Sharp Asymptotics for KPP Pulsating Front Speed-up and Diffusion Enhancement by Flows

TL;DR

This paper investigates how periodic incompressible flows can accelerate KPP front speeds and enhance effective diffusivity, providing asymptotic limits for these quantities as flow strength increases.

Contribution

It establishes the existence and asymptotic behavior of minimal front speed and effective diffusivity in the presence of strong periodic flows.

Findings

Limit of minimal front speed divided by flow strength as A→∞

Limit of effective diffusivity divided by flow strength squared as A→∞

Quantitative characterization of flow-induced front speed-up and diffusion enhancement

Abstract

We study KPP pulsating front speed-up and effective diffusivity enhancement by general periodic incompressible flows. We prove the existence of and determine the limits $c^*(A)/A$ and $D(A)/A^2$ as $A\to\infty$, where $c^*(A)$ is the minimal front speed and $D(A)$ the effective diffusivity.

Equations

$$T_t+Au\cdot \nabla T=\Delta T+f(T)$$

$$f\in C^{1,\epsilon }([0,1]),$$

$$\text{f(0)=f(1)=0 and f is non-increasing on (1−ε,1) for some ε>0},$$

$$0<f(s)\le s{f}^{\prime }(0)\text{ for s∈(0,1)}.$$

$$u\in C^{1,\epsilon }({\mathbb{T}}^n),\nabla \cdot u\equiv 0,{\int }_{{\mathbb{T}}^n}udx=0.$$

$$\underset{s\to -\infty }{\lim }U(s,x)=1,$$

$$\underset{s\to +\infty }{\lim }U(s,x)=0,$$

$$u(x)=(\alpha (x^{\prime }),0,\dots ,0)(x^{\prime }=(x_2,\dots ,x_n))\text{and}e=(1,0,\dots ,0).$$

$${\mathcal{I}}\equiv\big{\{}w\in H^{1}({\mathbb{T}}^{n})\,\,\,\big{|}\,\,\,\operatorname{Im}w=0\text{ and }u\cdot\nabla w=0\big{\}}$$

$$\underset{A\to \infty }{\lim }\frac{c_e^{\ast }(A)}{A}=\underset{\begin{array}{c}w\in \mathcal{I}\\ \Vert \nabla w{\Vert }_2^2\le f^{\prime }(0)\Vert w{\Vert }_2^2\end{array}}{\sup }\frac{{\int }_{{\mathbb{T}}^n}(u\cdot e)w^{2}dx}{\Vert w{\Vert }_2^2}.$$

$$\underset{f^{\prime }(0)\to 0}{\lim }\underset{A\to \infty }{\lim }\frac{c_e^{\ast }(A)}{2\sqrt{f^{\prime }(0)}A}=\underset{w\in \mathcal{I}}{\sup }\frac{{\int }_{{\mathbb{T}}^n}(u\cdot e)wdx}{\Vert \nabla w{\Vert }_2},$$

$$\underset{f^{\prime }(0)\to \infty }{\lim }\underset{A\to \infty }{\lim }\frac{c_e^{\ast }(A)}{A}=\underset{w\in \mathcal{I}}{\sup }\frac{{\int }_{{\mathbb{T}}^n}(u\cdot e)w^{2}dx}{\Vert w{\Vert }_2^2}\le \underset{x\in {\mathbb{T}}^n}{\max }\{u(x)\cdot e\}.$$

$${\Phi }_t+Au\cdot \nabla \Phi =\Delta \Phi .$$

$${\Psi }_t=\sum _{i,j=1}^n{\sigma }_{ij}(A)\frac{{\partial }^2\Psi }{\partial x_i\partial x_j}.$$

$$-\Delta {\chi }_{e,A}+Au\cdot \nabla {\chi }_{e,A}=Au\cdot e$$

$$e\cdot \sigma (A)e^{\prime }={\int }_{{\mathbb{T}}^n}(\nabla {\chi }_{e,A}+e)\cdot (\nabla {\chi }_{e^{\prime },A}+e^{\prime })dx=e\cdot e^{\prime }+{\int }_{{\mathbb{T}}^n}\nabla {\chi }_{e,A}\cdot \nabla {\chi }_{e^{\prime },A}dx.$$

