Enhanced dissipation and Hörmander's hypoellipticity

Albritton, Dallas; Beekie, Rajendra; Novack, Matthew

doi:10.48550/arxiv.2105.12308

Cited by 5 publications

(28 citation statements)

References 37 publications

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“…The estimate (6.8) is a direct consequence of the main theorem of [1]. We also refer the interested readers to [2] and [23].…”

Section: Proof Of Theoremmentioning

confidence: 94%

“…Recall our notation L † for the differential operator L † = 1 2 ν∆ + Au(y)∂ x subject to Dirichlet boundary conditions at y = 0, L. We also recall Theorem 1.1 in [1], which provides enhanced dissipation estimates for the solutions to passive scalar equations subject to shear flows and Dirichlet boundary conditions in the channel. The explicit estimate is identical to (6.8), so we omit the details.…”

Section: Proof Of Theoremmentioning

confidence: 99%

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Random search in fluid flow aided by chemotaxis

Gong¹,

He²,

Kiselev³

2021

Preprint

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In this paper, we consider the dynamics of a 2D target-searching agent performing Brownian motion under the influence of fluid shear flow and chemical attraction. The analysis is motivated by numerous situations in biology where these effects are present, such as broadcast spawning of marine animals and other reproduction processes or workings of the immune systems. We rigorously characterize the limit of the expected hit time in the large flow amplitude limit as corresponding to the effective one-dimensional problem. We also perform numerical computations to characterize the finer properties of the expected duration of the search. The numerical experiments show many interesting features of the process, and in particular existence of the optimal value of the shear flow that minimizes the expected target hit time and outperforms the large flow limit.

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“…The estimate (6.8) is a direct consequence of the main theorem of [1]. We also refer the interested readers to [2] and [23].…”

Section: Proof Of Theoremmentioning

confidence: 94%

Section: Proof Of Theoremmentioning

confidence: 99%

Random search in fluid flow aided by chemotaxis

Gong¹,

He²,

Kiselev³

2021

Preprint

View full text Add to dashboard Cite

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“…Remark 3.5. Quantitative arguments based on L 2 Hörmander inequalities can be found in [3,20] (completed concurrently with or after [16]). Thinking about hypoellipticity in terms of functional inequalities, rather than qualitative statements about regularity of solutions to PDEs, has other important advantages as well, for example, it is easier to adapt classical elliptic and parabolic PDE methods, such as De Giorgi or Moser iterations, into hypoelliptic equations [20,45,72].…”

Section: Uniform Hypoellipticitymentioning

confidence: 99%

Lower bounds on the Lyapunov exponents of stochastic differential equations

Bedrossian,

Blumenthal,

Punshon-Smith

2021

Preprint

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In this article, we review our recently introduced methods for obtaining strictly positive lower bounds on the top Lyapunov exponent of high-dimensional, stochastic differential equations such as the weaklydamped Lorenz-96 (L96) model or Galerkin truncations of the 2d Navier-Stokes equations. This hallmark of chaos has long been observed in these models, however, no mathematical proof had been made for either deterministic or stochastic forcing.The method we proposed combines (A) a new identity connecting the Lyapunov exponents to a Fisher information of the stationary measure of the Markov process tracking tangent directions (the so-called "projective process"); and (B) an L 1 -based hypoelliptic regularity estimate to show that this (degenerate) Fisher information is an upper bound on some fractional regularity. For L96 and GNSE, we then further reduce the lower bound of the top Lyapunov exponent to proving that the projective process satisfies Hörmander's condition. We review the recent contributions of the first and third author on the verification of this condition for the 2d Galerkin-Navier-Stokes equations in a rectangular, periodic box of any aspect ratio. Finally, we briefly contrast this work with our earlier work on Lagrangian chaos in the stochastic Navier-Stokes equations. We end the review with a discussion of some open problems.

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“…In this section, we give the main idea of the proof for the linearized QG equation in critical case. By taking s = 1 2 of (1.3), we get that…”

Section: Hypocoercivity and Non-local Enhancementmentioning

confidence: 99%

“…Generally speaking, the sheared velocity sends information to higher frequency, the diffusion term 'kill' the information in the higher frequency. The degeneracy rate of sheared velocity is corresponding to the lowest speed how the information moves to higher frequency, which leads to the different enhanced dissipation rates [1,4,12,32]. Also the stronger diffusion gives stronger enhanced dissipation [19].…”

Section: Introductionmentioning

confidence: 99%

Metastability for the dissipative quasi-geostrophic equation and the non-local enhancement

Li,

Zhao

2021

Preprint

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In this paper, we study the metastability for the 2-D linearized dissipative quasigeostrophic equation with small viscosity ν around the quasi steady state θsin = e −νt sin y. We proved the linear enhanced dissipation and obtained the dissipation rate. Moreover, the new non-local enhancement phenomenon was discovered and discussed. Precisely we showed that the non-local term re-enhances the enhanced diffusion effect by the shear-diffusion mechanism.

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Enhanced dissipation and Hörmander's hypoellipticity

Cited by 5 publications

References 37 publications

Random search in fluid flow aided by chemotaxis

Random search in fluid flow aided by chemotaxis

Lower bounds on the Lyapunov exponents of stochastic differential equations

Metastability for the dissipative quasi-geostrophic equation and the non-local enhancement

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