Abstract
We consider an elliptic eigenvalue problem with a fast cellular flow of amplitude A, in a two-dimensional domain with L2 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 bothA→∞ and L→∞, then a transition between the homogenization and averaging regimes occurs at A≈L4. When A≫L4, the principal Dirichlet eigenvalue is approximately constant. On the other hand, when A≪L4, the principal eigenvalue behaves like σ¯(A)/L2, where σ¯(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 Lp→L∞ estimates for elliptic equations with an incompressible drift. This provides effective sub- and super-solutions for our problem.
Highlights
Consider an advection diffusion equation of the form∂t φ + Av(x) · ∇φ − φ = 0, (1.1)where A is the non-dimensional strength of a prescribed vector field v(x)
We show that if both A → ∞ and L → ∞, a transition between the homogenization and averaging regimes occurs at A ≈ L4
The main focus of this paper is to study a transition between the two well-known regimes described above
Summary
Where A is the non-dimensional strength of a prescribed vector field v(x). Under reasonable assumptions when A → ∞, the solution φ becomes constant on the trajectories of v. Our first two results study the averaging to homogenization transition for the principal Dirichlet eigenvalue. If A L4, the stirring is strong enough to force the diffusion X to exit D almost immediately along separatrices In this case, we show that τ → 0 on separatrices, and is bounded everywhere else above by a constant independent of A and L. Since it is well known that the principal eigenvalue is bounded below by the inverse of the maximum expected exit time, the lower bound in Theorem 1.2 follows from Proposition 1.5. We believe that for other flows the transition from the averaging to the homogenization regime happens when the effective diffusivity σ (A) balances with the domain size L.
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