SHARP ASYMPTOTICS FOR KPP PULSATING FRONT SPEED-UP
AND DIFFUSION ENHANCEMENT BY FLOWS
ANDREJ ZLATOŠ
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)/A2 as A → ∞, where c∗(A) is the minimal front speed and D(A) the
effective diffusivity.
1. Introduction
We study reaction-diffusion fronts in the presence of strong incompressible flows. We
consider the PDE
Tt + Au · ∇T = ∆T + f(T ) (1.1)
on Rn, with T (t, x) ∈ [0, 1] the normalized temperature of a premixed combustible gas. The
non-linear reaction rate f is of Kolmogorov-Petrovskii-Piskunov (KPP) type [11]:
f ∈ C1,ε([0, 1]),
f(0) = f(1) = 0 and f is non-increasing on (1− ε, 1) for some ε > 0, (1.2)
0 < f(s) ≤ sf ′(0) for s ∈ (0, 1).
The 1-periodic flow u : Tn → Rn satisfies
u ∈ C1,ε(Tn), ∇ · u ≡ 0,
u dx = 0. (1.3)
That is, u is incompressible and mean-zero.
The number A ∈ 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 ∈ Rn, |e| = 1, is a solution of (1.1) of the form
T (t, x) = U(x · e− ct, x), with c the front speed, and U 1-periodic in x and such that
U(s, x) = 1,
U(s, x) = 0,
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 ≥ c∗e(A) (we suppress the
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.
http://arxiv.org/abs/0704.1163v2
2 ANDREJ ZLATOŠ
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]
u(x) = (α(x′), 0, . . . , 0) (x′ = (x2, . . . , xn)) and e = (1, 0, . . . , 0). (1.4)
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→ ∞. 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
w ∈ H1(Tn)
∣ Imw = 0 and u · ∇w = 0
(1.5)
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
c∗e(A)
= sup
‖∇w‖2
≤f ′(0)‖w‖2
(u · e)w2 dx
‖w‖22
. (1.6)
In particular, the limit exists. Moreover,
f ′(0)→0
c∗e(A)
f ′(0)A
= sup
(u · e)w dx
‖∇w‖2
, (1.7)
f ′(0)→∞
c∗e(A)
= sup
(u · e)w2 dx
‖w‖22
≤ max
{u(x) · e}. (1.8)
Remarks. 1. Inequality “≥” in (1.6) (with lim infA→∞ in place of limA→∞) 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 lim infA→∞ or lim supA→∞ in place of limA→∞.
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 ′(0)) is positive precisely when there exists w ∈ I such that
(u · e)w dx 6= 0 (take 1± εw 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.
PULSATING FRONT SPEED-UP AND DIFFUSION ENHANCEMENT 3
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
Φt + Au · ∇Φ = ∆Φ. (1.9)
As is well known, the long-time behavior of solutions to (1.9) is governed by the effective
diffusion equation
i,j=1
σij(A)
∂xi∂xj
Here σ(A) is a constant effective diffusivity matrix. If e ∈ Rn and we let χe,A be the mean-zero
solution of
−∆χe,A + Au · ∇χe,A = Au · e (1.10)
on Tn, then σ(A) is given by
e · σ(A)e′ =
(∇χe,A + e) · (∇χe′,A + e′)dx = e · e′ +
∇χe,A · ∇χe′,Adx.
The effective diffusivity for (1.9) in the direction e ∈ Rn, |e| = 1, is now
De(A) ≡ e · σ(A)e = 1 + ‖∇χe,A‖22. (1.11)
Again, mixing by flows enhances the effective diffusivity. It is easy to show that De(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 De(A)/A
2 as A→ ∞ 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
De(A)
= sup
(u · e)w dx
‖∇w‖2
. (1.12)
In particular, the limit exists. Moreover, there is w0 ∈ I which is a maximizer of (1.12) and
χe,A/A→ w0 in H1(Tn).
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→ ∞ limit (see also [13]).
2. We show that if (1.12) is positive, then the maximizers are precisely w = aw0 + b with
a, b ∈ R, a 6= 0.
3. If one considers the small diffusion problem φt = ε∆φ+u ·∇φ instead of (1.9), then the
corresponding effective diffusivity satisfies D̃e(ε) = εDe(ε
−1). Hence the limit limε→0 εD̃e(ε)
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).
4 ANDREJ ZLATOŠ
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 ψA ≡ χe,A/A, so that
−∆ψA + Au · ∇ψA = u · e (2.1)
Multiplying this by ψA and integrating over T
n we obtain using incompressibility of the flow,
‖∇ψA‖22 =
(u · e)ψAdx ≤ ‖u · e‖2‖ψA‖2. (2.2)
Poincaré inequality
‖w‖2 ≤ C‖∇w‖2 (2.3)
for some C <∞ and any mean-zero w then yields
‖ψA‖H1 ≤ C‖u · e‖2. (2.4)
It also follows from (2.2) that
De(A)
+ ‖∇ψA‖22 =
(u · e)ψAdx. (2.5)
Since ‖ψA‖H1 is uniformly bounded, there is a sequence Ak → ∞ such that ψAk converges
to some w0 ∈ H1(Tn), weakly in H1(Tn) and strongly in L2(Tn). Then ∆ψAk → ∆w0 and
∇ψAk → ∇w0 in the sense of distributions and (2.1) divided by Ak implies
u · ∇w0 = 0 (2.6)
in the sense of distributions. Since w0 ∈ H1(Tn), this equality holds almost everywhere and
w0 ∈ I. We also have
‖∇w0‖22 ≤ lim sup
‖∇ψAk‖
(u · e)w0dx =
∇ψA∇w0dx ≤ ‖∇ψA‖2‖∇w0‖2
where we used (2.2) in the second step, and (2.1) multiplied by w0 and integrated over T
(together with (2.6)) in the third step. Thus
‖∇w0‖2 ≤ ‖∇ψA‖2 (2.7)
as well as
lim sup
‖∇ψAk‖2 ≤ ‖∇w0‖2.
