A functional approach towards eigenvalue problems associated with incompressible flow

We propose a certain functional which is associated with principal eigenfunctions of the elliptic operator \begin{document}$ L_{A} = -\mathrm{div}(a(x)\nabla )+A\mathbf{V}\cdot\nabla +c(x) $\end{document} and its adjoint operator for general incompressible flow \begin{document}$ \mathbf{V} $\end{document} . The functional can be applied to establish the monotonicity of the principal eigenvalue \begin{document}$ \lambda_1(A) $\end{document} , as a function of the advection amplitude \begin{document}$ A $\end{document} , for the operator \begin{document}$ L_{A} $\end{document} subject to Dirichlet, Robin and Neumann boundary conditions. This gives a new proof of a conjecture raised by Berestycki, Hamel and Nadirashvili [ 5 ]. The functional can also be used to prove the monotonicity of the normalized speed \begin{document}$ c^{*}(A)/A $\end{document} for general incompressible flow, where \begin{document}$ c^{*}(A) $\end{document} is the minimal speed of traveling fronts. This extends an earlier result of Berestycki [ 3 ] for steady shear flow.


1.
Introduction. There have been extensive studies on the reaction-diffusion equations of the form w t = div(a(x)∇w) − AV · ∇w + wf (x, w), Of particular interest is the dependence of the principal eigenvalue λ 1 (A) and the minimal speed c * (A) on the advection amplitude A. Let us focus on the case where vector field V is divergence free, i.e., divV = 0 in Ω, while the case of gradient flow V = ∇m for some m ∈ C 2 (Ω) has been investigated by Chen and Lou in [8] and [9]. The purpose of this paper is to introduce a certain functional to prove the monotonicity of λ 1 (A) and c * (A)/A with respect to A.
Set L * A := −div(a(x)∇) − AV · ∇ + c(x) as the adjoint operator of L A . By u A , v A we further denote the principal eigenfunctions corresponding to L A and L * A with appropriate boundary conditions, respectively, which are normalized by Ω u 2 A dx = Ω u A v A dx = 1. In terms of operator L A and u A , v A , we now introduce functional J A by which is defined on some cone S. Such a functional turns out to be new, which is different from the general relative entropy introduced in [30]. A direct observation from the definition of functional J A leads to J A (u A ) = λ 1 (A) and a far less obvious result (see Lemma 2.2) says that functional J A attains its maximum at the principal eigenfunction u A and its scalar multiples. This is crucial in the proof of the monotonicity of λ 1 (A) and c * (A)/A and it also allows us to explore a new min-max characterization of λ 1 (A).
Assume that c ∈ C α (Ω) and the diffusion matrix a(x) is symmetric and uniformly elliptic C 1,α (Ω) matrix field satisfying ∃ 0 < γ 1 < γ 2 , such that γ 1 |ξ| 2 ≤ ξ T a(x)ξ ≤ γ 2 |ξ| 2 , ∀x ∈ Ω, ∀ξ ∈ R N , for some constant α ∈ (0, 1). Furthermore, we always assume that the vector field V ∈ C 1 (Ω) satisfying divV = 0 in Ω, whereas an additional assumption stating that V · n = 0 on ∂Ω is assumed for the case of 0 ≤ b < 1. Under these assumptions the Krein-Rutman Theorem guarantees the existence of the principal eigenvalue λ 1 (A) and it can be easily shown that λ 1 (A) is symmetric in A. Therefore, throughout this paper we shall assume A ≥ 0. For such flow V, Berestycki et al. investigated in [5] the asymptotic behavior of λ 1 (A) as A approaches infinity, and they identified a direct link between the limit of λ 1 (A) and the first integral set of V, defined as {ϕ ∈ H 1 (Ω) : ϕ = 0, V · ∇ϕ = 0 a.e. in Ω}, 0 ≤ b < 1, {ϕ ∈ H 1 0 (Ω) : ϕ = 0, V · ∇ϕ = 0 a.e. in Ω}, b = 1. More precisely, Berestycki et al. showed in [5] that for the operator L A defined on Ω with Dirichlet (b = 1) or Neumann (b = 0) boundary conditions, λ 1 (A) stays bounded as A → +∞ if and only if I 1 = ∅ or I 0 = ∅, respectively. Furthermore, they proved that for any A ≥ 0, That is, λ 1 (A) attains its minimum at A = 0 and its maximum at A = ∞. As mentioned in [5], λ 1 (A) is a nondecreasing function of |A| if V is an incompressible gradient flow. One of the goals of this paper is to give a new proof of the following result.
