Min-Max Variational Principle and Front Speeds in Random Shear Flows

0.16 MB·English

by  James Nolen

5 Min-Max Variational Principle and Front
0
0
Speeds in Random Shear Flows
2

n
a
J James Nolen Jack Xin
∗ †
5
2

]
P
A Abstract
.
h Speed ensemble of bistable (combustion) fronts in mean zero sta-
t
a tionary Gaussian shear flows inside two and three dimensional chan-
m
nels is studied with a min-max variational principle. In the small
[ root mean square regime of shear flows, a new class of multi-scale test

1 functions are found to yield speed asymptotics. The quadratic speed
v enhancement law holds with probability arbitrarily close to one under
5
the almost sure continuity (dimension two) and mean square H¨older
4
4 regularity (dimension three) of the shear flows. Remarks are made on
1 the conditions for the linear growth of front speed expectation in the
0
large root mean square regime.
5
0
/
h
t
a
m
:
v
i
X
r
a
∗Department of Mathematics, University of Texas at Austin, Austin, TX 78712
([e-mail]).
†Department of Mathematics and ICES (Institute of Computational Engineering and
Sciences), University of Texas at Austin, Austin, TX 78712 ([e-mail]).
1
1 Introduction
We consider propagation speeds of reaction-diffusion fronts in random shear
flows. The model equation is:
u = ∆ u+δB u+f(u), (1.1)
t x x
·∇
where x = (x ,x˜) D = R1 Ω, Ω = [0,L]n 1, n = 2,3; δ > 0 a scaling
1 −
∈ ×
parameter; the vector field B = (b(x˜,ω),0), b(x˜,ω) is a scalar stationary
Gaussian field with zero ensemble mean and almost surely continuous sample
paths; f is either one of the following:
(1) the combustion nonlinearity with ignition temperature cutoff: f(u) = 0,
u [0,µ], µ (0,1); f(u) > 0, u (µ,1), f(1) = 0, f C2([0,1];
∈ ∈ ∈ ∈
(2) the bistable nonlinearity: f(u) = u(1 u)(u µ), µ (0,1/2).
− − ∈
The boundary condition is zero Neumann at R1 ∂Ω. The initial data u
0
×
belongs to the set I . The set I for bistable f consists of bounded continuous
s s
functions with limits one and zero at x respectively. For combustion
1
∼ ±∞
f, one requires also that u decay to zero exponentially at . For each
0
−∞
realization of B, there exists a traveling front solution of the formu = Φ(x +
1
ct,x˜), with unique speed c and profile Φ [4]. For u I , u(x,t) converges to
0 s
∈
a traveling front at large times [14].
We are interested in the asymptotic behavior of c = c(δ) when δ is small
or large. Such speed asymptotics have been recently studied for both de-
terministic fronts [3, 5, 8, 10, 9, 11, 13] and KPP fronts in random shears
[12, 17]. See also [16] for related applied science literature. The bistable
(combustion) front speed c has a min-max variational characterization [7, 8].
Following the notation of [8], define the functional:
Lv + f(v) ∆v +δb(x˜)∂ v+f(v)
ψ(v) = ψ(v(x)) x1 . (1.2)
≡ ∂ v ≡ ∂ v
x1 x1
The strong form of the min-max variational formula of [8] for c(δ) is:
sup inf ψ(v(x)) = c(δ) = inf supψ(v(x)) (1.3)
v Kx D v Kx D
∈ ∈ ∈ ∈
where K is the set of admissible functions:
K = v C2(D) ∂ v > 0, 0 < v(x) < 1,v I .
{ ∈ | x1 ∈ s}
2
The variational principle (1.3) is a powerful tool for getting tight bounds
on c, if one can construct an admissible test function v that approximates
well the exact traveling front solution Φ. This is the case when δ is small and
the test function is a perturbation of the known traveling front solution when
δ = 0. In [8], a form of such test function is proposed for deterministic shear
flow, and the quadratic speed enhancement law ([13] and references therein)
is recovered if δ is small enough. To handle an ensemble of shear flows in the
random case, one must know which norm of b controls the smallness of δ by
analyzing an admissible test function v and the functional ψ(v). A new form
of multi-scale test function is found for this purpose, and it cures a delicate
divergence problem when perturbing the traveling front at δ = 0. It turns
out that the smallness of δ for the quadratic speed enhancement depends on
the maximum (H¨older) norm of b if n = 2(3).