These give
‖∇ψAk‖2 = ‖∇w0‖2,
which turns the weak H1-convergence into a strong one:
ψAk → w0 in H
1(Tn). (2.8)
PULSATING FRONT SPEED-UP AND DIFFUSION ENHANCEMENT 5
Let us assume w0 6≡ 0. Then ∇w0 6≡ 0 because each ψA is mean-zero. From (2.5) and
(2.8),
De(Ak)
= ‖∇w0‖22 =
(u · e)w0 dx =
(u · e)w0 dx
‖∇w0‖2
. (2.9)
Pick an arbitrary non-constant w ∈ I. If we multiply (2.1) by w and integrate, we obtain
(u · e)w dx
∇ψAk∇w dx
∇w0∇w dx
≤ ‖∇w0‖2‖∇w‖2. (2.10)
Hence
(u · e)w dx
‖∇w‖2
≤ ‖∇w0‖22 = lim
De(Ak)
with equality precisely when ∇w is a multiple of ∇w0 (and so w = aw0+ b). This also means
that w0 is a maximizer for (1.12).
If now Bk → ∞ is any sequence, then as above we can find a subsequence (which we again
call Bk) such that ψBk → w1 ∈ I. But then w1 must also maximize (1.12), thus w1 = aw0+b.
Moreover, b = 0 because ψA are mean-zero, and (2.9) with Bk in place of Ak forces a = 1.
Hence ψA → w0 in H1(Tn) and (1.12) follows.
Finally, if w0 ≡ 0 is the only limit point of ψA, then ψA → 0 in H1(Tn), and (1.12) follows
from (2.5) and (2.10). �
Notice that (2.5), (2.7), and (2.9) show that De(A) ≥ 1 + δA2, where δ is the limit in
(1.12).
We also note that in the special case of shear flows u(x) = (α(x′), 0, . . . , 0) equation (1.10)
becomes
−∆x′χe,A = Ae1α(x′)
with χe,A(x) = χe,A(x
′). Hence χe,A = Ae1∇x′(−∆x′)−1α and the limit in (1.12) equals
|e1|‖∇x′(−∆x′)−1α‖22. 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
Φt + Au · ∇Φ = ∇ · (a∇Φ) (2.11)
instead of (1.9) with 1-periodic and real symmetric uniformly elliptic matrix a and 1-periodic
flow u such that
a ∈ C2(Tn), u ∈ C1,α(Tn), ∇ · u ≡ 0, ū ≡
u dx. (2.12)
Then (1.10) and (1.11) are replaced by
−∇ · (a∇χe,A) + Au · ∇χe,A = A(u− ū) · e,
De(A) ≡ |||∇χe,A + e|||22,
with |||w|||22 ≡
∇w · (a∇w) dx. If we define
w ∈ I
w dx = 0
6 ANDREJ ZLATOŠ
then we have
Theorem 2.1. If a and u satisfy (2.12) and |e| = 1, then
De(A)
= sup
(u · e)w dx
|||∇w|||2
. (2.13)
In particular, the limit exists. Moreover, there is w0 ∈ I0 which is a maximizer of (2.13) and
χe,A/A→ w0 in H1(Tn).
3. KPP Front Speed-up
In this section we prove Theorem 1.1. We start with an auxiliary lemma. Let us define
κe(λ) ≡ sup
(u · e)w2 dx− ‖∇w‖22
‖w‖22
. (3.1)
Note that κe(λ) must be convex as it is a supremum of linear functions. Also, κe(λ) ≥ 0
because w ≡ 1 ∈ I.
Lemma 3.1. Assume the setting of Theorem 1.1. Then for each λ > 0, the supremum in
(3.1) is attained, the maximizer is unique up to multiplication, and
c∗e(A)
= inf
f ′(0) + κe(λ)
. (3.2)
Proof. It has been shown in [4] that the minimal front speed c∗e(A) can be computed using
the variational principle
c∗e(A) = inf
f ′(0) + λ2 + κ(λ;A)
. (3.3)
Here κ(λ;A) is the unique eigenvalue of the problem
∆ϕ− Au · ∇ϕ− 2λe · ∇ϕ+ λAu · eϕ = κ(λ;A)ϕ, ϕ > 0 (3.4)
on Tn, with a unique normalized eigenfunction ϕA(x;λ). Moreover, the function
µ(λ;A) ≡ λ2 + κ(λ;A)
is monotonically increasing and convex in λ ≥ 0, with µ(0;A) = 0 (see [3, 13]).
We now rewrite (3.3) and (3.4) as
c∗e(A)
= inf
f ′(0) + (λ/A)2 + κ(λ/A;A)
. (3.5)
∆ϕA − Au · ∇ϕA −
e · ∇ϕA + λu · eϕA = κ(λ/A;A)ϕA, ϕA > 0. (3.6)
We multiply (3.6) by ϕ−1A and integrate to obtain (using incompressibility of u)
0 ≤ ‖∇ lnϕA‖22 = κ(λ/A;A). (3.7)
PULSATING FRONT SPEED-UP AND DIFFUSION ENHANCEMENT 7
Similarly, multiplication by ϕA yields
κ(λ/A;A) + ‖∇ϕA‖22 = λ
(u · e)ϕ2Adx ≤ λ‖u · e‖∞. (3.8)
since ‖ϕA‖2 = 1. This again means that there is a sequence Ak → ∞ such that ϕAk
converges to some w0 ∈ H1(Tn), weakly in H1(Tn) and strongly in L2(Tn). The convergence
∆ϕAk → ∆w0 and ∇ϕAk → ∇w0 in the sense of distributions, boundedness of κ(λ/A;A) in
A, and (3.6) divided by A then imply (2.6) and so w0 ∈ I (note that ‖w0‖2 = ‖ϕAk‖2 = 1).
Now we multiply (3.6) by w0 and integrate to obtain (with o(1) = o(k
0) and using (3.8))
∇ϕAk∇w0dx+ λ
(u · e)w20dx+ o(1) = κ(λ/Ak;Ak) + o(1)
(u · e)w20dx− ‖∇ϕAk‖
2 + o(1).
Once again it follows that
‖∇w0‖22 ≤ lim sup
‖∇ϕAk‖
2 ≤ ‖∇w0‖2 lim sup
‖∇ϕAk‖2
and so as in Section 2,
ϕAk → w0 in H
1(Tn). (3.9)
(3.8) then yields
κ0 ≡ lim
κ(λ/Ak;Ak) = λ
(u · e)w20dx−
|∇w0|2 dx.