Then λ 1 (A) is non-decreasing for A ≥ 0. Furthermore, Here u 0 is the principal eigenfunction of L 0 satisfying The proof of Theorem 1.1 was first given by Godoy et al. [19] via a variant of the min-max formula derived in [18] for principal eigenvalues. Our proof relies heavily on properties of functional J A defined in (2) by identifying the definition cone S as Our proof avoids the min-max formula of principal eigenvalues for non-symmetric operators and gives an explicit expression for the derivatives of λ 1 (A). Theorem 1.1 implies that the strict monotonicity of λ 1 (A) with respect to the advection amplitude A relies on u 0 , the principal eigenfunction of operator L 0 . 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].
Our next result, as a corollary of Theorem 1.1, provides the boundedness and asymptotic behavior of λ 1 (A) for Robin boundary conditions, consistent with the main result in [5] for Neumann boundary conditions. Theorem 1.2. If 0 ≤ b < 1, the limit lim A→+∞ λ 1 (A) always exists, is finite and satisfies In particular, the principal eigenvalue λ 1 (A) of (3) with 0 ≤ b < 1, is uniformly bounded.
The proof of the boundedness for λ 1 (A) in Theorem 1.2 is essentially due to Berestycki et al. [5]. Nevertheless, the existence of the limit lim A→∞ λ 1 (A) for Robin boundary conditions appears to be new.

1.2.
A new min-max characterization of λ 1 (A). The characterization of the principal eigenvalue has always been an interesting and active topic, and we refer to Donsker and Varadhan, Nussbaum and Pinchover for some earlier works [13,15,37]. Employing the maximum principle, Protter and Weinberger [38] established a classical characterization of the principal eigenvalue λ 1 (A) given by the min-max formula This characterization is valid for general elliptic operators in both bounded and unbounded domains [37,38]. As a byproduct of properties of functional J A , we have the following characterization for λ 1 (A): (3), with divV = 0 in Ω and additionally V · n = 0 on ∂Ω for 0 ≤ b < 1, can be characterized as This min-max formula may not be valid for general second elliptic operators, and it reduces to the classical Rayleigh-Ritz formula when V = 0, by treating p 2 dx as some probability measure; See Remark 2 for details. Different from formula (5), the min-max characterization in Theorem 1.3 relies on the properties of functional J A . They however may be connected via a min-max theorem in [39]. Via functional J A we observe that the min-max formula attains the extremum when p 2 = u A v A .
1.3. Monotonicity of c * (A)/A. Another application of functional J A is concerned with the minimal speed c * (A) of traveling fronts for equation (1) with Neumann boundary condition (b = 0). As in [4,6,40], we here consider the general periodic setting described in Subsection 5.1. One interesting question is the dependence of the minimal speed c * (A) on amplitude A.
From the physical point of view, the presence of incompressible flow V in equation (1) generally improves mixing [22,23,40] and is thus expected to enhance c * (A). Difficulties however may arise because of the interplay between the stream lines of general incompressible flow V; See [29,40]. Some results focused on the case where V = (α(y), 0, . . . , 0), so-called shear flow in a straight cylinder. Examples are known for which the minimal speed c * (A) is asymptotically linear with respect to A in the sense of c * (A)/A → ρ > 0 as A → +∞ [3,22,23], while c * (A)/A → 0 could happen in general incompressible flow. We refer to [44] for the precise limit as A → +∞. Furthermore, c * (A) is increasing in A, c * (A)/A is decreasing in A for shear flow [3,33]  Theorem 1.4 extends the result proved by Berestycki [3] and Nadin [33] for shear flow. For shear flow, we can write L A as a symmetric operator by some manipulations. Theorem 1.4 henceforth can be proved by the Rayleigh-Ritz formula of the principal eigenvalues. However, this technique does not appear to work for general incompressible flow V. Heinze introduced in [23] an interesting change of variables to prove the monotonicity of c * (A)/A. Such change of variables then was applied by Nadin [32] to give a new characterization of principal eigenvalue for a nonsymmetric operator. Different from Heinze's argument, our proof of Theorem 1.4 relies heavily on functional J A defined by (2), which allows us to obtain an explicit expression (32) for the derivatives of c * (A)/A. Unlike the framework of bounded domain, some modifications for the definition cone S are required for the periodic setting here.
