The explosion problem in a flow

Berestycki, Henri; Kiselev, Alexander; Novikov, Alexei; Ryzhik, Lenya

doi:10.1007/s11854-010-0002-7

Cited by 32 publications

(42 citation statements)

References 34 publications

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“…There exists λ * (u) such that this problem has a solution for all λ λ * (u) and no solution for λ > λ * (u) (see [4,8,10] for u ≡ 0 and [1] for u ≡ 0). Surprisingly, it was shown numerically in [9] that in a long rectangle there are incompressible flows with λ * (u) < λ * (0).…”

Section: Introductionmentioning

confidence: 99%

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Exit Times of Diffusions with Incompressible Drift

Iyer¹,

Novikov²,

Ryzhik³

et al. 2010

SIAM J. Math. Anal.

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Abstract. Let Ω ⊂ R n be a bounded domain, and for x ∈ Ω let τ (x) be the expected exit time from Ω of a diffusing particle starting at x and advected by an incompressible flow u. We are interested in the question which flows maximize τ L ∞ (Ω) , that is, they are most efficient in the creation of hotspots inside Ω. Surprisingly, among all simply connected domains in two dimensions, the discs are the only ones for which the zero flow u ≡ 0 maximizes τ L ∞ (Ω) . We also show that in any dimension, among all domains with a fixed volume and all incompressible flows on them, τ L ∞ (Ω) is maximized by the zero flow on the ball.

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Section: Introductionmentioning

confidence: 99%

“…Closely related to the problem studied in the present paper is the following question. It is shown in [1] that for any p > n/2 there exists a constant C p (Ω) such that for any incompressible u tangential to ∂Ω and any f ∈ L p (Ω), the solution of…”

Section: Introductionmentioning

confidence: 99%

Exit Times of Diffusions with Incompressible Drift

Iyer¹,

Novikov²,

Ryzhik³

et al. 2010

SIAM J. Math. Anal.

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“…Now by rotating the initial profile η 0 appropriately, we obtain the linear decrease , and then choose L, A large enough so that cA −1/2 L −α/4 < c 2 4 , we see that θ 1 is forced to attain it's maximum on the inner boundary ∂B 1−2h , and the Lemma follows immediately.…”

Section: The Upper Boundmentioning

confidence: 86%

“…We remark that while Lemma 2.7 is stated for homogeneous Dirichlet boundary conditions, the proof in [4] goes through verbatim for periodic boundary conditions, provided, of course, we assume our solution is mean-zero. This justifies the application of Lemma 2.7 in this context.…”

Section: Lemma 41 Implies Thatmentioning

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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“…showed numerically in [14] that in a long thin rectangle with g(s) := e . In [3], Berestycki et. al.…”

Section: The Explosion Problemmentioning

confidence: 99%

Resolving the One-Dimensional Autonomous Flow-Free Explosion Problem

Murphy¹

2014

SIURO

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We introduce the explosion problem for semilinear elliptic and parabolic PDEs. The explosion problem is a question of existence of nonnegative solutions to −Lφ = λg(x, φ) and nonnegative global solutions to ∂tφ − Lφ = λg(x, t, φ). Under certain conditions there is a λ * > 0, called the explosion threshold, for which λ < λ * implies a solution exists, and λ > λ * implies no solution exists. In this paper we focus on the one-dimensional elliptic case. Our main result is the explicit calculation of the explosion threshold λ * for the one-dimensional autonomous flow-free problem −φ = λg(φ).

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The explosion problem in a flow

Cited by 32 publications

References 34 publications

Exit Times of Diffusions with Incompressible Drift

Exit Times of Diffusions with Incompressible Drift

From homogenization to averaging in cellular flows

Resolving the One-Dimensional Autonomous Flow-Free Explosion Problem

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