Let w ∈ I ∩ L∞(Tn), multiply (3.6) for A = Ak by w2/ϕAk and integrate to obtain (using
that ∇ϕA/ϕA = ∇ lnϕA are uniformly bounded in L2(Tn) by (3.7) and (3.8))
κ0‖w‖22 = λ
(u · e)w2dx+ lim
w2 − 2∇ϕAk
w∇w − 2λ
e · ∇ϕAk
w2 dx
(u · e)w2dx−
|∇w|2 dx. (3.10)
Since each w ∈ I is the H1-limit of wN(x) ≡ sgn(w(x))min{|w(x)|, N} ∈ I ∩ L∞(Tn), this
inequality extends to all w ∈ I. Hence κ0 = κe(λ) from (3.1), and w0 is a maximizer for
(3.1) (because ‖w0‖2 = 1). Moreover, if Bk → ∞ is any sequence with
κ(λ/Bk;Bk) ≡ κ1,
then repeating the above argument we find that there must be a subsequence (which we again
call Bk) such that ϕBk → w1 ∈ I in H1(Tn), ‖w1‖2 = 1. But then as before,
(u · e)w21dx−
|∇w1|2 dx = κ1 ≥
(u · e)w2dx−
|∇w|2 dx
‖w‖22
for any w ∈ I. Taking w = w0 we obtain κ1 = κe(λ), and so
κe(λ) = lim
κ(λ/A;A) = λ
(u · e)w20dx−
|∇w0|2 dx. (3.11)
8 ANDREJ ZLATOŠ
The function κe(λ) is convex, monotonically increasing, and non-negative, as it is the
pointwise limit of functions µ(λ/A;A) = (λ/A)2+κ(λ/A;A) which have the same properties.
This also implies that the convergence in (3.11) is uniform on each bounded interval of λ.
We then have
f ′(0) + µ(λ/A;A)
= inf
f ′(0) + κe(λ)
(≤ is immediate, whereas ≥ uses convexity of µ(λ/A;A) once more). This proves (3.2).
We are left with showing that any maximizer of (3.1) is a multiple of w0. Denote ϕk ≡ ϕAk
and notice that (3.9) shows that (after passing to a subsequence — we will repeat this without
mentioning it below), ∇ϕk(x) → ∇w0(x) and ϕk(x) → w0(x) for a.e. x. Next (3.7) and (2.3)
imply that if ck is the average of lnϕk, then lnϕk − ck → ω strongly in L2 and weakly in H1.
But then lnϕk(x)− ck → ω(x) for a.e. x. Since lnϕk(x) → lnw0(x) for a.e. x, it follows that
ck → c and ω = lnw0 − c. We thus obtain lnw0 ∈ H1 which means w0(x) 6= 0 for a.e. x, and
so for a.e. x,
∇ϕk(x)
ϕk(x)
→ ∇w0(x)
w0(x)
. (3.12)
Let now w 6≡ 0 be a maximizer of (3.1) and let us first assume w ≥ 0 almost everywhere.
Then (3.10) for wN and wN → w in H1 show
lim sup
wN −∇wN
But then (3.12) and pointwise convergence of wN and ∇wN to w and ∇w, respectively, give
for a.e. x,
∇w0(x)
w0(x)
w(x) = ∇w(x).
We now let wε(x) ≡ max{w(x), ε} so that lnwε ∈ H1 and
∇ lnwε(x) =
∇ lnw0(x) wε(x) > ε,
0 wε(x) = ε.
This and lnw0 ∈ H1 means that ‖∇ lnwε‖2 is bounded, and again we must have lnwεk−cεk →
ω strongly in L2, weakly in H1, and pointwise almost everywhere. But lnwεk(x) → lnw(x),
so again cεk → c and lnw ∈ H1. Hence w(x) > 0 for a.e. x, and so ∇ lnw(x) = ∇ lnw0(x)
for a.e. x. This means lnw − lnw0 is constant, that is, w is a multiple of w0.
If w is an arbitrary maximizer of (3.1), then both w±(x) ≡ max{±w(x), 0} ∈ I must be
maximizers of (3.1) (or ≡ 0). But then w±(x) > 0 for a.e. x, meaning that one of them is
zero while the other is a multiple of w0. �
Proof of Theorem 1.1. Inequality “≥” in (1.6) is immediate from (3.1) and (3.2). To prove
the opposite inequality it is sufficient to find λ such that the unique normalized non-negative
maximizer w0(λ) of (3.1) satisfies γ(λ) ≡ ‖∇w0(λ)‖22 = f ′(0).
To this end notice that if λ = 0, then w0(λ) ≡ 1 and so γ(0) = 0. Also, γ must be
continuous. Indeed — let λk → λ∞ < ∞ and denote wk ≡ w0(λk). Then (3.8) and (3.9)
imply that wk are uniformly bounded in H
1. Thus a subsequence (again called wk) converges
PULSATING FRONT SPEED-UP AND DIFFUSION ENHANCEMENT 9
strongly in L2 and weakly in H1 to some ω. Obviously ω ∈ I as well as λk
(u · e)w2k dx →
(u · e)ω2 dx and ‖∇ω‖2 ≤ lim inf ‖∇wk‖2. But ‖∇ω‖2 < lim inf ‖∇wk‖2 is impossible
(otherwise wk would not maximize (3.1) for large k) and so for a subsequence,
κe(λk) = λk
(u · e)w2k dx− ‖∇wk‖22 → λ∞
(u · e)ω2 dx− ‖∇ω‖22.
Since κe is continuous, this means ω = w∞. We have thus proved that every sequence
λk → λ∞ has a subsequence with γ(λkj) → γ(λ∞), that is, γ is continuous with γ(0) = 0.
Let now Γ ≡ supλ≥0 γ(λ). If f ′(0) ∈ (0,Γ), then there is λ > 0 with γ(λ) = f ′(0) and (1.6)
is proved. If, on the other hand, Γ <∞ and f ′(0) ≥ Γ, then (3.2) is bounded above by
lim inf
(u · e)w0(λ)2 dx− ‖∇w0(λ)‖22 + f ′(0)
= lim inf
(u · e)w0(λ)2 dx,
which does not exceed the right hand side of (1.6) due to ‖∇w0(λ)‖22 ≤ Γ ≤ f ′(0)‖w0(λ)‖22.