The rest of this paper is organized as follows: In Section 2, we shall give some properties of functional J A on bounded domain. Section 3 is devoted to the proof of Theorems 1.1 and 1.2. In Section 4 we establish the new min-max characterization of λ 1 (A). We then use functional J A to prove the monotonicity of c * (A)/A in Sections 5 and 6 under the general periodic setting. Finally, the implications of our method/results and some open questions will be discussed in Section 7.
2. Functional J A on bounded domain. We shall present some properties of functional J A defined by (2) on S b in this section. Before proceeding further, we point out again that throughout this paper, u A and v A normalized by Ω u 2 A dx = Ω u A v A dx = 1 are the principal eigenfunctions corresponding to L A and L * A , respectively, with general boundary conditions. Precisely, u A > 0 in Ω satisfies (3) and v A satisfies Due to the slight difference between the definitions of functional J A in the cases of 0 ≤ b < 1 and b = 1, we divide this section into two subsections.
2.1. Neumann and Robin boundary conditions: 0 ≤ b < 1. Recalling the regularity requirements of coefficients c, V and matrix field a(x), Sobolev embedding theorem implies that u A , v A ∈ C 2,α (Ω) and u A , v A ∈ S b for 0 ≤ b < 1. We emphasize here that the constant b is confined to 0 ≤ b < 1 unless otherwise specified, and the incompressible flow V satisfies divV = 0 in Ω with V · n = 0 on ∂Ω in this subsection. We now recall the functional associated to operator L A with Neumann or Robin boundary conditions, defined on S b as in Section 1, Proof. For any ϕ ∈S b , the Fréchet derivation J A (ω) can be written as In view of ϕ ∈S b , direct calculation gives which completes the proof of Lemma 2.1.
Next we establish a crucial property of functional J A .
Proposition 2.2. For any ω ∈ S b , the following formula holds: Proof. First, for any ω ∈ S b , a simple but useful observation leads to Using equality (7), the Fréchet derivation J A (u A ) can be rewritten as To prove Proposition 2.2, some elementary but a bit tedious manipulations are needed. Together with equality (7), direct calculation yields where we have used the symmetry of matrix field a(x) and the boundary conditions of ω and u A . By straightforward calculations we have u A log ω The assertion of Proposition 2.2 thus follows.
The following result is an immediate consequence of Proposition 2.2.  [5]. It is perhaps worth pointing out that in this case, the functional J A shall be defined on S 1 and the extra assumption V · n = 0 on ∂Ω is not needed for further discussions. Hopf Boundary Lemma implies that ∇u A · n < 0 and ∇v A ·n < 0 on ∂Ω, and thus u A , v A ∈ S 1 so that J A (u A ) and J A (v A ) are well defined. Moreover, the adjoint operator of L A subject to Dirichlet boundary conditions can be written as (7).

Proof. A simple observation leads to
With the same argument as in the Neumann or Robin boundary conditions, Lemma 2.1 also holds true in this case, i.e., J A (u A )ϕ = 0 for all ϕ ∈S 1 . Based on this fact, the formula in Proposition 2.2 remains true. As the proof is similar, thus it is omitted. Therefore, the properties of functional J A listed in subsection 2.1 hold for all 0 ≤ b ≤ 1.
3. Monotonicity and boundedness of λ 1 (A). Our goal of this section is to show Theorems 1.1 and 1.2.
For the proof of part (ii), we assume that u 0 ∈ I b . Differentiate equation (3) with respect to A and denote ∂u A ∂A = u A for the sake of brevity, we obtain Multiply (9) by v A and integrate the result in Ω, together with the definition of v A we have Observe that u 0 = v 0 for A = 0. This leads to Here we used that V is divergence free together with V · n = 0 on ∂Ω for 0 ≤ b < 1 and u 0 = 0 on ∂Ω for b = 1.