$$D_e(A)\equiv e\cdot \sigma (A)e=1+\Vert \nabla {\chi }_{e,A}{\Vert }_2^2.$$

$$\lim_{A\to\infty}\frac{D_{e}(A)}{A^{2}}=\sup_{w\in{\mathcal{I}}}\bigg{(}\frac{\int_{{\mathbb{T}}^{n}}(u\cdot e)w\,dx}{\|\nabla w\|_{2}}\bigg{)}^{2}.$$

$$-\Delta {\psi }_A+Au\cdot \nabla {\psi }_A=u\cdot e$$

$$\Vert \nabla {\psi }_A{\Vert }_2^2={\int }_{{\mathbb{T}}^n}(u\cdot e){\psi }_{A}dx\le \Vert u\cdot e{\Vert }_2\Vert {\psi }_A{\Vert }_2.$$

$$\Vert w{\Vert }_2\le C\Vert \nabla w{\Vert }_2$$

$$\Vert {\psi }_A{\Vert }_{H^1}\le C\Vert u\cdot e{\Vert }_2.$$

$$\frac{D_e(A)}{A^2}=\frac{1}{A^2}+\Vert \nabla {\psi }_A{\Vert }_2^2=\frac{1}{A^2}+{\int }_{{\mathbb{T}}^n}(u\cdot e){\psi }_{A}dx.$$

$$u\cdot \nabla w_0=0$$

$$\Vert \nabla w_0{\Vert }_2^2\le \underset{k\to \infty }{\mathrm{limsup}}\Vert \nabla {\psi }_{A_k}{\Vert }_2^2={\int }_{{\mathbb{T}}^n}(u\cdot e)w_{0}dx={\int }_{{\mathbb{T}}^n}\nabla {\psi }_A\nabla w_{0}dx\le \Vert \nabla {\psi }_A{\Vert }_2\Vert \nabla w_0{\Vert }_2$$

$$\Vert \nabla w_0{\Vert }_2\le \Vert \nabla {\psi }_A{\Vert }_2$$

$$\underset{k\to \infty }{\mathrm{limsup}}\Vert \nabla {\psi }_{A_k}{\Vert }_2\le \Vert \nabla w_0{\Vert }_2.$$

$$\underset{k\to \infty }{\lim }\Vert \nabla {\psi }_{A_k}{\Vert }_2=\Vert \nabla w_0{\Vert }_2,$$

$${\psi }_{A_k}\to w_0\text{in H1(Tn)}.$$

$$\lim_{k\to\infty}\frac{D_{e}(A_{k})}{A_{k}^{2}}=\|\nabla w_{0}\|_{2}^{2}=\int_{{\mathbb{T}}^{n}}(u\cdot e)w_{0}\,dx=\bigg{(}\frac{\int_{{\mathbb{T}}^{n}}(u\cdot e)w_{0}\,dx}{\|\nabla w_{0}\|_{2}}\bigg{)}^{2}.$$

$$\bigg{|}\int_{{\mathbb{T}}^{n}}(u\cdot e)w\,dx\bigg{|}=\bigg{|}\lim_{k\to\infty}\int_{{\mathbb{T}}^{n}}\nabla\psi_{A_{k}}\nabla w\,dx\bigg{|}=\bigg{|}\int_{{\mathbb{T}}^{n}}\nabla w_{0}\nabla w\,dx\bigg{|}\leq\|\nabla w_{0}\|_{2}\|\nabla w\|_{2}.$$

$$\bigg{(}\frac{\int_{{\mathbb{T}}^{n}}(u\cdot e)w\,dx}{\|\nabla w\|_{2}}\bigg{)}^{2}\leq\|\nabla w_{0}\|_{2}^{2}=\lim_{k\to\infty}\frac{D_{e}(A_{k})}{A_{k}^{2}},$$

Full text

Department of Mathematics, University of Chicago, Chicago, IL 60637; email: zlatos@math.uchicago.edu

The author acknowledges partial support by the NSF through the grant DMS-0632442

1. Introduction

We study reaction-diffusion fronts in the presence of strong incompressible flows. We consider the PDE

[TABLE]

on ${\mathbb{R}}^{n}$, with $T(t,x)\in[0,1]$ the normalized temperature of a premixed combustible gas. The non-linear reaction rate $f$ is of Kolmogorov-Petrovskii-Piskunov (KPP) type [11]:

[TABLE]

The 1-periodic flow $u:{\mathbb{T}}^{n}\to{\mathbb{R}}^{n}$ satisfies

[TABLE]

That is, $u$ is incompressible and mean-zero.