Since (1.8) is immediate from (1.6), we are left with proving (1.7). Let us consider any
w ∈ I with ‖w‖2 = 1 and ‖∇w‖22 ≤ f ′(0). Let w̄ ≡
w dx ∈ [−1, 1] and ω ≡ w − w̄. Then
‖ω‖22 ≤ C‖∇ω‖22 ≤ Cf ′(0), and so
w̄2 + 2w̄ω + ω2 dx = w̄2 +O(
f ′(0))
as f ′(0) → 0. Hence w̄ = 1 +O(
f ′(0)) and we have
(u · e)w2 dx = 2
(u · e)ω dx+O(f ′(0)) ≤ 2
f ′(0)
(u · e)ω dx
‖∇ω‖2
+O(f ′(0))
with equality when ‖∇ω‖22 = f ′(0). Picking first w that maximizes (1.6) and then ω that
maximizes (1.7) with ‖∇ω‖22 = f ′(0) (and adjusting w̄ accordingly) finishes the proof. �
Note that in the case of shear flows (1.4) equation (3.6) becomes
∆x′ϕ+ λαϕ = κ(λ/A;A)ϕ, ϕ > 0. (3.13)
with ϕ(x) = ϕ(x′). As a result κ(λ/A;A) = κ(λ; 1) = κe(λ) and w0(λ) = ϕ, and (3.5) shows
that c∗e(A)/A is non-increasing. This has been proved in [2]. If the limit is γ, then (3.5) gives
c∗e(A)
− γ ≤
f ′(0)
(which has been already observed in [9]). Here one uses convexity of κe and κe(0) = 0 to
show that the infimum in (3.5) is achieved at some λ ≤ λA ≡
f ′(0)A, as well as
f ′(0) + κe(λ)
≥ min
λ∈(0,λA)
f ′(0) + κe(λ)
κe(λA)
Moreover, if the infimum in (3.2) is achieved at a finite λ, then (3.5) gives that
c∗e(A)
− γ = O(A2).
10 ANDREJ ZLATOŠ
This condition is satisfied for all f ′(0) < Γ, where Γ is from the proof of Lemma 3.1, that
is, it is the supremum over λ > 0 of the Ḣ1 norms of the principal eigenfunctions of (3.13).
This is because of (3.2), the definition of κe, and the fact that
κe(λ)
= sup
(u · e)w2 dx
‖w‖22
Finally, we note that Γ < ∞ is possible — in the shear flow case it holds when there is
an open set U ⊆ Tn−1 such that α(x′) = maxTn−1 α for all x′ ∈ U . Then any w ∈ H1(Tn)
supported on T × U and independent of x1 belongs to I and maximizes (1.6) whenever
f ′(0) ≥ ‖∇w‖22/‖w‖22. Thus the limit in (1.6) need not be strictly increasing with f ′(0)
(which happens precisely when Γ <∞).
In the more general case when the second order term and the non-linearity depend on x,
we consider
Tt + Au · ∇T = ∇ · (a∇T ) + f(x, T ) (3.14)
with a 1-periodic real symmetric uniformly elliptic matrix and u 1-periodic such that
a ∈ C2(Tn), u ∈ C1,ε(Tn), ∇ · u ≡ 0,
u dx = 0. (3.15)
The non-linearity f is 1-periodic in x and satisfies for some ε > 0
f ∈ C1,δ(Tn × [0, 1]),
f(x, 0) = f(x, 1) = 0 and f(x, ·) is non-increasing on (1− ε, 1) for each x ∈ Tn, (3.16)
0 < f(x, s) ≤ sf ′s(x, 0) for (s, x) ∈ (0, 1)× Tn.
We let ζ(x) ≡ f ′s(x, 0) > 0 and ζ0 ≡
ζ(x) dx. Equations (3.3) and (3.4) are then replaced
by (see [4])
c∗e(A) = inf
κ(λ, f ;A)
∇ · (a∇ϕ)− Au · ∇ϕ− 2λe · a∇ϕ+ [λAu · e+ ζ + λ2e · ae− λ∇ · (ae)]ϕ = κ(λ, f ;A)ϕ.
If we now define
κe(λ, f) ≡ sup
(λu · e+ ζ)w2 dx− |||∇w|||22
‖w‖22
then (3.2) becomes
c∗e(A, f)
= inf
κe(λ, f)
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
c∗e(A)
= sup
|||∇w|||2
ζw2 dx
(u · e)w2 dx
‖w‖22
PULSATING FRONT SPEED-UP AND DIFFUSION ENHANCEMENT 11
In particular, the limit exists. Moreover,
c∗e(A, αf)
= sup
(u · e)w dx
|||∇w|||2
c∗e(A, αf)
= sup
(u · e)w2 dx
‖w‖22
≤ max
{u(x) · e}.
References
[1] B. Audoly, H. Berestycki and Y. Pomeau, Réaction diffusion en écoulement stationnaire rapide, C. R.
Acad. Sci. Paris 328, Série IIb, 2000, 255–262.
[2] H. Berestycki, The influence of advection on the propagation of fronts in reaction-diffusion equations, in
Nonlinear PDEs in Condensed Matter and Reactive Flows, NATO Science Series C, 569, H. Berestycki
and Y. Pomeau eds, Kluwer, Doordrecht, 2003.
[3] H. Berestycki and F. Hamel, Front propagation in periodic excitable media, Comm. Pure and Appl.
Math., 55, 2002, 949–1032.
[4] H. Berestycki, F. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems, I -
Periodic framework, J. European Math. Soc. 7, 2005, 173–213.
[5] H. Berestycki, F. Hamel and N. Nadirashvili, Elliptic eigenvalue problems with large drift and applica-
tions to nonlinear propagation phenomena, Comm. Math. Phys. 253, 2005, 451–480.
[6] R.N. Bhattacharya, V.K. Gupta and H.F. Walker, Asymptotics of solute dispersion in periodic porous
media, SIAM J. Appl. Math. 49, 1989, 86–98.
[7] P. Constantin, A. Kiselev, A. Oberman and L. Ryzhik, Bulk burning rate in passive-reactive diffusion,
Arch. Ration. Mech. Anal. 154, 2000, 53–91.