. To establish this assertion, recall the definition of L A and L * A to rewrite equality (10) as A direct application of Corollary 2.3 and positive definiteness of a(x) yields Hence, λ 1 (A) = λ 1 (0). The Claim is proved. Before proceeding further to show ∂λ1 ∂A (A) > 0 for all A > 0, let us calculate ∂A 2 (0) firstly. Differentiate equation (9) with respect to A again, and applying the notation Setting A = 0 in (11) and multiplying it by u 0 and integrating the result in Ω, it follows from ∂λ1 On the other hand, multiplying equation (9) by u 0 and setting A = 0, we have which in turn implies that 1 2 We are now in a position to prove Theorem 1.1. According to the above Claim, it suffices to prove that λ 1 (A) > λ 1 (0) for every A > 0. If λ 1 (Â) = λ 1 (0) for somê so the variational argument of principal eigenvalue λ 1 (0) implies that u 0 = cu 0 for some constant c. Setting A = 0 and then substituting equality u 0 = cu 0 into equation (9), we can conclude that V · ∇u 0 ≡ 0 in Ω, which is a contradiction. This completes the proof.
We now proceed to prove Theorem 1.2.
Proof of Theorem 1.2. It suffices to establish the following result: Claim. Assume that I b = ∅. Then λ 1 (A) is uniformly bounded and The idea of the proof for the Claim comes from Theorem 2.2 in [5] and we shall sketch the proof for the sake of completeness. Note that u A > 0 inΩ by Hopf Boundary Lemma for case of 0 ≤ b < 1. Choose any function ω ∈ I b and multiply the equation of u A by ω 2 u A , then integration by parts implies that b An interesting observation, in analogy with the proof of Theorem 2.2 in [5], gives which leads to the Claim by combining equality (13) and I b = ∅. It turns out that I b = ∅ always holds for 0 ≤ b < 1, since it at least follows that c ∈ I b for any constant c. Together with the above Claim, the monotonicity of λ 1 (A) in Theorem 1.1 readily implies that the limit lim A→∞ λ 1 (A) always exists and is finite. The proof of Theorem 1.2 is complete.
Remark 1. (Necessity of the assumption V · n = 0 on ∂Ω): We now remark that the additional assumption V · n = 0 on ∂Ω is necessary for 0 ≤ b < 1, while not necessary for b = 1, corresponding to zero Dirichlet boundary condition.
• For b = 1, zero Dirichlet boundary condition implies u A = v A = 0 on ∂Ω and the adjoint operator of L A can be written as L * A = −div(a(x)∇) − AV · ∇ + c(x) without the additional assumption, whence Theorem 1.1 remains true as the properties of J A in Section 2 hold without this assumption as stated in subsection 2.2.
Consider the same example as in Remark 2.5 of [5], Here we consider the special case where b = 0 and the incompressible flow V = 1 does not satisfy the assumption V · n = 0 at x = 0, 1. Chen and Lou's result in [8] together with the facts inf p∈L 2 (Ω), Ω p 2 =1 Ω it is straightforward to derive the following min-max characterization of λ 1 (A): However, the min-max characterization in Theorem 1.3 is somewhat different. The following result is the key of the proof of Theorem 1.3: Proof of Theorem 1.3. We first choose p 2 = u A v A and apply Lemma 4.1 to deduce On the other hand, for any p ∈ L 2 (Ω) satisfying Ω p 2 = 1, it is easy to see that which implies that Hence equality (6) holds. The proof of Theorem 1.3 is now complete.

Remark 2. (Reduce to the classical Rayleigh-Ritz formula)
The classical Rayleigh-Ritz formula is actually implicity contained in the min-max formula in Theorem 1.3 if L A is self-adjoint, i.e., V = 0 or A = 0. It can be deduced from an important result in [14]. More specifically, viewing µ = p 2 (x)dx in (6) as a positive measure satisfying the mild assumption µ λ for the Borel measure λ and noting that dµ dλ = p 2 (x), Theorem 5 in [14] leads to which reduces the formula in Theorem 1.3 to the classical Rayleigh-Ritz formula.

5.