The number $A\in{\mathbb{R}}$ is the flow amplitude. We will consider the case of strong flows (i.e., large $A$) and their influence on the speed of propagation of pulsating fronts for (1.1). This problem has recently seen increased activity and has been addressed by various authors — see, e.g., [1, 2, 5, 7, 9, 10, 13, 14].

A pulsating front in the direction $e\in{\mathbb{R}}^{n}$, $|e|=1$, is a solution of (1.1) of the form $T(t,x)=U(x\cdot e-ct,x)$, with $c$ the front speed, and $U$ 1-periodic in $x$ and such that

[TABLE]

uniformly in $x$. It is well known [4] that in the KPP case there is $c_{e}^{*}(A)$, called the minimal pulsating front speed, such that pulsating fronts exist precisely for $c\geq c_{e}^{*}(A)$ (we suppress the $u$ and $f$ dependence in our notation). We note that $c_{e}^{*}(A)$ also determines the propagation speed of solutions to the Cauchy problem with general compactly supported initial data [4, 15].

Mixing by flows (coupled to diffusion) typically increases the speed of pulsating fronts for (1.1). The minimal front speed $c_{e}^{*}(A)$ can grow at most linearly with $A$ [5] and does so for shear (unidirectional) flows [1, 2, 7, 9]

[TABLE]

The same is true for so-called percolating flows which possess infinite channels [7], contrasting with the case of cellular flows when, at least in two dimensions, $c_{e}^{*}(A)=O(A^{1/4})$ [1, 7, 9, 13] (see also [14] for a three-dimensional example).

We are interested here in all flows which maximally (i.e., linearly) enhance the minimal front speed for (1.1) and our goal is to determine the asymptotic rate of this front speed-up — to prove the existence and evaluate the limit of $c_{e}^{*}(A)/A$ as $A\to\infty$. For shear flows, this limit has been known to exist [2] and has been determined in [9], but both problems have been open in general.

We thus consider general periodic flows (1.3) and let

[TABLE]

be the set of real-valued first integrals of the flow $u$. We then have the following main result.

Theorem 1.1**.**

If $u$ and $f$ satisfy (1.2) and (1.3) and $|e|=1$, then

[TABLE]

*In particular, the limit exists. Moreover, *

[TABLE]

Remarks. 1. Inequality “$\geq$” in (1.6) (with $\liminf_{A\to\infty}$ in place of $\lim_{A\to\infty}$) has been proved in [5], and [9] showed equality in the case of shear flows (1.4).

2. (1.8) already appeared in [5], with either $\liminf_{A\to\infty}$ or $\limsup_{A\to\infty}$ in place of $\lim_{A\to\infty}$. For shear flows (1.4) the inequality becomes an equality [9] due to (1.6) and continuity of $u$.

3. Notice that (1.6) (for any $f^{\prime}(0)$) is positive precisely when there exists $w\in{\mathcal{I}}$ such that $\int_{{\mathbb{T}}^{n}}(u\cdot e)w\,dx\neq 0$ (take $1\pm\varepsilon w$ in (1.6)). This is also the condition for positivity of (1.7) and (1.12) below.

4. The result extends directly to the more general case of $x$-dependent and 1-periodic reaction and second-order term (see Theorem 3.2). We perform the proof in the simpler setting above for the sake of transparency.