[8] A. Fannjiang and G. Papanicolaou, Convection enhanced diffusion for periodic flows, SIAM Jour. Appl.
Math., 54, 1994, 333–408.
[9] S. Heinze, Large convection limits for KPP fronts, Max Planck Institute for Mathematics Preprint Nr.
21/2005, 2005.
[10] A. Kiselev and L. Ryzhik, Enhancement of the traveling front speeds in reaction-diffusion equations with
advection, Ann. Inst. H. Poincaré Anal. Non Linéaire 18, 2001, 309–358.
[11] A.N. Kolmogorov, I.G. Petrovskii and N.S. Piskunov, Étude de l’équation de la chaleur de matière et
son application à un problème biologique, Bull. Moskov. Gos. Univ. Mat. Mekh. 1 (1937), 1–25.
[12] P. Kramer and A. Majda, Simplified models for turbulent diffusion: theory, numerical modelling, and
physical phenomena, Phys. Rep. 314, 1999, 237–574.
[13] A. Novikov and L. Ryzhik, Bounds on the speed of propagation of the KPP fronts in a cellular flow,
Arch. Rat. Mech. Anal., to appear.
[14] L. Ryzhik and A. Zlatoš, KPP Pulsating front speed-up by flows, preprint.
[15] H. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic
habitat, Jour. Math. Biol., 45, 2002, 511–548.
	1. Introduction
	2. Effective Diffusivity Enhancement
	3. KPP Front Speed-up
	References

  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.

Introduction
We study reaction-diffusion fronts in the presence of strong incompressible flows. We
consider the PDE
Tt + Au · ∇T = ∆T + f(T ) (1.1)
on Rn, with T (t, x) ∈ [0, 1] the normalized temperature of a premixed combustible gas. The
non-linear reaction rate f is of Kolmogorov-Petrovskii-Piskunov (KPP) type [11]:
f ∈ C1,ε([0, 1]),
f(0) = f(1) = 0 and f is non-increasing on (1− ε, 1) for some ε > 0, (1.2)
0 < f(s) ≤ sf ′(0) for s ∈ (0, 1).
The 1-periodic flow u : Tn → Rn satisfies
u ∈ C1,ε(Tn), ∇ · u ≡ 0,
u dx = 0. (1.3)
That is, u is incompressible and mean-zero.
The number A ∈ 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 ∈ Rn, |e| = 1, is a solution of (1.1) of the form
T (t, x) = U(x · e− ct, x), with c the front speed, and U 1-periodic in x and such that
U(s, x) = 1,
U(s, x) = 0,
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 ≥ c∗e(A) (we suppress the
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.
http://arxiv.org/abs/0704.1163v2
2 ANDREJ ZLATOŠ
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]
u(x) = (α(x′), 0, . . . , 0) (x′ = (x2, . . . , xn)) and e = (1, 0, . . . , 0). (1.4)
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→ ∞. 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
w ∈ H1(Tn)
∣ Imw = 0 and u · ∇w = 0
(1.5)
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
c∗e(A)
= sup
‖∇w‖2
≤f ′(0)‖w‖2
(u · e)w2 dx
‖w‖22
. (1.6)
In particular, the limit exists. Moreover,
f ′(0)→0
c∗e(A)
f ′(0)A
= sup
(u · e)w dx
‖∇w‖2
, (1.7)
f ′(0)→∞
c∗e(A)
= sup
(u · e)w2 dx
‖w‖22
≤ max
{u(x) · e}. (1.8)
Remarks. 1. Inequality “≥” in (1.6) (with lim infA→∞ in place of limA→∞) 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 lim infA→∞ or lim supA→∞ in place of limA→∞.
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 ′(0)) is positive precisely when there exists w ∈ I such that
(u · e)w dx 6= 0 (take 1± εw 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.
PULSATING FRONT SPEED-UP AND DIFFUSION ENHANCEMENT 3
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
Φt + Au · ∇Φ = ∆Φ. (1.9)
As is well known, the long-time behavior of solutions to (1.9) is governed by the effective
diffusion equation
i,j=1
σij(A)
∂xi∂xj
Here σ(A) is a constant effective diffusivity matrix. If e ∈ Rn and we let χe,A be the mean-zero
solution of
−∆χe,A + Au · ∇χe,A = Au · e (1.10)
on Tn, then σ(A) is given by
e · σ(A)e′ =
(∇χe,A + e) · (∇χe′,A + e′)dx = e · e′ +
∇χe,A · ∇χe′,Adx.
The effective diffusivity for (1.9) in the direction e ∈ Rn, |e| = 1, is now
De(A) ≡ e · σ(A)e = 1 + ‖∇χe,A‖22. (1.11)
Again, mixing by flows enhances the effective diffusivity. It is easy to show that De(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 De(A)/A
2 as A→ ∞ 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
De(A)
= sup
(u · e)w dx
‖∇w‖2
. (1.12)
In particular, the limit exists. Moreover, there is w0 ∈ I which is a maximizer of (1.12) and
χe,A/A→ w0 in H1(Tn).
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→ ∞ limit (see also [13]).
2. We show that if (1.12) is positive, then the maximizers are precisely w = aw0 + b with
a, b ∈ R, a 6= 0.
3. If one considers the small diffusion problem φt = ε∆φ+u ·∇φ instead of (1.9), then the
corresponding effective diffusivity satisfies D̃e(ε) = εDe(ε
−1). Hence the limit limε→0 εD̃e(ε)
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).
