Exit Times of Diffusions with Incompressible Drift

Iyer, Gautam; Novikov, Alexei; Ryzhik, Lenya; Zlatoš, Andrej

doi:10.1137/090776895

Cited by 17 publications

(17 citation statements)

References 12 publications

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“…Interpreting this in the context of convection-enhanced diffusion, Theorem 1.1 suggests that larger advection amplitude generally produces faster mixing for reactiondiffusion-advection equation (1) as long as u 0 ∈ I b . In this sense, Theorem 1.1 seems to refine the well-known statement that mixing by an incompressible flow enhances diffusion in various contexts [10,17,20,21,26,27,40,44].…”

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confidence: 58%

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A functional approach towards eigenvalue problems associated with incompressible flow

Liu¹,

Lou²

2020

Discrete &Amp; Continuous Dynamical Systems - A

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We propose a certain functional which is associated with principal eigenfunctions of the elliptic operator L A = −div(a(x)∇) + AV · ∇ + c(x) and its adjoint operator for general incompressible flow V. The functional can be applied to establish the monotonicity of the principal eigenvalue λ 1 (A), as a function of the advection amplitude A, for the operator L A subject to Dirichlet, Robin and Neumann boundary conditions. This gives a new proof of a conjecture raised by Berestycki, Hamel and Nadirashvili [5]. The functional can also be used to prove the monotonicity of the normalized speed c * (A)/A for general incompressible flow, where c * (A) is the minimal speed of traveling fronts. This extends an earlier result of Berestycki [3] for steady shear flow.which model various physical, chemical, and biological processes: On unbounded domains [17,44], compact manifolds [10], and bounded domains with appropriate boundary conditions [1,5,7,34]. Function w represents the density of a population or a substance diffusing with diffusion matrix a(x), reacting through the nonlinearity wf , and advected by the stationary fluid flow V in heterogeneous media.Let Ω be a bounded region of R N with smooth boundary ∂Ω, and n(x) be the outward unit normal vector at x ∈ ∂Ω. Consider equation (1) defined on Ω and suppose that w satisfies bw +(1−b)[a(x)∇w]·n = 0 on ∂Ω with parameter b ∈ [0, 1]. The stability of steady state w ≡ 0 and the minimal speed c * (A) of traveling fronts for equation (1) are associated with the principal eigenvalue, denoted as λ 1 (A), for the linear eigenvalue problem L A u := −div(a(x)∇u) + AV · ∇u + c(x)u = λ 1 (A)u, subject to appropriate boundary conditions, where c(x) = −f (x, 0). 2010 Mathematics Subject Classification. Primary: 35J20, 35J60, 35P15.

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confidence: 58%

“…Together with the definition (18) of v A and the boundary condition of u A in (25), the Claim follows from (24) and (26). Formula (20) is therefore verified.…”

Section: Shuang Liu and Yuan Loumentioning

confidence: 64%

A functional approach towards eigenvalue problems associated with incompressible flow

Liu¹,

Lou²

2020

Discrete &Amp; Continuous Dynamical Systems - A

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“…However, as will be seen in section 2.2, stirring always lowers the integrated mean exit time for our problem. Note that this result is true for the L 1 -norm but not, for instance, the L ∞ -norm as demonstrated by [10], who proved that for any two-dimensional, simply connected domain different from a disk, there always exists a flow that increases the largest exit time compared to the pure conduction case (see Theorem 1.1 in [10]).…”

mentioning

confidence: 99%

Optimal Heat Transfer and Optimal Exit Times

Marcotte¹,

Doering²,

Thiffeault³

et al. 2018

SIAM J. Appl. Math.

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A heat exchanger can be modeled as a closed domain containing an incompressible fluid. The moving fluid has a temperature distribution obeying the advection-diffusion equation, with zero temperature boundary conditions at the walls. Starting from a positive initial temperature distribution in the interior, the goal is to flux the heat through the walls as efficiently as possible. Here we consider a distinct but closely related problem, that of the integrated mean exit time of Brownian particles starting inside the domain. Since flows favorable to rapid heat exchange should lower exit times, we minimize a norm of the exit time. This is a time-independent optimization problem that we solve analytically in some limits, and numerically otherwise. We find an (at least locally) optimal velocity field that cools the domain on a mechanical time scale, in the sense that the integrated mean exit time is independent on molecular diffusivity in the limit of large-energy flows.

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“…Next, we show that the streamlines above enclose a large enough region to encompass most of the mass of ϕ 2 . Finally, we use the drift independent apriori estimates in [4,11] to obtain the desired lower bound on λ.…”

Section: The Lower Boundmentioning

confidence: 99%

From homogenization to averaging in cellular flows

Iyer

Komorowski

Novikov

et al. 2014

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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We consider an elliptic eigenvalue problem with a fast cellular flow of amplitude A, in a twodimensional domain with L 2 cells. For fixed A, and L → ∞, the problem homogenizes, and has been well studied. Also well studied is the limit when L is fixed, and A → ∞. In this case the solution equilibrates along stream lines.In this paper, we show that if both A → ∞ and L → ∞, then a transition between the homogenization and averaging regimes occurs at A ≈ L 4 . When A ≫ L 4 , the principal Dirichlet eigenvalue is approximately constant. On the other hand, when A ≪ L 4 , the principal eigenvalue behaves likeσ(A)/L 2 , whereσ(A) ≈ √ AI is the effective diffusion matrix. A similar transition is observed for the solution of the exit time problem. The proof in the homogenization regime involves bounds on the second correctors. Miraculously, if the slow profile is quadratic, these estimates can be obtained using drift independent L p → L ∞ estimates for elliptic equations with an incompressible drift. This provides effective sub and super-solutions for our problem.

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Exit Times of Diffusions with Incompressible Drift

Cited by 17 publications

References 12 publications

A functional approach towards eigenvalue problems associated with incompressible flow

A functional approach towards eigenvalue problems associated with incompressible flow

Optimal Heat Transfer and Optimal Exit Times

From homogenization to averaging in cellular flows

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