Full text of DOI:10.1002/cpa.10006

Stability of One-Dimensional Boundary Layers
by Using Green’s Functions
EMMANUEL GRENIER
AND
FRÉDÉRIC ROUSSET
Ecole Normale Supérieure de Lyon
Abstract
The aim of this paper is to investigate the stability of one-dimensional boundary
layers of parabolic systems as the viscosity goes to 0 in the noncharacteristic case
and, more precisely, to prove that spectral stability implies linear and nonlinear
stability of approximate solutions. In particular, we replace the smallness con-
dition obtained by the energy method [10, 13] by a weaker spectral condition.
ꢀ
c 2001 John Wiley & Sons, Inc.
1
Introduction
Let
1.1)
ε
ε
ε
(
∂ u + ∂ f (u ) − ε∂ u = 0
t
x
xx
ε
be a one-dimensional system of conservation laws. We assume ε > 0, u (t, x) ∈
N
R is a vector-valued function of t ≥ 0 and x ≥ 0. Let
A(u) = ∇ f (u)
be an N × N matrix. As usual we will assume that A can be symmetrized: There
−1
exist smooth matrices P and D, D being diagonal, such that A = P DP . We
will consider Dirichlet boundary conditions
ε
(1.2)
u (t, 0) = 0 for all t ≥ 0 .
System (1.1)–(1.2) must also be completed with initial conditions.
In this paper we are interested in the inviscid limit ε → 0 of (1.1) and (1.2). It
is well known that in the general case a boundary layer appears near x = 0 that can
√
be of size ε (characteristic boundary layers) or ε (noncharacteristic boundary);
see, for instance, [14]. We will focus here on the noncharacteristic case, which
was formerly investigated in particular in [10, 13], and will assume throughout this
paper that the eigenvalues of A(u) at x = 0 are bounded away from 0; see (5.1).
Communications on Pure and Applied Mathematics, Vol. LIV, 1343–1385 (2001)
ꢀ
c 2001 John Wiley & Sons, Inc.

1
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E. GRENIER AND F. ROUSSET
The first step of these studies is to construct an approximate solution of the form
M
M
X
X
(1.3)
uapp(t, x) =
εj uint, j (t, x) +
εj ub, j (t, ε x)
−1
j=0
j=0
int, j
b, j
(
with M large) where u (t, x) describes the interior behavior and u (t, X) with
−1
b, j
X = ε x are boundary layer profiles. We assume that u are rapidly decreasing
int, j
b, j
int, j
b, j
in X and that u (t, 0) + u (t, 0) = 0 for all t ≥ 0, and that u
and u are
b,0
smooth functions. It turns out that u is a solution of
ꢀ
ꢁ
∂ f u (t, 0) + u (t, X) = ∂ ub,0(t, X)
int,0
b,0
(
1.4)
with
1.5)
X
X X
ub, j (t, 0) = −u (t, 0) , ub, j →X→+∞ 0 .
int, j
(
This system of ordinary differential equations has a solution only for particular
int, j
int, j
values of u (t, 0), namely, if and only if u (t, 0) ∈ C where C depends only
on f and is locally a manifold near 0 ∈ C (see also [20] for the study in the vicinity
int,0
of an arbitrary point). The limit equation for u
is then simply
∂t uint,0 + ∂ f (u ) = 0 ,
int,0
(
1.6)
1.7)
x
(
uint,0(t, 0) ∈ C .
The existence of solutions to this system and the existence of the higher-order
profiles are quite standard, and we will not go into detail; see [10, 13] for more
details.
The second step in the study of the inviscid limit of (1.1) and (1.2) is to prove
app
that the true solution is close to the approximate one, namely, to control u − u
M
by ε . One way is to use energy methods and weighted Sobolev norms. This has
been done in the general case in [13] under a smallness assumption on u (t, 0)+
int,0
b,0
u
(t, x/ε), which leads to the following:
THEOREM 1.1 (Stability under smallness condition) [13] Let M be large enough,
app
and let u be an approximate solution of the form (1.3) on [0, T ], with
uapp + ∂ f (u ) − ε∂xx u = R
app
app
app
where kR kL
app
2
≤ Cε
M/2
.
∂
t
x
ε
app
Let u be the solution of (1.1)–(1.2) with initial data u (0, ·). Then
ε
app
∞
2
u − u → 0 in L ([0, T ], L )
provided
1.8)
Z
1
b,0
(
sup
|z∂zu (t, z)|dz ≤ C0
0
≤t≤T
0
where C depends only on f .
0