Functional J A on unbounded periodic domain. We now turn to refine functional J A defined in (2) for unbounded periodic domain by identifying the definition cone S. Different from S b in Section 2 for bounded domain, some modifications are required. To this end, we first present general periodic setting considered in [4,6,40].
Let N ≥ 1 be the space dimension and d be an integer satisfying 1 ≤ d ≤ N . Write x = (x 1 , . . . , x d ) and y = (y d+1 , . . . , y N ) so that for all z = (x, y) ∈ Ω, |y| ≤ R for some And Ω is further assumed to be periodic with respect to x variables in the following sense: there exists L = (L 1 , L 2 , . . . , L d ) such that where {e i } 1≤i≤N is the canonical basis of R N . Denote the periodicity cell of Ω by This framework includes several types of simple geometrical configurations: the whole space R N corresponding to d = N in [4,43], the infinite cylinders to d = 1 in [3,23], and the infinite slabs to 1 < d < N . For the coefficients of equation (1) with z = (x, y), we assume that the diffusion matrix a is symmetric and uniformly elliptic C 1,α (Ω) matrix field satisfying a(x, y) is L−periodic with respect to x; ∃ 0 < γ 1 < γ 2 , such that γ 1 |ξ| 2 ≤ ξ T a(x, y)ξ ≤ γ 2 |ξ| 2 , ∀(x, y) ∈ Ω, ∀ξ ∈ R N . Also, the vector field V ∈ C 1 (Ω) is assumed to satisfy which means the underlying flow V is tangent to ∂Ω and the first d components have been normalized as in [3,4,6,22,40]. Finally, the reaction wf (z, w) in (1) is the KPP nonlinearity satisfying the conventional condition described, for example, in (1.5) of [6] or (2.4) of [40].
Under the above framework, of particular interest is the minimal speed of the traveling fronts. For the case of A = 0, i.e., in the presence of incompressible flow V, Berestycki et al. [4,6] established a variational formula for the minimal speed c * (A) in any direction e ∈ R d : To proceed further, set u A = e −λe·x ω A . Then u A > 0 satisfies Similarly, let v A be a positive eigenfunction associated with the eigenvalue µ 1 (A, λ) of the adjoint problem of (17), namely, We further normalize u A and v A without loss of generality such that C u 2 A = C u A v A = 1. We then identify S = S 0 ∩ (S λ per+ ∪ S λ per− ), i.e., define functional J A on periodicity cell C by

Functional J
We show a crucial property of J A , which is different from Proposition 2.2.
Proposition 5.1. For all ω ∈ S λ per− , the following formula holds: Proof. As in (7), we begin with rewriting functional J A as Here we used the assumption ω ∈ S λ per+ ∪ S λ per− to remove boundary integrals by noting the periodicity of u A v A (a∇ log ω).
By (21), we may write the Fréchet derivation J A (u A ) in the form of for all ϕ ∈ S λ per+ . Following the same arguments in Lemma 2.1, we can conclude from the definition (19) To obtain formula (20), using (21) and by direct calculation we have where we have used v A ∈ S λ per− and u A ∈ S λ per+ , and the symmetry of matrix field a. We now turn to verify formula (20). Choose where we used (23) to remove J A (u A )ϕ, and G is given by To assert formula (20), it is now desirable to show Claim. G = − ∂µ1 ∂λ . To establish our assertion, multiplying (17) by v A e · x, we derive by virtue of V · n = 0 on ∂Ω that Again integrating by parts and inferring from the definition (18) which leads directly to To proceed further, we now turn to calculate ∂µ1 ∂λ . Differentiate (17) with respect to λ and denote ∂u A ∂λ = u A for the sake of brevity, we obtain We may multiply this equation by v A to derive that 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.
6. Monotonicity of c * (A)/A. Our goal of this section is to prove Theorem 1.4. Before proceeding further, we require some necessary properties of principal eigenvalue µ 1 (A, λ) to verify the differentiability of c * (A).
6.1. Concavity of µ 1 (A, λ) with respect to λ. This subsection is devoted to describing some properties related to the convexity of µ 1 (A, λ) as a function of λ.