Our main results are that the quadratic speed enhancement law holds
with probability arbitrarily close to one, under the almost sure continuity
(n = 2) or mean square H¨older regularity (n = 3) conditions of the shear. As
b can be arbitrarily large (though with small probability), our admissible test
functions are not valid for the entire ensemble. We are unable to discuss the
expectation of front speed. This is in contrast to the Kolmogorov-Petrovsky-
Piskunov (KPP) case (e.g. when f(u) = u(1 u)), where there is less
−
restriction on test functions and the minimal speed variational principle is
minimization over functions on the cross section Ω [12].
The paper is organized as follows. In section 2, the new multi-scale test
function is presented and used to estimate the front speed and identify the
smallness of δ in terms of proper norms of b. In section 3, probabilistic esti-
mates based on Borell’s inequality and Karhunen-Lo`eve expansion of mean
zero Gaussian fields help to control the maximum (H¨older) norms of b, and
lead to the quadratic speed enhancement laws. Remarks are made on the
linear front speed growth law when δ is large.
2 Deterministic Front Speed Asymptotics
Let us consider b to be a given H¨older continuous deterministic function,
and assume that < b > 1 b(y)dy = 0. Otherwise, the integral mean
≡ Ω Ω
| |
contributes a linear term δ < b > to the front speed. Let U = U(x +c t) be
R 1 0
the traveling fronts when δ = 0, satisfying the equation:
U c U +f(U) = 0,
′′ 0 ′
−
3
U( ) = 0, U(+ ) = 1, U > 0, U approaches zero and one at exponential
′
−∞ ∞
rates.
In the small δ regime, define the test function
v(x) = U(ξ)+δ2u˜(ξ,x˜), (2.1)
where the variable:
ξ = (1+αδ2)x +δχ, (2.2)
1
with α a constant to be determined and χ = χ(x˜) solution of:
∆ χ = b, x˜ Ω,
x˜
− ∈
subject to zero Neumann boundary condition at ∂Ω. We normalize χ by
setting χdy = 0. The test function (2.1) reduces to the one in [8] if
Ω
α = 0. However, we shall see that α = 0 is essential in suppressing certain
R 6
divergence arising in the evaluation of ψ(v), similar to curing secular growth
in multiple-scale perturbation expansions.
For H¨older continuous b(x˜), χ is C2+p, p (0,1). Assume that u˜ is C2,
∈
and decays sufficiently fast at ξ infinities. A straightforward computation
shows that:
Lv +f(v) = U +f(U)
′′
+δ2 2αU +∆ u˜+U χ 2 +f (U)u˜
′′ ξ,x˜ ′′ ′
|∇ |
+δ3(bαU +2 u˜ χ)
(cid:0) ′ x˜ ξ (cid:1)
∇ ·∇
+δ4(α2U +2αu˜ +∆ u˜ χ 2)
′′ ξξ ξ
|∇ |
+f(v) f(U) f (U)δ2u˜
′
− −
+δ5αbu˜ +δ6α2u˜ , (2.3)
ξ ξξ
where f(v) f(U) f (U)δ2u˜ = O(δ4u˜2) and the O(δ) terms cancel by choice
′
− −
of ∆χ = b. On the other hand, up to an undetermined constant γ, we have:
−
(c +δ2γ) ∂ v = c U +δ2(c αU +c u˜ +γU )
0 x1 0 ′ 0 ′ 0 ξ ′
+δ4(γαU +c αu˜ +γu˜ )
′ 0 ξ ξ
+δ6(αγu˜ ). (2.4)
ξ
All terms in (2.3) and (2.4) are evaluated at (ξ,x˜). Let us choose u˜ to solve
the equation:
L¯u˜ = ∆ u˜ c u˜ +f (U)u˜ = χ 2U 2αU +γU +c αU , (2.5)
ξ,x˜ 0 ξ ′ ′′ ′′ ′ 0 ′
− −|∇ | −
4
subject to zero Neumann boundary condition at ∂Ω, and exponential decay
at ξ infinities. The solvability of (2.5) will be discussed later. Then we would
have
Lv +f(v) = (∂ v)(c +δ2γ)+R . (2.6)
x1 0 1
The remainder R is:
1
R = δ3A+δ4B +δ5D +δ6E +O(δ4u˜2), (2.7)
1
where
A = bαU +2 u˜ χ