It has been shown in [13, 14] that, at least in two dimensions, there is a close relationship between the minimal front speeds for (1.1) and the effective diffusivity in the homogenization theory for the related advection-diffusion problem

[TABLE]

As is well known, the long-time behavior of solutions to (1.9) is governed by the effective diffusion equation

[TABLE]

Here $\sigma(A)$ is a constant effective diffusivity matrix. If $e\in{\mathbb{R}}^{n}$ and we let $\chi_{e,A}$ be the mean-zero solution of

[TABLE]

on ${\mathbb{T}}^{n}$, then $\sigma(A)$ is given by

[TABLE]

The effective diffusivity for (1.9) in the direction $e\in{\mathbb{R}}^{n}$, $|e|=1$, is now

[TABLE]

Again, mixing by flows enhances the effective diffusivity. It is easy to show that $D_{e}(A)$ can grow at most quadratically with $A$, and flows that achieve this are said to maximally enhance diffusion (see [6, 8, 12] and references therein). It turns out that our method applies to the problem of determining the asymptotic rate of this enhancement as well, and we find the limit $D_{e}(A)/A^{2}$ as $A\to\infty$ for general periodic flows. To the best of the author’s knowledge, existence of this limit has not been known before.

Theorem 1.2**.**

If $u$ satisfies (1.3) and $|e|=1$, then

[TABLE]

In particular, the limit exists. Moreover, there is $w_{0}\in{\mathcal{I}}$ which is a maximizer of (1.12) and $\chi_{e,A}/A\to w_{0}$ in $H^{1}({\mathbb{T}}^{n})$.

Remarks. 1. It follows that the left hand side of (1.12) is the square of the left hand side of (1.7). This has been established in two dimensions by Ryzhik and the author [14], even without the $A\to\infty$ limit (see also [13]).

2. We show that if (1.12) is positive, then the maximizers are precisely $w=aw_{0}+b$ with $a,b\in{\mathbb{R}}$, $a\neq 0$.

3. If one considers the small diffusion problem $\phi_{t}=\varepsilon\Delta\phi+u\cdot\nabla\phi$ instead of (1.9), then the corresponding effective diffusivity satisfies $\tilde{D}_{e}(\varepsilon)=\varepsilon D_{e}(\varepsilon^{-1})$. Hence the limit $\lim_{\varepsilon\to 0}\varepsilon\tilde{D}_{e}(\varepsilon)$ also equals (1.12).

4. Again, there is a straightforward extension to the case of $x$-dependent second order term and even non-mean-zero flows (see Theorem 2.1).

We prove Theorem 1.2 in Section 2 and Theorem 1.1 in Section 3. The generalizations to the case of $x$-dependent second-order and reaction terms are Theorems 2.1 and 3.2 below.

2. Effective Diffusivity Enhancement

Proof of Theorem 1.2.

Let $\psi_{A}\equiv\chi_{e,A}/A$, so that

[TABLE]

Multiplying this by $\psi_{A}$ and integrating over ${\mathbb{T}}^{n}$ we obtain using incompressibility of the flow,

[TABLE]

Poincaré inequality

[TABLE]

for some $C<\infty$ and any mean-zero $w$ then yields

[TABLE]

It also follows from (2.2) that

[TABLE]

Since $\|\psi_{A}\|_{H^{1}}$ is uniformly bounded, there is a sequence $A_{k}\to\infty$ such that $\psi_{A_{k}}$ converges to some $w_{0}\in H^{1}({\mathbb{T}}^{n})$, weakly in $H^{1}({\mathbb{T}}^{n})$ and strongly in $L^{2}({\mathbb{T}}^{n})$. Then $\Delta\psi_{A_{k}}\to\Delta w_{0}$ and $\nabla\psi_{A_{k}}\to\nabla w_{0}$ in the sense of distributions and (2.1) divided by $A_{k}$ implies

[TABLE]

in the sense of distributions. Since $w_{0}\in H^{1}({\mathbb{T}}^{n})$, this equality holds almost everywhere and $w_{0}\in{\mathcal{I}}$. We also have

[TABLE]

where we used (2.2) in the second step, and (2.1) multiplied by $w_{0}$ and integrated over ${\mathbb{T}}^{n}$ (together with (2.6)) in the third step. Thus

[TABLE]

as well as

[TABLE]

These give

[TABLE]

which turns the weak $H^{1}$-convergence into a strong one:

[TABLE]

Let us assume $w_{0}\not\equiv 0$. Then $\nabla w_{0}\not\equiv 0$ because each $\psi_{A}$ is mean-zero. From (2.5) and (2.8),

[TABLE]

Pick an arbitrary non-constant $w\in{\mathcal{I}}$. If we multiply (2.1) by $w$ and integrate, we obtain

[TABLE]

Hence

[TABLE]

with equality precisely when $\nabla w$ is a multiple of $\nabla w_{0}$ (and so $w=aw_{0}+b$). This also means that $w_{0}$ is a maximizer for (1.12).