4 ANDREJ ZLATOŠ
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 ψA ≡ χe,A/A, so that
−∆ψA + Au · ∇ψA = u · e (2.1)
Multiplying this by ψA and integrating over T
n we obtain using incompressibility of the flow,
‖∇ψA‖22 =
(u · e)ψAdx ≤ ‖u · e‖2‖ψA‖2. (2.2)
Poincaré inequality
‖w‖2 ≤ C‖∇w‖2 (2.3)
for some C <∞ and any mean-zero w then yields
‖ψA‖H1 ≤ C‖u · e‖2. (2.4)
It also follows from (2.2) that
De(A)
+ ‖∇ψA‖22 =
(u · e)ψAdx. (2.5)
Since ‖ψA‖H1 is uniformly bounded, there is a sequence Ak → ∞ such that ψAk converges
to some w0 ∈ H1(Tn), weakly in H1(Tn) and strongly in L2(Tn). Then ∆ψAk → ∆w0 and
∇ψAk → ∇w0 in the sense of distributions and (2.1) divided by Ak implies
u · ∇w0 = 0 (2.6)
in the sense of distributions. Since w0 ∈ H1(Tn), this equality holds almost everywhere and
w0 ∈ I. We also have
‖∇w0‖22 ≤ lim sup
‖∇ψAk‖
(u · e)w0dx =
∇ψA∇w0dx ≤ ‖∇ψA‖2‖∇w0‖2
where we used (2.2) in the second step, and (2.1) multiplied by w0 and integrated over T
(together with (2.6)) in the third step. Thus
‖∇w0‖2 ≤ ‖∇ψA‖2 (2.7)
as well as
lim sup
‖∇ψAk‖2 ≤ ‖∇w0‖2.
These give
‖∇ψAk‖2 = ‖∇w0‖2,
which turns the weak H1-convergence into a strong one:
ψAk → w0 in H
1(Tn). (2.8)
PULSATING FRONT SPEED-UP AND DIFFUSION ENHANCEMENT 5
Let us assume w0 6≡ 0. Then ∇w0 6≡ 0 because each ψA is mean-zero. From (2.5) and
(2.8),
De(Ak)
= ‖∇w0‖22 =
(u · e)w0 dx =
(u · e)w0 dx
‖∇w0‖2
. (2.9)
Pick an arbitrary non-constant w ∈ I. If we multiply (2.1) by w and integrate, we obtain
(u · e)w dx
∇ψAk∇w dx
∇w0∇w dx
≤ ‖∇w0‖2‖∇w‖2. (2.10)
Hence
(u · e)w dx
‖∇w‖2
≤ ‖∇w0‖22 = lim
De(Ak)
with equality precisely when ∇w is a multiple of ∇w0 (and so w = aw0+ b). This also means
that w0 is a maximizer for (1.12).
If now Bk → ∞ is any sequence, then as above we can find a subsequence (which we again
call Bk) such that ψBk → w1 ∈ I. But then w1 must also maximize (1.12), thus w1 = aw0+b.
Moreover, b = 0 because ψA are mean-zero, and (2.9) with Bk in place of Ak forces a = 1.
Hence ψA → w0 in H1(Tn) and (1.12) follows.
Finally, if w0 ≡ 0 is the only limit point of ψA, then ψA → 0 in H1(Tn), and (1.12) follows
from (2.5) and (2.10). �
Notice that (2.5), (2.7), and (2.9) show that De(A) ≥ 1 + δA2, where δ is the limit in
(1.12).
We also note that in the special case of shear flows u(x) = (α(x′), 0, . . . , 0) equation (1.10)
becomes
−∆x′χe,A = Ae1α(x′)
with χe,A(x) = χe,A(x
′). Hence χe,A = Ae1∇x′(−∆x′)−1α and the limit in (1.12) equals
|e1|‖∇x′(−∆x′)−1α‖22. 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
Φt + Au · ∇Φ = ∇ · (a∇Φ) (2.11)
instead of (1.9) with 1-periodic and real symmetric uniformly elliptic matrix a and 1-periodic
flow u such that
a ∈ C2(Tn), u ∈ C1,α(Tn), ∇ · u ≡ 0, ū ≡
u dx. (2.12)
Then (1.10) and (1.11) are replaced by
−∇ · (a∇χe,A) + Au · ∇χe,A = A(u− ū) · e,
De(A) ≡ |||∇χe,A + e|||22,
with |||w|||22 ≡
∇w · (a∇w) dx. If we define
w ∈ I
w dx = 0
6 ANDREJ ZLATOŠ
then we have
Theorem 2.1. If a and u satisfy (2.12) and |e| = 1, then
De(A)
= sup
(u · e)w dx
|||∇w|||2
. (2.13)
In particular, the limit exists. Moreover, there is w0 ∈ I0 which is a maximizer of (2.13) and
χe,A/A→ w0 in H1(Tn).
3. KPP Front Speed-up
In this section we prove Theorem 1.1. We start with an auxiliary lemma. Let us define
κe(λ) ≡ sup
(u · e)w2 dx− ‖∇w‖22
‖w‖22
. (3.1)
Note that κe(λ) must be convex as it is a supremum of linear functions. Also, κe(λ) ≥ 0
because w ≡ 1 ∈ I.
Lemma 3.1. Assume the setting of Theorem 1.1. Then for each λ > 0, the supremum in
(3.1) is attained, the maximizer is unique up to multiplication, and
c∗e(A)
= inf
f ′(0) + κe(λ)
. (3.2)
Proof. It has been shown in [4] that the minimal front speed c∗e(A) can be computed using
the variational principle
c∗e(A) = inf
f ′(0) + λ2 + κ(λ;A)
. (3.3)
Here κ(λ;A) is the unique eigenvalue of the problem
∆ϕ− Au · ∇ϕ− 2λe · ∇ϕ+ λAu · eϕ = κ(λ;A)ϕ, ϕ > 0 (3.4)
on Tn, with a unique normalized eigenfunction ϕA(x;λ). Moreover, the function
µ(λ;A) ≡ λ2 + κ(λ;A)
is monotonically increasing and convex in λ ≥ 0, with µ(0;A) = 0 (see [3, 13]).
We now rewrite (3.3) and (3.4) as
c∗e(A)
= inf
f ′(0) + (λ/A)2 + κ(λ/A;A)
. (3.5)
∆ϕA − Au · ∇ϕA −
e · ∇ϕA + λu · eϕA = κ(λ/A;A)ϕA, ϕA > 0. (3.6)
We multiply (3.6) by ϕ−1A and integrate to obtain (using incompressibility of u)
0 ≤ ‖∇ lnϕA‖22 = κ(λ/A;A). (3.7)
PULSATING FRONT SPEED-UP AND DIFFUSION ENHANCEMENT 7
Similarly, multiplication by ϕA yields
κ(λ/A;A) + ‖∇ϕA‖22 = λ
(u · e)ϕ2Adx ≤ λ‖u · e‖∞. (3.8)
since ‖ϕA‖2 = 1. This again means that there is a sequence Ak → ∞ such that ϕAk
converges to some w0 ∈ H1(Tn), weakly in H1(Tn) and strongly in L2(Tn). The convergence
∆ϕAk → ∆w0 and ∇ϕAk → ∇w0 in the sense of distributions, boundedness of κ(λ/A;A) in
A, and (3.6) divided by A then imply (2.6) and so w0 ∈ I (note that ‖w0‖2 = ‖ϕAk‖2 = 1).