′ x˜ ξ
∇ · ∇
B = α2U +2αu˜ + ∆ u˜ χ 2 γαU c αu˜ γu˜
′′ ξξ ξ ′ 0 ξ ξ
|∇ | − − −
D = αbu˜
ξ
E = α2u˜ αγu˜ . (2.8)
ξξ ξ
−
The constant γ is determined by a solvability condition for (2.5). The
right hand side must be orthogonal to the function U (ξ)e c0ξ which spans
′ −
the kernel of the adjoint operator L¯ v = ∆ v +c v +f (U)v. Thus,
∗ ξ,x˜ 0 ξ ′
1
γ (U )2e c0ξdξ = ( χ 2 +2α)U U c α(U )2 e c0ξdξdx˜
′ − ′′ ′ 0 ′ −
Ω |∇ | −
ZR | | ZD
c (cid:0) (cid:1)
= 0 χ 2 +2α (U )2e c0ξdξ c α (U )2e c0ξdξ
′ − 0 ′ −
2 h|∇ | i −
ZR ZR
where the braket denotes the integral average over Ω. We have performed
integration by parts once, the boundary terms decay to zero and the integral
(U )2e c0ξdξ converges for both bistable and combustion type nonlineari-
R ′ −
ties. It follows that equation (2.5) is solvable when
R
c c
γ = 0 χ 2dx˜ = 0 χ 2 , (2.9)
2 Ω |∇ | 2 h|∇ | i
| | ZΩ
regardless of the choice of α. For such γ, we may choose α = 1 χ 2 so
−2h|∇ | i
γ +c α = 0 and the u˜ equation becomes:
0
L¯u˜ = ∆ u˜ c u˜ +f (U)u˜ = χ 2 χ 2 U . (2.10)
ξ,x˜ 0 ξ ′ ′′
− h|∇ | i−|∇ |
(cid:0) (cid:1)
Note that the right hand side of (2.10) has zero integral average over Ω.
5
The solution u˜ of (2.10) can be expanded in eigenfunctions of the Lapla-
cian ∆ :
x˜
u˜(ξ,x˜) = u (ξ)φ (x˜),
j j
j∈Z+Xn−1∪{0}
χ 2 = a φ (x˜),
x˜ j j
|∇ |
j∈Z+Xn−1∪{0}
where Zn 1 denotes nonnegative integer vectors in Rn 1 with at least one
+− −
positive component; ∆ φ = λ φ with zero Neumann boundary condition
x˜ j j j
−
on ∂Ω. The eigenfunctions φj are normalized so that φj L∞ = 1, and
k k
the eigenvalues λ = O( j 2) for large j . Clearly, φ = 1 and λ = 0,
j 0 0
| | | |
a = χ 2 . The functions u solve the equations
0 j
h|∇ | i
L u = u λ u c u +f (U)u = (a δ a )U . (2.11)
j j ′j′ − j j − 0 ′j ′ j − j − j,0 0 ′′
The j = 0 equation has zero right hand side, and we take u 0. If
0
≡
α were zero, equation (2.5) implies that the right hand side of the j = 0
equation would be a U +γU . The general solution (zeroth-mode) for u
0 ′′ ′ 0
−
is a0ξU (ξ)+const.U (ξ). The linear factor ξ would render v(x) unable to
− 2 ′ ′
stay in the set K due to divergence at large ξ (and x ).
1
Now consider the equations for j = 0. As shown in [15], the operators L
j
6
are invertible in part due to λ > 0 for j = 0. As in Lemma 2.3 of [15], we
j
6
have the regularity estimate:
u C a , j = 0, (2.12)
j C2(R) 1 j
k k ≤ | | 6
where the constant C is independent of j and depends only on U and its
1
derivatives.
Next we show that for j = 0, the ratio uj is bounded by C a , where C
6 U′ 2| j| 2
is a positive constant independent of j. It suffices to prove a uniform bound
if the a factor is one. Consider the function w = βU u for some positive
j j ′ j
−
constant β. For some r > 0 sufficiently large, f (U) 0 whenever ξ r .
0 ′ 0
≤ | | ≥
Choose β as
C
1
β β , (2.13)
1
≥ ≡ min U (ξ)
ξ r0 ′
| |≤
so that by (2.12), w > 0 in the region ξ r . By equation (2.11) for u
j 0 j
| | ≤
and the fact that U solves L U = λ U < 0, we have
′ j ′ j ′
−
L w = βλ U +U . (2.14)
j j j ′ ′′
−
6
Using the fact that U is bounded by a constant multiple of U , and that
′′ ′
inf λ > 0,wemayfurtherincreaseβ ifnecessary toensurethatL w < 0
j=0 j j j
6
in the region ξ > r . Maximum principle implies that w > 0 for all ξ R.