If now $B_{k}\to\infty$ is any sequence, then as above we can find a subsequence (which we again call $B_{k}$) such that $\psi_{B_{k}}\to w_{1}\in{\mathcal{I}}$. But then $w_{1}$ must also maximize (1.12), thus $w_{1}=aw_{0}+b$. Moreover, $b=0$ because $\psi_{A}$ are mean-zero, and (2.9) with $B_{k}$ in place of $A_{k}$ forces $a=1$. Hence $\psi_{A}\to w_{0}$ in $H^{1}({\mathbb{T}}^{n})$ and (1.12) follows.

Finally, if $w_{0}\equiv 0$ is the only limit point of $\psi_{A}$, then $\psi_{A}\to 0$ in $H^{1}({\mathbb{T}}^{n})$, and (1.12) follows from (2.5) and (2.10). ∎

Notice that (2.5), (2.7), and (2.9) show that $D_{e}(A)\geq 1+\delta A^{2}$, where $\delta$ is the limit in (1.12).

We also note that in the special case of shear flows $u(x)=(\alpha(x^{\prime}),0,\dots,0)$ equation (1.10) becomes

[TABLE]

with $\chi_{e,A}(x)=\chi_{e,A}(x^{\prime})$. Hence $\chi_{e,A}=Ae_{1}\nabla_{x^{\prime}}(-\Delta_{x^{\prime}})^{-1}\alpha$ and the limit in (1.12) equals $|e_{1}|\|\nabla_{x^{\prime}}(-\Delta_{x^{\prime}})^{-1}\alpha\|_{2}^{2}$. This can be found, e.g., in [8, Lemma 7.3].

As mentioned above, the result easily extends to the case of $x$-dependent second order term and a non-mean-zero flow. We consider

[TABLE]

instead of (1.9) with 1-periodic and real symmetric uniformly elliptic matrix $a$ and 1-periodic flow $u$ such that

[TABLE]

Then (1.10) and (1.11) are replaced by

[TABLE]

with $|||w|||_{2}^{2}\equiv\int_{{\mathbb{T}}^{n}}\nabla w\cdot(a\nabla w)\,dx$. If we define

[TABLE]

then we have

Theorem 2.1**.**

If $a$ and $u$ satisfy (2.12) and $|e|=1$, then

[TABLE]

In particular, the limit exists. Moreover, there is $w_{0}\in{\mathcal{I}}_{0}$ which is a maximizer of (2.13) and $\chi_{e,A}/A\to w_{0}$ in $H^{1}({\mathbb{T}}^{n})$.

3. KPP Front Speed-up

In this section we prove Theorem 1.1. We start with an auxiliary lemma. Let us define

[TABLE]

Note that $\kappa_{e}(\lambda)$ must be convex as it is a supremum of linear functions. Also, $\kappa_{e}(\lambda)\geq 0$ because $w\equiv 1\in{\mathcal{I}}$.

Lemma 3.1**.**

Assume the setting of Theorem 1.1. Then for each $\lambda>0$, the supremum in (3.1) is attained, the maximizer is unique up to multiplication, and

[TABLE]

Proof.

It has been shown in [4] that the minimal front speed $c_{e}^{*}(A)$ can be computed using the variational principle

[TABLE]

Here $\kappa(\lambda;A)$ is the unique eigenvalue of the problem

[TABLE]

on ${\mathbb{T}}^{n}$, with a unique normalized eigenfunction $\varphi_{A}(x;\lambda)$. Moreover, the function

[TABLE]

is monotonically increasing and convex in $\lambda\geq 0$, with $\mu(0;A)=0$ (see [3, 13]).