STABILITY OF BOUNDARY LAYERS
1345
ε
app
In fact, we have an even better convergence, since basically u − u
→ 0
∞
s
in every space of the type L ([0, T ], H ) provided M is sufficiently large with
respect to s.
The main drawback of this result is the smallness assumption (1.8). Note, how-
ever, that the theorem is false if we authorize large boundary layers (see a sim-
ple example of instability in [13]) and in some cases the boundary layer can be
linearly unstable, even in physical cases [21]. Therefore a smallness assumption
cannot be avoided in general, but condition (1.8) is too strong and the constant C0
is far from being optimal. An explicit example of such a gap between condition
(1.8) and instability is given by the Ekman layers, which appear in the theory of
rotating viscous flows (not a hyperbolic system, but a very close and physically
relevant problem). For rotating fluids the smallness condition can be expressed in
terms of Reynolds numbers. Condition (1.8) gives stability for Reynolds less than
4, whereas linear (and nonlinear) instability only appear for Reynolds larger than
∼
55 [18]. In between the boundary layers are linearly stable, but this cannot be
proven through an L -estimate.
2
More precisely, let us consider the linearized equations of (1.1)–(1.2) in the
boundary layer, frozen at some time τ
ꢀ
ꢁ
0
int,0
b,0
(
1.9)
∂t v + ∂X f (u (τ, 0) + u (τ, X))v − ∂ v = 0 ,
X X
(1.10)
v(t, 0) = 0 .
If uint,0 is small enough, then v = 0 is a stable solution of (1.9)–(1.10), and this can
int,0
be proven through energy estimates. If u
is too large, v = 0 may be unstable;
int,0
see [12] for related works. However, there is a range of u
where 0 is stable for
(
1.9)–(1.10), but where this cannot be proven by energy methods. The reason is that
2
all the solutions of (1.9)–(1.10) go to 0 in the L -norm (this is the same as saying
that v = 0 is stable as t → +∞), but not in a monotonic way: Their energy may
increase before ultimately decreasing and going to 0. This is not possible if (1.8)
is satisfied, since then all the linear perturbations have a decreasing energy, but this
is possible in an intermediate regime, where (1.8) is not satisfied, but where linear
instabilities have not yet appeared. In such cases one could expect that Theorem 1.1
would be true. To establish this fact is the aim of the paper.
Let us formalize a little the notion of stability of 0 for (1.9)–(1.10). Let us
introduce the linear operator
ꢀ
ꢁ
0
int,0
b,0
(1.11)
L v = −∂X f (u (τ, 0) + u (τ, X))v + ∂
v
X X
τ
with Dirichlet boundary condition. Let Sp(L ) be the set of the eigenvalues of L .
τ
τ
We will say that L is spectrally stable if all the eigenvalues of L lie in the open
τ
τ
half-plane <λ < 0.
ε
app
One natural condition for the convergence of u − u would be that for every
τ, L is spectrally stable. It is, however, not quite sufficient, because 0 plays a
τ
particular role and can be a “generalized eigenvalue” if we follow [16, 25]. To