0
| | ∈
Repeating the argument for u and taking into account the a factor give:
j j
−
u
j
| | C a , (2.15)
3 j
U ≤ | |
′
where the constant C depends only on U, and (2.15) holds uniformly in
3
j = 0. Inequality (2.15) says that there is no resonance when inverting the
6
operator L to find u , for j = 0, in other words u decays same as U at
j j j ′
6
infinities.
To improve the estimate above, we move the term f (U)u to the right
′ j
hand side of (2.11) which is then bounded by C a U , for C independent
3′| j| ′ 3′
of j. The left hand side operator becomes u λ u c u . An explicit
′j′ − j j − 0 ′j
formula can be written for u , and the asymptotics of λ imply that
j j
(i)
u (ξ,x˜) a
| j | C | j| , (2.16)
U (ξ) ≤ 4(1+ j 2)
′
| |
for a constant C depending only on U, where i = 0,1,2 denotes the order
4
of ξ derivatives.
Suppose that b is H¨older continuous, then Schauder estimates give [6]:
χ 2 C b 2, (2.17)
k|∇ | k1+p ≤ k kp
for some p (0,1), C = C(Ω), is the standard H¨older norm. For
p
∈ k · k
rectangular cross section Ω (dimension n 1), the φ ’s are trigonometric
j
−
functions. Then
a C b 2 (1+ j ) (1+p), p (0,1), (2.18)
| j| ≤ 5 k kp | | − ∈
for some constant C [18]. Combining (2.16), (2.17), and (2.18), we see that
5
the eigenfunction expansion of u˜
u˜(ξ,x˜) = u (ξ)φ (x˜) (2.19)
j j
j∈XZ+n−1
7
converges uniformly in (ξ,x˜). Moreover,
(i)
u˜ (ξ,x˜)
| ξ | u(i) (ξ) φ (x˜)/U
U (ξ) ≤ | j | | j| ′
′ j∈XZ+n−1
C C b 2 (1+ j ) (3+p) = C b 2, (2.20)
≤ 4 5k kp | | − 6k kp
j∈XZ+n−1
if n = 2,3. The mixed derivative term is bounded as:
(1)
u˜ (ξ,x˜) u (ξ)
|∇x˜ ξ | | j | j φ (x˜)
j
U (ξ) ≤ U | || |
′ j∈XZ+n−1 ′
C b 2 (1+ j ) (2+p) = C b 2, (2.21)
≤ 5k kp | | − 7k kp
j∈XZ+n−1
if n = 2,3. In case n = 2, Ω is an interval, then b L (Ω) suffices, because
∞
∈
χ is Lipschitz, so a O( b 2 j p) for some p (0,1) which replaces
x˜ j −
∇ | | ≤ k k | | ∈
estimate (2.18). The exponent in ∞(2.21) goes down to 1+p, yet enough for
convergence on j Z1. As a result, for n = 2, i = 0,1,2, the estimates:
∈ +
(i)
u˜ (ξ,x˜) u˜ (ξ,x˜)
| ξ | C b 2 , |∇x˜ ξ | C b 2 , (2.22)
8 9
U (ξ) ≤ k k U (ξ) ≤ k k
′ ∞ ′ ∞
hold.
As a consequence of these estimates, let us verify the admissibility of
test function v. Clearly, v approaches zero (one) exponentially at x =
1
−∞
(+ ). As
∞
v = (1+δ2α)U +δ2(1+δ2α)u˜ ,
x1 ′ ξ
we have for n = 3:
1 1 1 C 1
v U δ2 u˜ δ2 6 b 2 U U > 0, (2.23)
x1 ≥ 2 ′ − 2 | ξ| ≥ 2 − 2 k kp ′ ≥ 4 ′
(cid:20) (cid:21)
if:
1 1
δ2 min , (C b 2) 1 . (2.24)
≤ 2 α 2 6k kp −
(cid:18) | | (cid:19)
For n = 2, in view of (2.22), v 1 U if
x1 ≥ 4 ′
1 1
δ2 min , (C b 2 ) 1 . (2.25)
8 −
≤ 2 α 2 k k
∞
(cid:18) | | (cid:19)
8
It follows from the monotonicity of v in x and its asymptotics near x
1 1
infinities that 0 < v < 1 for all x ,x˜.