We now rewrite (3.3) and (3.4) as

[TABLE]

and

[TABLE]

We multiply (3.6) by $\varphi_{A}^{-1}$ and integrate to obtain (using incompressibility of $u$)

[TABLE]

Similarly, multiplication by $\varphi_{A}$ yields

[TABLE]

since $\|\varphi_{A}\|_{2}=1$. This again means that there is a sequence $A_{k}\to\infty$ such that $\varphi_{A_{k}}$ converges to some $w_{0}\in H^{1}({\mathbb{T}}^{n})$, weakly in $H^{1}({\mathbb{T}}^{n})$ and strongly in $L^{2}({\mathbb{T}}^{n})$. The convergence $\Delta\varphi_{A_{k}}\to\Delta w_{0}$ and $\nabla\varphi_{A_{k}}\to\nabla w_{0}$ in the sense of distributions, boundedness of $\kappa(\lambda/A;A)$ in $A$, and (3.6) divided by $A$ then imply (2.6) and so $w_{0}\in{\mathcal{I}}$ (note that $\|w_{0}\|_{2}=\|\varphi_{A_{k}}\|_{2}=1$).

Now we multiply (3.6) by $w_{0}$ and integrate to obtain (with $o(1)=o(k^{0})$ and using (3.8))

[TABLE]

Once again it follows that

[TABLE]

and so as in Section 2,

[TABLE]

(3.8) then yields

[TABLE]

Let $w\in{\mathcal{I}}\cap L^{\infty}({\mathbb{T}}^{n})$, multiply (3.6) for $A=A_{k}$ by $w^{2}/\varphi_{A_{k}}$ and integrate to obtain (using that $\nabla\varphi_{A}/\varphi_{A}=\nabla\ln\varphi_{A}$ are uniformly bounded in $L^{2}({\mathbb{T}}^{n})$ by (3.7) and (3.8))

[TABLE]

Since each $w\in{\mathcal{I}}$ is the $H^{1}$-limit of $w_{N}(x)\equiv\operatorname*{sgn}(w(x))\min\{|w(x)|,N\}\in{\mathcal{I}}\cap L^{\infty}({\mathbb{T}}^{n})$, this inequality extends to all $w\in{\mathcal{I}}$. Hence $\kappa_{0}=\kappa_{e}(\lambda)$ from (3.1), and $w_{0}$ is a maximizer for (3.1) (because $\|w_{0}\|_{2}=1$). Moreover, if $B_{k}\to\infty$ is any sequence with

[TABLE]

then repeating the above argument we find that there must be a subsequence (which we again call $B_{k}$) such that $\varphi_{B_{k}}\to w_{1}\in{\mathcal{I}}$ in $H^{1}({\mathbb{T}}^{n})$, $\|w_{1}\|_{2}=1$. But then as before,

[TABLE]

for any $w\in{\mathcal{I}}$. Taking $w=w_{0}$ we obtain $\kappa_{1}=\kappa_{e}(\lambda)$, and so

[TABLE]

The function $\kappa_{e}(\lambda)$ is convex, monotonically increasing, and non-negative, as it is the pointwise limit of functions $\mu(\lambda/A;A)=(\lambda/A)^{2}+\kappa(\lambda/A;A)$ which have the same properties. This also implies that the convergence in (3.11) is uniform on each bounded interval of $\lambda$. We then have

[TABLE]

($\leq$ is immediate, whereas $\geq$ uses convexity of $\mu(\lambda/A;A)$ once more). This proves (3.2).

We are left with showing that any maximizer of (3.1) is a multiple of $w_{0}$. Denote $\varphi_{k}\equiv\varphi_{A_{k}}$ and notice that (3.9) shows that (after passing to a subsequence — we will repeat this without mentioning it below), $\nabla\varphi_{k}(x)\to\nabla w_{0}(x)$ and $\varphi_{k}(x)\to w_{0}(x)$ for a.e. $x$. Next (3.7) and (2.3) imply that if $c_{k}$ is the average of $\ln\varphi_{k}$, then $\ln\varphi_{k}-c_{k}\to\omega$ strongly in $L^{2}$ and weakly in $H^{1}$. But then $\ln\varphi_{k}(x)-c_{k}\to\omega(x)$ for a.e. $x$. Since $\ln\varphi_{k}(x)\to\ln w_{0}(x)$ for a.e. $x$, it follows that $c_{k}\to c$ and $\omega=\ln w_{0}-c$. We thus obtain $\ln w_{0}\in H^{1}$ which means $w_{0}(x)\neq 0$ for a.e. $x$, and so for a.e. $x$,

[TABLE]