1
346
E. GRENIER AND F. ROUSSET
formalize this notion, we introduce the so-called Evans function E(τ, s) associated
with L . The Evans function is a Wronskian of solutions of
τ
(1.12)
sv + L v = 0 .
τ
More precisely, we consider a continuous basis v (s, X), . . . , v (s, X) of the
1
N
solutions of (1.12) satisfying v(0) = 0 and a continuous basis v
(s, X), . . . ,
N+1
v (s, X) of the solutions of (1.12) satisfying v → 0 as X → +∞. We define
2
N
E(τ, s) =
ꢂ
ꢃ
v (s, X ) · · ·
v (s, X )
v
(s, X0) · · ·
v (s, X )
2N 0
1
0
N
0
N+1
det
∂
v (s, X ) · · · ∂ v (s, X ) ∂ v
(s, X ) · · · ∂ v (s, X )
X
1
0
X
N
0
X
N+1
0
X
2N
0
where X is some fixed number. As usual for Wronskians, up to an exponential
0
renormalization factor, the value of E does not depend on X , and more generally
0
only the zeros of E have a meaning.
If there exists s
6
= 0 with <s ≥ 0 such that E(τ, s) = 0, then s ∈ Sp L (more
τ
precisely, s is an eigenvalue of L ). However, E(τ, 0) could vanish with 0 not an
τ
eigenvalue (neutral case). It is likely that the neutral case is linearly unstable, and
we refer to [25] for a proof in the nearby case of viscous shock profiles where the
∂
neutral case is ∂s E(τ, 0) = 0 since the translation invariance implies that 0 is an
eigenvalue.
We will say that L is “Evans stable” if E(τ, s) has no zero in <s ≥ 0. In
τ
particular “Evans stable” implies “spectrally stable,” but the converse can be false.
int,0
b,0
Values of u
such that the solution u of (1.4)–(1.5) leads to Evans-stable
L are called Evans-stable. Their set is denoted by C . The theory of Evans
τ
s
functions of viscous perturbations and discrete approximations of systems of con-
servation laws has been much developed recently, and we refer in particular to
[1, 2, 3, 9, 11, 21, 22, 26]. A good introduction is provided by the lecture notes
[24].
The aim of this paper is to prove the following:
app
THEOREM 1.2 (Stability under spectral condition) Let M ≥ 3 and let u be an
approximate solution of the form (1.3) on [0, T ] with
app
app
app
app
app
M
∂
u
+ ∂ f (u ) − εu = R
where kR kL
1
≤ Cε .
t
x
0
int,0
Let us assume that f (u (t, x)) is diagonalizable in a smooth way and that the
int
eigenvalues λ (t, 0) are uniformly bounded away from 0. Let us also assume that
j
there exists ω > 0 and a constant C such that
b,0
b,0
(1.13)
|u (t, X)| + |∂t u (t, X)| + |∂ ub,0(t, X)| ≤ Ce−ωX
X
ε
for X ≥ 0. Let L be defined by (1.11). Let u be the solution of (1.1)–(1.2) with
τ
app
initial data u (0, ·). Then
ε
app
∞
1,1
u − u → 0 in L ([0, T ], W
)

STABILITY OF BOUNDARY LAYERS
1347
provided
1.14)
(
uint,0(t, 0) ∈ Cs for all t ≥ 0 .
Note that this theorem contains Theorem 1.1 as a particular case (up to a nonrel-
2
1
evant change of L -type to L -type Sobolev spaces), and that it is sharp: If (1.14) is
not satisfied, linear and nonlinear instabilities do appear as in [6]. It seems possible
∞
M−2,1
M−2
to improve the convergence and get L ([0, T ], W
) with speed ε .
The proof relies on a very precise study of the Green’s function of the linearized
system of (1.1)–(1.2) near u , and the main difficulty is to prove the following:
app
THEOREM 1.3 (Green’s function) Under the assumptions of Theorem 1.2 and
(1.14), the solution of
0
app
∂
t
v + ∂ ( f (u )v) − ε∂ v = w
x
xx
with initial condition v (x) is given by
0
Z
ZZ
ε
ε
v(t, x) = G (0, y; t, x)v (y)dy +
G (τ, y; t, x)w(τ, y)dy dτ
0
ε
where the Green’s function G , which is independent of v and w, satisfies
0
Z
Z
+∞
T
+∞
ε
(
1.15)
1.16)
|G (τ, y; t, x)|dx dt ≤ C (T ) ,
G
0
0
Z
Z
T
C (T )
G
ε
(
|∂ G (τ, y; t, x)|dx dt ≤
√
,
x
ε
0
0
C (T ) being independent of ε and y and locally bounded in T .
G
The proof of Theorem 1.3 cannot be obtained through energy estimates. It
requires a very careful analysis of all the phenomena that occur in the system: cre-
ation and propagation of waves by the hyperbolic part, interactions of the waves
app
with u , and creation and damping of the waves in the boundary layer. The diffi-
cult part is the precise description of what happens when a wave coming from the
interior of the domain hits the boundary. Two phenomena occur: A part is damped
by viscosity and a second part is reflected and reenters the domain, going to the
right. A key point of the analysis is that it takes some time for this reflected wave
app
to interact with u and create new waves that go to the left; hence no resonance
phenomena take place.
The Green’s function is obtained through an iterative procedure (fixed point
1
2
argument), starting from four elementary approximate Green’s functions G , G ,
3
4
1
G , G ; G describes the creation and propagation of ingoing waves in the vicin-
2
ity of the boundary, G the propagation of ingoing waves created away from the
3
4
boundary layer, G the propagation of outgoing waves, and G the dynamics of the
boundary layer. The technical mathematical calculation part of the proof consists
1
4
of a careful analysis of how error terms from G , . . . , G are treated at the next
1
4
step of the iteration by G , . . . , G . Because of this we have to estimate a large

