consider both the space-periodic $${\mathbb{T}^2}$$T2 setting and the case of a bounded channel $${\mathbb{T} \times [0,1]}$$T×[0,1] with no-flux boundary conditions. In the infinite Péclet number limit (diffusivity $${\nu\to 0}$$ν→0), our work quantifies the enhanced dissipation effect due to the shear. We also obtain hypoelliptic regularization, showing that solutions are instantly Gevrey regular even with only partial diffusion. The proofs rely on localized spectral gap inequalities and ideas from hypocoercivity with an augmented energy functional with weights replaced by pseudo-differential operators (of a rather simple form). As an application, we study small noise inviscid limits of invariant measures of stochastic perturbations of passive scalars, and show that the classical Freidlin scaling between noise and diffusion can be modified. In particular, although statistically stationary solutions blow up in $${H^1}$$H1 in the limit $${\nu \to 0}$$ν→0, we show that viscous invariant measures still converge to a unique inviscid measure.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Finance

View Paper: https://www.semanticscholar.org/paper/3ce5e3380572d12f613e1a099e3bf6d800373533

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Citations in the corpus

Enhanced dissipation and Hörmander's hypoellipticity (2021, methodology): “• In the Neumann setting, we improve the timescale from T ∼ ν−1/2 in [8] to T ∼ ν−1/3 when N = 0, and we remove the additional logarithmic factors when N ∈ N0.”
On the stability threshold for the 3D Couette flow in Sobolev regularity (2015, background): “4 for more discussion or previous works such as [45, 20, 34, 12, 4, 5, 17, 11, 6] ([20] are the first to the authors’ knowledge to observe that this is important for understanding (1.”