Let now $w\not\equiv 0$ be a maximizer of (3.1) and let us first assume $w\geq 0$ almost everywhere. Then (3.10) for $w_{N}$ and $w_{N}\to w$ in $H^{1}$ show

[TABLE]

But then (3.12) and pointwise convergence of $w_{N}$ and $\nabla w_{N}$ to $w$ and $\nabla w$, respectively, give for a.e. $x$,

[TABLE]

We now let $w_{\varepsilon}(x)\equiv\max\{w(x),\varepsilon\}$ so that $\ln w_{\varepsilon}\in H^{1}$ and

[TABLE]

This and $\ln w_{0}\in H^{1}$ means that $\|\nabla\ln w_{\varepsilon}\|_{2}$ is bounded, and again we must have $\ln w_{\varepsilon_{k}}-c_{\varepsilon_{k}}\to\omega$ strongly in $L^{2}$, weakly in $H^{1}$, and pointwise almost everywhere. But $\ln w_{\varepsilon_{k}}(x)\to\ln w(x)$, so again $c_{\varepsilon_{k}}\to c$ and $\ln w\in H^{1}$. Hence $w(x)>0$ for a.e. $x$, and so $\nabla\ln w(x)=\nabla\ln w_{0}(x)$ for a.e. $x$. This means $\ln w-\ln w_{0}$ is constant, that is, $w$ is a multiple of $w_{0}$.

If $w$ is an arbitrary maximizer of (3.1), then both $w_{\pm}(x)\equiv\max\{\pm w(x),0\}\in{\mathcal{I}}$ must be maximizers of (3.1) (or $\equiv 0$). But then $w_{\pm}(x)>0$ for a.e. $x$, meaning that one of them is zero while the other is a multiple of $w_{0}$. ∎

Proof of Theorem 1.1.

Inequality “$\geq$” in (1.6) is immediate from (3.1) and (3.2). To prove the opposite inequality it is sufficient to find $\lambda$ such that the unique normalized non-negative maximizer $w_{0}(\lambda)$ of (3.1) satisfies $\gamma(\lambda)\equiv\|\nabla w_{0}(\lambda)\|_{2}^{2}=f^{\prime}(0)$.

To this end notice that if $\lambda=0$, then $w_{0}(\lambda)\equiv 1$ and so $\gamma(0)=0$. Also, $\gamma$ must be continuous. Indeed — let $\lambda_{k}\to\lambda_{\infty}<\infty$ and denote $w_{k}\equiv w_{0}(\lambda_{k})$. Then (3.8) and (3.9) imply that $w_{k}$ are uniformly bounded in $H^{1}$. Thus a subsequence (again called $w_{k}$) converges strongly in $L^{2}$ and weakly in $H^{1}$ to some $\omega$. Obviously $\omega\in{\mathcal{I}}$ as well as $\lambda_{k}\int(u\cdot e)w_{k}^{2}\,dx\to\lambda_{\infty}\int(u\cdot e)\omega^{2}\,dx$ and $\|\nabla\omega\|_{2}\leq\liminf\|\nabla w_{k}\|_{2}$. But $\|\nabla\omega\|_{2}<\liminf\|\nabla w_{k}\|_{2}$ is impossible (otherwise $w_{k}$ would not maximize (3.1) for large $k$) and so for a subsequence,

[TABLE]

Since $\kappa_{e}$ is continuous, this means $\omega=w_{\infty}$. We have thus proved that every sequence $\lambda_{k}\to\lambda_{\infty}$ has a subsequence with $\gamma(\lambda_{k_{j}})\to\gamma(\lambda_{\infty})$, that is, $\gamma$ is continuous with $\gamma(0)=0$.

Let now $\Gamma\equiv\sup_{\lambda\geq 0}\gamma(\lambda)$. If $f^{\prime}(0)\in(0,\Gamma)$, then there is $\lambda>0$ with $\gamma(\lambda)=f^{\prime}(0)$ and (1.6) is proved. If, on the other hand, $\Gamma<\infty$ and $f^{\prime}(0)\geq\Gamma$, then (3.2) is bounded above by

[TABLE]

which does not exceed the right hand side of (1.6) due to $\|\nabla w_{0}(\lambda)\|_{2}^{2}\leq\Gamma\leq f^{\prime}(0)\|w_{0}(\lambda)\|_{2}^{2}$.