1
348
E. GRENIER AND F. ROUSSET
number of elementary integrals involving Gaussians and exponentials in terms of
cutoff parameters T and ε.
The first step is to construct the Green’s function of (1.1) and (1.2) “away from
the boundary.” We use an approximate Green’s function as in [19] and [23] (The-
orem 3.1). The second step (Theorem 4.1) is to construct the Green’s function “in
the boundary layer,” namely, the Green’s function of ∂ − L . We will follow very
t
τ
closely the work of P. Howard and K. Zumbrun [25], who build a similar Green’s
function in the case of shocks. The main idea is to use the Laplace transform to
reduce the problem to the study of systems of second-order ordinary differential
equations, a study fulfilled with the help of the Wronskian of the system (the so-
called Evans function). We also refer the reader to [16] for a different approach.
Note that this construction is quite classical in physics, for instance, in fluid me-
chanics (see, for instance, [4, 8] and the corresponding chapters in [7]) or in plasma
physics (Landau damping effect [17]).
The third step is to combine these two Green’s functions (and in fact to split
them into four different approximate Green functions) to build the Green’s func-
tion of the linearized system of (1.1) and (1.2) following the strategy detailed in
Section 2 (Theorem 2.1). This requires following very precisely the behavior of
the various waves that propagate. Once we have linearized stability, nonlinear sta-
bility is standard (see, for instance, [16]) and is detailed in the last section.
2
Iterative Constructions of Green’s Functions
Let us first describe in an abstract manner a general way to construct Green’s
functions of parabolic-type problems, starting from approximate Green’s functions
ε
that solve approximate problems. Let L be a sequence of operators (we do not
detail their form), and let us consider the problem
ε
(
2.1)
2.2)
∂ u + L u = u˜ ,
t
(
u(0, x) = u (x) ,
0
d
u˜(t, x) and u (x) being given functions with values in R . We will skip the ε-
0
dependence in the notation in this section.
We split u˜ in N different parts, each part being treated by a different approx-
∞
imate Green function. Let 5 (for 1 ≤ k ≤ N) be smooth functions C (R ×
k
+
d
R , L(R )) such that
+
X
d
¯
k5 (t, x)vk ≤ Ckvk ∀x ≥ 0, ∀v ∈ R , and
5 = Id .
k
k
k
Let S (τ, y; t, x) (1 ≤ k ≤ N) be N different kernels (which in general depend
k
on ε), which define N different approximate solutions, corresponding to each 5
k
Z Z
t
Skv(t, x) =
Sk(τ, y; t, x)v(τ, y)dy dτ .
0

STABILITY OF BOUNDARY LAYERS
1349
Basically S will solve (2.1) in different areas of space or will solve only a part of
k
the equations. We assume that for all k,
Z
Z
Z
Z
T
T
√
sup
|S (τ, y; t, x)|dx dt + ε sup
k
|∂ S (τ, y; t, x)|dx dt
x k
0
≤τ≤T,y
0
0≤τ≤T,y
0
is bounded uniformly in ε for some T > 0 and that S = 0 for t < τ. We then
k
define
N
X
S =
Sk5k
k=1
d
and assume that ∀e ∈ R , ∀y > 0,
N
X
lim
S (τ, y; t, x)5 (τ, y)e = δ e .
k
k
y
+
t→τ
k=1
Let R be the associated errors defined by
k
R v = ∂ (S v) + L(S v) − v .
k
t
k
k
We will assume that the R are of the form
k
Z Z
t
R v(t, x) =
Rk(τ, y; t, x)v(τ, y)dy dτ .
k
0
We define
N
X
R =
R 5 .
k
k
k=1
Let S and RM be the M iterates of S and R, with corresponding kernels S
M
th
(M)
(
M)
and R
.
THEOREM 2.1 If
Z
Z
T
(
1)
sup
|R (τ, y; t, x)|dx dt < +∞
τ,y
0
and if there exists M ≥ 1 and C < 1 such that
0
Z
Z
T
(
M)
(2.3)
sup
|R (τ, y; t, x)|dx dt ≤ C < 1 ,
0
τ,y
0
ε
then there exists a Green’s function G (τ, y; t, x) for (2.1)–(2.2) on [0, T ] such
that
Z
Z
Z
Z
T
T
√
ε
ε
sup
|G (τ, y; t, x)|dxdt + ε sup
|∂ G (τ, y; t, x)|dx dt
x
0
≤τ≤T,y
0
0≤τ≤T,y
0
is bounded uniformly in ε. The solution of (2.1)–(2.2) is given by
Z
R
Z Z
R
t
ε
ε
u(t, x) = G (0, y; t, x)u (y)dy +
G (τ, y; t, x)u˜(τ, y)dy dτ .
0
0