Now we multiply (3.6) by w0 and integrate to obtain (with o(1) = o(k
0) and using (3.8))
∇ϕAk∇w0dx+ λ
(u · e)w20dx+ o(1) = κ(λ/Ak;Ak) + o(1)
(u · e)w20dx− ‖∇ϕAk‖
2 + o(1).
Once again it follows that
‖∇w0‖22 ≤ lim sup
‖∇ϕAk‖
2 ≤ ‖∇w0‖2 lim sup
‖∇ϕAk‖2
and so as in Section 2,
ϕAk → w0 in H
1(Tn). (3.9)
(3.8) then yields
κ0 ≡ lim
κ(λ/Ak;Ak) = λ
(u · e)w20dx−
|∇w0|2 dx.
Let w ∈ I ∩ L∞(Tn), multiply (3.6) for A = Ak by w2/ϕAk and integrate to obtain (using
that ∇ϕA/ϕA = ∇ lnϕA are uniformly bounded in L2(Tn) by (3.7) and (3.8))
κ0‖w‖22 = λ
(u · e)w2dx+ lim
w2 − 2∇ϕAk
w∇w − 2λ
e · ∇ϕAk
w2 dx
(u · e)w2dx−
|∇w|2 dx. (3.10)
Since each w ∈ I is the H1-limit of wN(x) ≡ sgn(w(x))min{|w(x)|, N} ∈ I ∩ L∞(Tn), this
inequality extends to all w ∈ I. Hence κ0 = κe(λ) from (3.1), and w0 is a maximizer for
(3.1) (because ‖w0‖2 = 1). Moreover, if Bk → ∞ is any sequence with
κ(λ/Bk;Bk) ≡ κ1,
then repeating the above argument we find that there must be a subsequence (which we again
call Bk) such that ϕBk → w1 ∈ I in H1(Tn), ‖w1‖2 = 1. But then as before,
(u · e)w21dx−
|∇w1|2 dx = κ1 ≥
(u · e)w2dx−
|∇w|2 dx
‖w‖22
for any w ∈ I. Taking w = w0 we obtain κ1 = κe(λ), and so
κe(λ) = lim
κ(λ/A;A) = λ
(u · e)w20dx−
|∇w0|2 dx. (3.11)
8 ANDREJ ZLATOŠ
The function κe(λ) is convex, monotonically increasing, and non-negative, as it is the
pointwise limit of functions µ(λ/A;A) = (λ/A)2+κ(λ/A;A) which have the same properties.
This also implies that the convergence in (3.11) is uniform on each bounded interval of λ.
We then have
f ′(0) + µ(λ/A;A)
= inf
f ′(0) + κe(λ)
(≤ is immediate, whereas ≥ uses convexity of µ(λ/A;A) once more). This proves (3.2).
We are left with showing that any maximizer of (3.1) is a multiple of w0. Denote ϕk ≡ ϕAk
and notice that (3.9) shows that (after passing to a subsequence — we will repeat this without
mentioning it below), ∇ϕk(x) → ∇w0(x) and ϕk(x) → w0(x) for a.e. x. Next (3.7) and (2.3)
imply that if ck is the average of lnϕk, then lnϕk − ck → ω strongly in L2 and weakly in H1.
But then lnϕk(x)− ck → ω(x) for a.e. x. Since lnϕk(x) → lnw0(x) for a.e. x, it follows that
ck → c and ω = lnw0 − c. We thus obtain lnw0 ∈ H1 which means w0(x) 6= 0 for a.e. x, and
so for a.e. x,
∇ϕk(x)
ϕk(x)
→ ∇w0(x)
w0(x)
. (3.12)
Let now w 6≡ 0 be a maximizer of (3.1) and let us first assume w ≥ 0 almost everywhere.
Then (3.10) for wN and wN → w in H1 show
lim sup
wN −∇wN
But then (3.12) and pointwise convergence of wN and ∇wN to w and ∇w, respectively, give
for a.e. x,
∇w0(x)
w0(x)
w(x) = ∇w(x).
We now let wε(x) ≡ max{w(x), ε} so that lnwε ∈ H1 and
∇ lnwε(x) =
∇ lnw0(x) wε(x) > ε,
0 wε(x) = ε.
This and lnw0 ∈ H1 means that ‖∇ lnwε‖2 is bounded, and again we must have lnwεk−cεk →
ω strongly in L2, weakly in H1, and pointwise almost everywhere. But lnwεk(x) → lnw(x),
so again cεk → c and lnw ∈ H1. Hence w(x) > 0 for a.e. x, and so ∇ lnw(x) = ∇ lnw0(x)
for a.e. x. This means lnw − lnw0 is constant, that is, w is a multiple of w0.
If w is an arbitrary maximizer of (3.1), then both w±(x) ≡ max{±w(x), 0} ∈ I must be
maximizers of (3.1) (or ≡ 0). But then w±(x) > 0 for a.e. x, meaning that one of them is
zero while the other is a multiple of w0. �
Proof of Theorem 1.1. Inequality “≥” in (1.6) is immediate from (3.1) and (3.2). To prove
the opposite inequality it is sufficient to find λ such that the unique normalized non-negative
maximizer w0(λ) of (3.1) satisfies γ(λ) ≡ ‖∇w0(λ)‖22 = f ′(0).