The concavity of the map λ → µ 1 (A, λ) has been already proved in [4], but we focus here on the strict concavity. We further prove ∂ 2 µ1 ∂λ 2 (A, λ) < 0, which appears to be new and it is a key to derive the differentiability of c * (A) with respect to A. Lemma 6.1. The principal eigenvalue µ 1 (A, λ) is strictly convave in λ > 0.
Proof. As already noted, the proof of the concavity of µ 1 (A, λ) has been carried out by the general lines of [4]. Here we just outline it for completeness and because it will lead us to the strict concavity.
We are now in a position to establish a stronger result that is the main objective of this subsection.
Again setting λ = λ 0 and regarding ∇ϕ A = ∇ϕ A = 0 at λ 0 , we arrive at This actually implies ∇ϕ A = 0 and thus ∂ 3 µ1 ∂λ 3 (A, λ 0 ) = 0. To prove this assertion, there is no loss of generality in assuming ∂ 3 µ1 ∂λ 3 (A, λ 0 ) ≥ 0. The boundary condition of ϕ A then asserts that ϕ A is a constant by the strong maximum principle and Hopf boundary lemma again.
Repeating the above procedure, we may conclude that ∇ϕ (n) A = 0 and ∂ n µ 1 ∂λ n (A, λ 0 ) = 0 for all n ≥ 2. Notice the well-known fact that µ 1 (A, λ) is analytic. See, e.g. Proposition 2.20 in [7]. The Taylor expansion of µ 1 (A, λ) at λ 0 reads which contradicts to the strict concavity in Lemma 6.1. Proof. Differentiating (17) with respect to A and writing ∂u A ∂A = u A to obtain a∇u A · n = 0 on ∂Ω and Multiplying both sides of the above equation by v A and integrating by parts gives Recalling the definitions of L A , L * A , and functional J A , we further derive by appealing to Proposition 5.1 with ω = v A ∈ S λ per− particularly. As already noted in Subsection 5.1, there exists a unique point λ * (A) ∈ (0, +∞) satisfying (16), whence the map A → λ * (A) is well defined. At λ * (A), by the argument of extreme point we have which establishes, along with (16), the following On the other hand, consider function F (A, λ) = λ ∂µ1 ∂λ (A, λ) − µ 1 (A, λ). Since F (A, λ * (A)) = 0 and ∂F (A,λ) ∂λ = ∂ 2 µ1 ∂λ 2 (A, λ) = 0 by Proposition 6.2, the implicit function theorem implies λ * (A), and hence c * (A), are differentiable with respect to A. In view of equalities (16), (30), and (31), direct calculation leads to This yields the desired result To complete the proof of Theorem 1.4, it suffices to show that d(c * (A)/A) dA < 0. Suppose not, there exists A 0 > 0 so that u A0 = cv A0 for some c > 0. The normalization that C u 2 A = 1 and C u A v A = 1 imply readily that u A = v A . Then one can apply the periodicity conditions of u A and v A to find λ * (A 0 ) = 0, which is a contradiction. Therefore, the function c * (A)/A is strictly decreasing as A > 0 increases, which proves Theorem 1.4.
We now consider problems (3) with gradient flow V 1 = ∇m for some m ∈ C 2 (Ω), where the principal eigenvalue λ 1 (A), in analogy with equality (1.2) in [5], can be written as Another open question is to determine the limit value of λ 1 (A) for incompressible flow V with Robin boundary conditions as A → +∞, though the existence of the limit has been shown in Theorem 1.2. The results for Dirichlet and Neumann boundary conditions in [5] show that the limit of λ 1 (A) can be determined by the variational principle (4). In view of Theorem 1.2, it seems plausible to conjecture that for 0 ≤ b < 1, which would reduce to the results in [5] for the case b = 0. On the other hand, the limit value of λ 1 (A) with the gradient flow V 1 = ∇m has been established by Chen and Lou [8] for Neumann boundary conditions, which can be stated as There are a substantial body of literatures concerning the asymptotic behavior of the principal eigenvalue of elliptic operators for small diffusion rates; See [9,11,12,16,42]. For the principal eigenvalue of operator L D = −D∆+V·∇+c(x), Chen and Lou [9] investigated its asymptotic behavior as D → 0 when V is a gradient flow. Much less seems to be known when V is a general incompressible flow; See [2,41].