1
350
E. GRENIER AND F. ROUSSET
PROOF: We construct the solution of (2.1)–(2.2) by successive approximations.
First we lift the initial data by
Z
X
(
0)
u (t, x) =
S (0, y; t, x)5 (y)u (y)dy
k
k
0
k
with the corresponding error
Z
X
(
0)
u˜ (t, x) =
R (0, y; t, x)5 u (y)dy + u˜(t, x)
k
k 0
k
(
we also add the u˜-term). Then we define by iteration
(i+1)
(i)
(i+1)
(i)
u
= Su˜
and u˜
= Ru˜
.
(
M)
By definition of R
we have
Z Z
t
u˜ i+M)(t, x) =
(
R(M)(τ, y; t, x)u˜ (τ, y)dy dτ ;
(i)
0
hence
ku˜ i+M)k
(
1
1
≤ C ku˜ k
(i)
1
1
L ([0,T ],L )
0
L ([0,T ],L )
and moreover
√
ku i+1)k
(
1
1
+ k ε∂ u(i+1)k
1
1
≤ Cku˜ k
(i)
1
1
,
L ([0,T ],L )
x
L ([0,T ],L )
L ([0,T ],L )
P
(
i)
which leads to the convergence of the serial
constructed in a similar way by starting with Dirac initial data δ e at time τ. In this
u . The Green’s function can be
y
case the first approximate solution is
X
X
S (0, y; t, x)5 (0, y)e with error u˜ =
R (0, y; t, x)5 (0, y)e .
k
k
k
k
k
k
We then apply the previous construction to u = 0 and u˜.
ꢀ
0
Let us now detail two elementary cases of applying Theorem 2.1:
(
1) N = 1 and M = 1. A particular case arises when
Z
sup |R (τ, y; t, x)|dx ≤ C(τ, t)
1
y
where C is locally bounded in τ and t. Condition (2.3) is then satisfied
provided T is small enough.
(
2) N = M = 1, with the estimate
Z
√
C(τ, t)
sup |R (τ, y; t, x)|dx ≤ C(τ, t) + ε√
1
y
t − τ
where C is locally bounded. Then (2.3) is satisfied provided T is small
enough.

STABILITY OF BOUNDARY LAYERS
1351
M
The main drawback of Theorem 2.1 is the need to compute kR k. This can be
achieved through the following matrix computation: Let us define for 0 ≤ T ≤ T ,
1
2
Z
Z
T2
σ
k,l
=
sup
|5 (t, x)R (τ, y; t, x)|dt dx .
k
l
T1≤τ≤T2,y∈supp 5l T1
THEOREM 2.2 Let
M(T , T ) = (σ (T , T ))
,
1
2
k,l
1
2
1≤k≤N,1≤l≤N
M
and let T > 0. If there exists η such that T − T ≤ η implies M (T , T ) → 0 as
2
1
1
2
M → +∞, then the conclusion of Theorem 2.1 holds true on [0, T ].
1
1
M
PROOF: Let u ∈ L ([0, T ], L ). Let us bound k5 R uk
(
1
. First
k
L [0,η],L )
0
¯
k5 R uk = k5 ukL
L1
1
≤ CkukL
1
;
k
k
next
N
X
M
R 5 RM−1u ;
5
k
R u = 5k
l
l
l=1
therefore
N
N
X
X
M
(5 R )(5 RM−1u) ≤
σ k5 RM−1ukL
k5 R uk ≤
1
.
k
k
l
l
k,l
l
l=1
l=1
Therefore by iteration,




M
k5 R uk
L1
k5 uk
L1
1
1
.
.
.