To this end notice that if λ = 0, then w0(λ) ≡ 1 and so γ(0) = 0. Also, γ must be
continuous. Indeed — let λk → λ∞ < ∞ and denote wk ≡ w0(λk). Then (3.8) and (3.9)
imply that wk are uniformly bounded in H
1. Thus a subsequence (again called wk) converges
PULSATING FRONT SPEED-UP AND DIFFUSION ENHANCEMENT 9
strongly in L2 and weakly in H1 to some ω. Obviously ω ∈ I as well as λk
(u · e)w2k dx →
(u · e)ω2 dx and ‖∇ω‖2 ≤ lim inf ‖∇wk‖2. But ‖∇ω‖2 < lim inf ‖∇wk‖2 is impossible
(otherwise wk would not maximize (3.1) for large k) and so for a subsequence,
κe(λk) = λk
(u · e)w2k dx− ‖∇wk‖22 → λ∞
(u · e)ω2 dx− ‖∇ω‖22.
Since κe is continuous, this means ω = w∞. We have thus proved that every sequence
λk → λ∞ has a subsequence with γ(λkj) → γ(λ∞), that is, γ is continuous with γ(0) = 0.
Let now Γ ≡ supλ≥0 γ(λ). If f ′(0) ∈ (0,Γ), then there is λ > 0 with γ(λ) = f ′(0) and (1.6)
is proved. If, on the other hand, Γ <∞ and f ′(0) ≥ Γ, then (3.2) is bounded above by
lim inf
(u · e)w0(λ)2 dx− ‖∇w0(λ)‖22 + f ′(0)
= lim inf
(u · e)w0(λ)2 dx,
which does not exceed the right hand side of (1.6) due to ‖∇w0(λ)‖22 ≤ Γ ≤ f ′(0)‖w0(λ)‖22.
Since (1.8) is immediate from (1.6), we are left with proving (1.7). Let us consider any
w ∈ I with ‖w‖2 = 1 and ‖∇w‖22 ≤ f ′(0). Let w̄ ≡
w dx ∈ [−1, 1] and ω ≡ w − w̄. Then
‖ω‖22 ≤ C‖∇ω‖22 ≤ Cf ′(0), and so
w̄2 + 2w̄ω + ω2 dx = w̄2 +O(
f ′(0))
as f ′(0) → 0. Hence w̄ = 1 +O(
f ′(0)) and we have
(u · e)w2 dx = 2
(u · e)ω dx+O(f ′(0)) ≤ 2
f ′(0)
(u · e)ω dx
‖∇ω‖2
+O(f ′(0))
with equality when ‖∇ω‖22 = f ′(0). Picking first w that maximizes (1.6) and then ω that
maximizes (1.7) with ‖∇ω‖22 = f ′(0) (and adjusting w̄ accordingly) finishes the proof. �
Note that in the case of shear flows (1.4) equation (3.6) becomes
∆x′ϕ+ λαϕ = κ(λ/A;A)ϕ, ϕ > 0. (3.13)
with ϕ(x) = ϕ(x′). As a result κ(λ/A;A) = κ(λ; 1) = κe(λ) and w0(λ) = ϕ, and (3.5) shows
that c∗e(A)/A is non-increasing. This has been proved in [2]. If the limit is γ, then (3.5) gives
c∗e(A)
− γ ≤
f ′(0)
(which has been already observed in [9]). Here one uses convexity of κe and κe(0) = 0 to
show that the infimum in (3.5) is achieved at some λ ≤ λA ≡
f ′(0)A, as well as
f ′(0) + κe(λ)
≥ min
λ∈(0,λA)
f ′(0) + κe(λ)
κe(λA)
Moreover, if the infimum in (3.2) is achieved at a finite λ, then (3.5) gives that
c∗e(A)
− γ = O(A2).
10 ANDREJ ZLATOŠ
This condition is satisfied for all f ′(0) < Γ, where Γ is from the proof of Lemma 3.1, that
is, it is the supremum over λ > 0 of the Ḣ1 norms of the principal eigenfunctions of (3.13).
This is because of (3.2), the definition of κe, and the fact that
κe(λ)
= sup
(u · e)w2 dx
‖w‖22
Finally, we note that Γ < ∞ is possible — in the shear flow case it holds when there is
an open set U ⊆ Tn−1 such that α(x′) = maxTn−1 α for all x′ ∈ U . Then any w ∈ H1(Tn)
supported on T × U and independent of x1 belongs to I and maximizes (1.6) whenever
f ′(0) ≥ ‖∇w‖22/‖w‖22. Thus the limit in (1.6) need not be strictly increasing with f ′(0)
(which happens precisely when Γ <∞).
In the more general case when the second order term and the non-linearity depend on x,
we consider
Tt + Au · ∇T = ∇ · (a∇T ) + f(x, T ) (3.14)
with a 1-periodic real symmetric uniformly elliptic matrix and u 1-periodic such that
a ∈ C2(Tn), u ∈ C1,ε(Tn), ∇ · u ≡ 0,
u dx = 0. (3.15)
The non-linearity f is 1-periodic in x and satisfies for some ε > 0
f ∈ C1,δ(Tn × [0, 1]),
f(x, 0) = f(x, 1) = 0 and f(x, ·) is non-increasing on (1− ε, 1) for each x ∈ Tn, (3.16)
0 < f(x, s) ≤ sf ′s(x, 0) for (s, x) ∈ (0, 1)× Tn.
We let ζ(x) ≡ f ′s(x, 0) > 0 and ζ0 ≡
ζ(x) dx. Equations (3.3) and (3.4) are then replaced
by (see [4])
c∗e(A) = inf
κ(λ, f ;A)
∇ · (a∇ϕ)− Au · ∇ϕ− 2λe · a∇ϕ+ [λAu · e+ ζ + λ2e · ae− λ∇ · (ae)]ϕ = κ(λ, f ;A)ϕ.
If we now define
κe(λ, f) ≡ sup
(λu · e+ ζ)w2 dx− |||∇w|||22
‖w‖22
then (3.2) becomes
c∗e(A, f)
= inf
κe(λ, f)
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
c∗e(A)
= sup
|||∇w|||2
ζw2 dx
(u · e)w2 dx
‖w‖22
PULSATING FRONT SPEED-UP AND DIFFUSION ENHANCEMENT 11
In particular, the limit exists. Moreover,
c∗e(A, αf)
= sup
(u · e)w dx
|||∇w|||2
c∗e(A, αf)
= sup
(u · e)w2 dx
‖w‖22
≤ max
{u(x) · e}.
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	1. Introduction
	2. Effective Diffusivity Enhancement
	3. KPP Front Speed-up
	References
