A functional approach towards eigenvalue problems associated with incompressible flow

Abstract

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.

Monotonicity of the principal eigenvalue for a linear time-periodic parabolic operator

“…in the stability analysis for equilibria [3,4,12,14]. Of particular interest is to understand the dependence of λ(D) on the parameters [15,16,19,20]. The present paper continues our previous studies in [17,18] on the principal eigenvalues for time-periodic parabolic operators, where the dependence of λ(D) on frequency and advection rate were investigated.…”

“…This section is devoted to the proofs of Theorems 1.1 and 1.2. The proofs are based upon some functional, which was first introduced in [8] for an elliptic eigenvalue problem.…”

“…For this simple-looking ODE eigenvalue problem, identifying the sign of λ 1 is not yet an easy task. In the recent works [5,6,16,21,23], the asymptotics of λ 1 for large α or small D in this case has been extensively studied. However, when V or/and ∂ x m depend both on the spatio-temporal variables, much less has been known for the behaviors of λ 1 ; the mathematical difficulty mainly comes from the lack of variational structure for problem (1.1).…”

Selected Topics on Reaction-Diffusion-Advection Models from Spatial Ecology

“…For the reaction-diffusion models discussed in this paper, determining the complete structure of the nodal set for Λ 1 (α, β) poses a challenging problem, both analytically and numerically [52]. To this end, some asymptotic behaviors of Λ 1 were studied in [26,27,99,139] in order to have a better understanding for this problem. Recently, the asymptotic behaviors for the principal eigenvalues of time-periodic parabolic operators were also studied, and we refer to [64,69,71,101,102,103,137].…”