.
.
.


M




≤ C¯M (0, η)
.
M
k5 R uk
L1
k5 ukL
1
N
N
Hence, as in the proof of Theorem 2.1, we get a Green’s function on [0, η]. We
can iterate the method and construct in a similar way Green’s functions on [η, 2η],
[
2η, 3η], and so on. By convolution we then get a Green’s function on the whole
interval [0, T ].
ꢀ
3
Green’s Functions on the Whole Line
Let us solve in this section the equation
3.1)
∂ u + (A(t, x)u) − ε∂ u = u˜(t, x)
with u(0, ·) = u (x), where u˜ and u are given functions and x ∈ R. We as-
(
t
x
xx
0
0
sume that A(t, x) is diagonalizable with eigenvalues λ (t, x), . . . , λ (t, x) and
1
N
that there exists P(t, x) smooth and invertible and D(t, x) smooth and diagonal
with A(t, x) = P(t, x)D(t, x)P− (t, x).
1

1
352
E. GRENIER AND F. ROUSSET
ε
THEOREM 3.1 Equation (3.1) has a Green’s function G such that
Z
Z
R
Z
Z
R
T
T
√
ε
ε
sup
|G (τ, y; t, x)|dx dτ + ε sup
|∂ G (τ, y; t, x)|dx dτ
x
0
≤τ≤T,y
0
0≤τ≤T,y
0
is bounded uniformly in ε. The solution of (3.1) is given by
Z
R
Z Z
R
t
ε
ε
u(t, x) = G (0, y; t, x)u (y)dy +
G (τ, y; t, x)u˜(τ, y)dy dτ .
0
0
PROOF: First let
v(t, x) = P−1(t, x)u(t, x) .
We have
3.2)
where v˜ = P u˜ and
−1
(
∂ v + D(t, x)∂ v − ε∂ v = v˜ + 2εP ∂ P∂ v + Bv + v˜
t
x
xx
x
x
−1
B = εP− ∂xx P − P−1∂t P − P−1∂x (P D) .
1
(
3.3)
th
Let us introduce the i characteristic X (τ, y; t) defined by
i
X (τ, y; τ) = y and ∂ X (τ, y; t) = λ (t, X (τ; y, t)) ,
i
t
i
i
i
characteristic of λ passing through y at t = τ.
i
We approximate the Green’s function of (3.1) by viscous transport on each
characteristic
ꢂ
ꢃ
N
X
2
1
(x − X (τ, y; t))
j
(
3.4) G (τ, y; t, x) = √
exp −
1
t≥τ
π
j
T
4
πε(t − τ)
4ε(t − τ)
j=1
th
where π is the projection on the j vector of the canonical basis. For t > τ,
j
(
3.5)
∂ G + D∂ G − ε∂ G = R
t
T
x
T
xx
T
where
R = diag(λ (t, x) − λ (t, X (τ, y; t)))∂ G .
i
i
i
x
T
Hence
−
1
˜
G + D∂ G − ε∂ G − 2εP ∂ P∂ G − BG = R
∂
t
T
x
T
xx
T
x
x
T
T
where
−1
˜
R = R − 2εP ∂ P∂ G − BG .
x
x
T
T
The main point is that, for Lipschitz-continuous λ , there exists a constant C such
i
that
Z
R
r
ε
˜
|
R(τ, y; t, x)|dx ≤ C + C
t − τ
uniformly in y and locally in τ and t. We end the proof by using an example of
application (ii) of Theorem 2.1.
ꢀ