1
With v being an admissible test function, we see from (2.6) that
Lv +f(v) R
ψ(v(x)) = = c +δ2γ + 1 , (2.26)
0
∂ v ∂ v
x1 x1
with R defined by (2.7). By (2.24) or (2.25), ∂ v 1U . Also, each term
1 x1 ≥ 4 ′
in R is bounded by a multiple of U . It follows that for δ satisfying (2.24) or
1 ′
(2.25), R1 4R1 C δ3, where C = C (U, b ),p (0,1), if n = 3,
|∂x1v| ≤ | U′| ≤ 10 10 10 k kp ∈
C = C (U, b ) if n = 2. We have shown:
10 10
k k∞
Proposition 2.1 Let n = 2 or 3, and let the shear flow profile b(x˜) be
H¨older continuous with exponent p (0,1) over rectangular domain Ω
∈ ⊂
Rn 1. There is a positive constant δ depending only on U and the H¨older
− 0
(maximum) norm of b for n = 3 (n = 2) such that if δ δ :
0
≤
c δ2
c(δ) = c + 0 χ 2dy +O(δ3). (2.27)
0
2 Ω |∇ |
| | ZΩ
A specific form of δ follows from (2.24) and (2.25).
0
At large δ, numerical evidence suggested that lim c(δ)/δ exists for
δ
→∞
bistable and combustion nonlinearities [11]. Such a limit holds for the KPP
nonlinearity [3].
3 Random Front Speed Asymptotics
Let us consider b = b(x˜,ω) as a stationary mean zero Gaussian field with
almost surely continuous sample paths in dimension one and two for n = 2, 3
respectively. Weareinterested intherestrictionofbover Ω. Inordertoapply
¯
results of the previous section on each realization of b, let us write b = b+b ,
1
where¯b = Ω 1 b(x˜,ω)dx˜. Correspondingly, c = c(δ,ω) = c δ¯b+c (ω),
| |− Ω 0− 2
where c depends on b . As the size of δ depends on either the maximum
2 1 0
R
norm or the H¨older norm of b which is an unbounded random variable, there
1
is no uniform way to choose a δ for all realizations. On the other hand, the
0
probability of the occurance of very large maximum (H¨older) norm of b
1
is often small, so the random speed asymptotics may hold with probibility
arbitrarily close to one. Let us consider n = 2 first.
9
Theorem 3.1 (Two dimensional channel) Let D = R1 [0,L], Ω =
×
[0,L], and b(x˜) be the restriction on Ω of a mean zero stationary, Gaus-
sian random process with almost surely continuous sample paths. Let f be
a bistable or combustion nonlinearity. Then for each small ǫ (0,1/4)
∈
and q (0,1), there is a deterministic constant δ = δ (ǫ,q) such that if
0 0
∈
δ (0,δ ),
0
∈
1 x˜ 2
Prob c(δ,ω) c +δ¯b δ2 b (y)dy κδ3 q < ǫ.
0 1 −
− − 2h i ≥
((cid:12)(cid:12) (cid:18)Z0 (cid:19) (cid:12)(cid:12) )
(cid:12) (cid:12)
(cid:12) (cid:12)
where the constant κ > 0 is independent of δ, ǫ, and q. As ǫ 0, the
(cid:12) (cid:12)
→
constant δ can be chosen to satisfy
0
2
δ0(ǫ,q) C log(ǫ) −q (3.1)
≥ | |
Proof: Recall the inequality (Lemma 3.1, p. 62, [2]):
E[sup b(x˜)] Esup b(x˜) E b(0) +2E[sup b(x˜)], (3.2)
≤ | | ≤ | |
Ω Ω Ω
and the Borell inequality on mean zero Gaussian process with almost surely
continuous sample paths (Theorem 2.1, p. 43, [2]):
µ E[sup b(x˜)] <
≡ ∞
Ω
P( sup b(x˜) µ > λ) 2e λ2/(2σ2), (3.3)
−
| − | ≤
Ω
where σ2 = E[b2]. With the choice of λ = 2σ2log(ǫ/4), inequality (3.3)
−
implies that
p
P( b > λ+µ) P(supb(x˜) µ > λ)+P( infb(x˜) µ > λ)
k k∞ ≤ Ω − − Ω −
4e λ2/(2σ2) ǫ. (3.4)
−
≤ ≤
We now see that for ǫ > 0 sufficiently small, λ > 1 and there is a constant
C independent of ǫ such that if
11
4
1 q 1
δ < δ (ǫ) = C C (3.5)
0 11 11
µ+λ ≤ µ+λ
(cid:18) (cid:19)
10