Since (1.8) is immediate from (1.6), we are left with proving (1.7). Let us consider any $w\in{\mathcal{I}}$ with $\|w\|_{2}=1$ and $\|\nabla w\|_{2}^{2}\leq f^{\prime}(0)$. Let $\bar{w}\equiv\int_{{\mathbb{T}}^{n}}w\,dx\in[-1,1]$ and $\omega\equiv w-\bar{w}$. Then $\|\omega\|_{2}^{2}\leq C\|\nabla\omega\|_{2}^{2}\leq Cf^{\prime}(0)$, and so

[TABLE]

as $f^{\prime}(0)\to 0$. Hence $\bar{w}=1+O(\sqrt{f^{\prime}(0)})$ and we have

[TABLE]

with equality when $\|\nabla\omega\|_{2}^{2}=f^{\prime}(0)$. Picking first $w$ that maximizes (1.6) and then $\omega$ that maximizes (1.7) with $\|\nabla\omega\|_{2}^{2}=f^{\prime}(0)$ (and adjusting $\bar{w}$ accordingly) finishes the proof. ∎

Note that in the case of shear flows (1.4) equation (3.6) becomes

[TABLE]

with $\varphi(x)=\varphi(x^{\prime})$. As a result $\kappa(\lambda/A;A)=\kappa(\lambda;1)=\kappa_{e}(\lambda)$ and $w_{0}(\lambda)=\varphi$, and (3.5) shows that $c_{e}^{*}(A)/A$ is non-increasing. This has been proved in [2]. If the limit is $\gamma$, then (3.5) gives

[TABLE]

(which has been already observed in [9]). Here one uses convexity of $\kappa_{e}$ and $\kappa_{e}(0)=0$ to show that the infimum in (3.5) is achieved at some $\lambda\leq\lambda_{A}\equiv\sqrt{f^{\prime}(0)}A$, as well as

[TABLE]

Moreover, if the infimum in (3.2) is achieved at a finite $\lambda$, then (3.5) gives that

[TABLE]

This condition is satisfied for all $f^{\prime}(0)<\Gamma$, where $\Gamma$ is from the proof of Lemma 3.1, that is, it is the supremum over $\lambda>0$ of the $\dot{H}^{1}$ norms of the principal eigenfunctions of (3.13). This is because of (3.2), the definition of $\kappa_{e}$, and the fact that

[TABLE]

Finally, we note that $\Gamma<\infty$ is possible — in the shear flow case it holds when there is an open set $U\subseteq{\mathbb{T}}^{n-1}$ such that $\alpha(x^{\prime})=\max_{{\mathbb{T}}^{n-1}}\alpha$ for all $x^{\prime}\in U$. Then any $w\in H^{1}({\mathbb{T}}^{n})$ supported on ${\mathbb{T}}\times U$ and independent of $x_{1}$ belongs to ${\mathcal{I}}$ and maximizes (1.6) whenever $f^{\prime}(0)\geq\|\nabla w\|_{2}^{2}/\|w\|_{2}^{2}$. Thus the limit in (1.6) need not be strictly increasing with $f^{\prime}(0)$ (which happens precisely when $\Gamma<\infty$).

In the more general case when the second order term and the non-linearity depend on $x$, we consider

[TABLE]

with $a$ 1-periodic real symmetric uniformly elliptic matrix and $u$ 1-periodic such that

[TABLE]

The non-linearity $f$ is 1-periodic in $x$ and satisfies for some $\varepsilon>0$

[TABLE]

We let $\zeta(x)\equiv f^{\prime}_{s}(x,0)>0$ and $\zeta_{0}\equiv\int_{{\mathbb{T}}^{n}}\zeta(x)\,dx$. Equations (3.3) and (3.4) are then replaced by (see [4])

[TABLE]

If we now define

[TABLE]

then (3.2) becomes

[TABLE]

and mimicking the above proofs one obtains the following extension of Theorem 1.1.

Theorem 3.2**.**

If $a$, $u$, and $f$ satisfy (3.15) and (3.16) and $|e|=1$, then

[TABLE]

*In particular, the limit exists. Moreover, *

[TABLE]