STABILITY OF BOUNDARY LAYERS
1353
Note that in the original u-variable, (3.4) gives an approximate Green’s function of
the form
P(t, x)G (τ, y; t, x)P−1(τ, y) .
T
4
Green’s Function of a Pure Boundary Layer
4.1 Preliminaries
The aim of this section is to derive estimates on the Green’s function of the
following half-line problem:
(4.1)
(4.2)
(4.3)
∂ u + (A(x)u) − ∂ u = u˜ ,
t
x
xx
u(t, 0) = 0 for all t ,
u(0, x) = u (x) ,
0
where u and u˜ are given functions. We will assume that
0
˜
A(x) = A + A(x)
∞
(
4.4)
with
−
ωx
−ωx
,
˜
˜
kA(x)k ≤ Ce
,
k∂ A(x)k ≤ Ce
x
for some ω > 0. We will also assume that A can be diagonalized, with real
∞
eigenvalues λ , . . . , λ and eigenvectors r , . . . , r . We will get estimates on the
1
N
1
N
Green’s function G(t, y, x) of (4.1)–(4.3) under the assumption that
L = uxx − (A(x)u)x
is Evans-stable. The proof is strongly inspired by [16] and [25].
The method is, thanks to inverse Laplace transform, to write G(t, y, x) as an
integral on a path in the complex plane of the Green’s function G (y, x) of the
λ
elliptic problem



(L − λ)u = f
u(0) = 0
(4.5)


u(+∞) = 0 .
The advantage of working on Green’s functions instead of semigroup and resol-
vent is that the Green’s function G can be continued analytically into the essential
λ
spectrum of L. Poles of G are eigenvalues of L. Hence we can integrate on com-
λ
plex paths in {<eλ < 0} to get better estimates using Cauchy’s formula. Residues
due to poles in the right half-plane are sources of instability. Consequently, we will
suppose
(H1)
L has no eigenvalue of nonnegative real part.
Next, note that 0 is not an eigenvalue of L; hence, we need another kind of hypoth-
esis for 0. To explain this, we need to study more carefully the eigenvalue problem

1
354
E. GRENIER AND F. ROUSSET
associated to (4.5):


(L − λ)u = 0
u(0) = 0



u(+∞) = 0 .
This problem can be set as a first-order system
0
(
4.6)
w = A(x, λ)w ,
u
w
w
I
I
with conditions w (0) = 0, w(+∞) = 0, where w = (
0
) = ( ), and
u
I I
ꢂ
ꢃ
0
In
A(x, λ) =
.
0
A (x) + λI A(x)
n
Let A (λ) = A(+∞, λ). We still have
∞
−
ωx
(
4.7)
kA(x, λ) − A (λ)k ≤ Ce
, the constant C being independent of λ.
∞
ꢀ
ꢁ
0
In
where A (λ) =
∞
λIn A∞
The eigenvalues of A (λ) are
∞
q
2
j
λj ꢁ λ + 4λ
µ
j,ꢁ(λ) =
,
2
which are holomorphic in
ꢄ
ꢅ
2
λ
j

1
= λ ∈ C : λ ∈/ R or λ > − inf
.
j
4
Associated eigenvectors are
ꢂ
ꢃ
rj
(λ)rj
(4.8)
R
=
.
j,ꢁ
µ
j,ꢁ
When λ belongs to the right-hand side of the parabola
ꢆ
ꢇ
2
P = −β − 2iβλ , β ∈ R
j
j
and is not on the parabola, we have <eµ (λ) > 0 and <eµ (λ) < 0. In a vicinity
j+
j−
of 0, we find the following behavior:
(
4.9)
µ
µ
j,−(λ) ∼ λj
if λ < 0 ,
j
λ
(4.10)
j,−(λ) ∼ −λ if λ > 0 .
j
j
Consequently, as in [25], we can choose 8 (x, λ), . . . , 8 (x, λ) solutions of (4.6)
1
n
depending analytically on λ ∈ D(0, r), for r sufficiently small, such that
µj,−(λ)x
−ωx
(4.11)
8 (x, λ) = e
(R (λ) + O(e )) .
j
j,−
Note that 8 is decreasing when λ is on the right-hand side of the parabola P , but
j
j
may grow when λ crosses the parabola. We can also easily choose 8 , . . . , 8
solutions of (4.6) depending analytically on λ such that
n+1
2n
I
i
8
(x, 0) = 0 .

STABILITY OF BOUNDARY LAYERS
1355
Now we define a Wronskian for (4.6), also called an Evans function:
E(x, λ) = det(8 (x, λ), . . . , 8 (x, λ), 8 (x, λ), . . . , 8 (x, λ)) .
1
n
n+1
2n
When λ is on the right-hand side of all the parabolas P , λ is an eigenvalue of L if
j
and only if E(0, λ) = 0. Our second hypothesis will be
(
H2)
E(0, 0)
6= 0 .
This means that λ = 0 is not a kind of “generalized” eigenvalue.
Next, we choose r sufficiently small so that
(
4.12)
E(0, λ)
6= 0 ∀λ ∈ D(0, r) ,
since D is continuous at zero. Then we define

= {λ ∈ C : −θ ≤ arg(λ + η) ≤ θ}
π
for some η > 0 and θ ∈ ] , π[.
2
Thanks to (H1) and since L is sectorial, we can choose η and θ such that when λ
0
belongs to \D(0, r) :=  , λ is not an eigenvalue of L and λ is on the right-hand
side of all the parabolas P .
j
The aim of this section is to prove the following:
THEOREM 4.1 Suppose (H1) and (H2); then the Green’s function of (4.1) satisfies
ꢂ
ꢂ
ꢃꢃ
X
2
1
(x − λ t)
j
(
4.13)
4.14)
G(y; t, x) =
O √ exp −
rj
t
Ct
j,λj >0
ꢂ
ꢂ
ꢃꢃ
2
1
(x − y)
−
σt
+
O √ e exp −
,
t
Ct
ꢂ
ꢂ
ꢃꢃ
X
2
1
(x − λ t)
j
(
∂x G(y; t, x) =
O
exp −
rj
t
Ct
j,λj >0
ꢂ
ꢂ
ꢃꢃ
2
1
(x − y)
−σt
+
O
e
exp −
,
t
Ct
all the constants being locally bounded in y, uniformly bounded in t, x, and σ, and
C > 0.
This theorem is completely similar to the main result of [25], which deals with
shocks. The situation is in fact simpler since 0 is not an eigenvalue, but for the sake
of completeness we detail its proof.
Physically the theorem says that a small perturbation of the boundary layer
creates waves that propagate with (positive) characteristic speeds and enter into
the domain. These waves are damped by viscosity and have the shape of moving
Gaussians. There is also a remainder term that describes the behavior in small
time and also the long-time decay of the part of the perturbation remaining in the
boundary layer. The proof of this theorem can be skipped in a first approach.

1
356
E. GRENIER AND F. ROUSSET
4.2 Expression and Estimates of Gλ
We first give an expression of the Green’s function G (y, x) for λ ∈ D(0, r).
λ
We show
PROPOSITION 4.2
ꢂ
ꢃ
ꢃ
N
X
G (y, x)
λ
=
=
d (y, λ, ·)8 (x, λ) ,
x > y ,
j
j
∂
∂
G (y, x)
x
λ
j=1
ꢂ
2
N
X
G (y, x)
λ
−
d (y, λ, ·)8 (x, λ) , x < y ,
j
j
G (y, x)
x
λ
j=N+1
where d and 8 depend analytically on λ ∈ D(0, r). Moreover, d (y, λ) is uni-
j
j
j
formly bounded for y bounded, λ ∈ D(0, r), and 8 has the asymptotic behav-
j
ior (4.11).
I
PROOF: We use the functions 8 defined in the preliminaries as a basis of
j
n
solutions of (4.5). For e ∈ R , let G (y, x)e be the solution of (4.5) with f = δ e.
λ
y
We look for such a solution under the form
2
N
X
I
(
4.15)
4.16)
G (y, x)e =
α 8 (x) for x < y ,
j
j
λ
j=N+1
N
X
I
(
G (y, x)e =
β 8 (x)
for x > y ,
λ
j
j
j=1
where α and β are computed below. The continuity of G e at x = y and the
j
j
λ
condition on the jump of its derivative lead to the following system:
N
2N
X
X
I
j
I
j
β 8 (y) −
α 8 (y) = 0 ,
j
j
j=1
j=N+1
N
2N
X
X
I
0
I
0
β (8 ) (y) −
α (8 (y)) = e .
j
j
j
j
j=1
j=N+1
Let
M(λ, y) = (8 (y), . . . , 8 (y)) ;
1
2N
M is nonsingular for λ ∈ D(0, r) since its determinant E(λ, y) does not vanish.
We immediately get
t
−1
t
(4.17)
(β , . . . , β , −α , . . . , −α ) = (M(λ, y)) (0, e) ,
1
N
1
N
−
1
which ends the construction of G (y, x)e. Since M(λ, y) is bounded for λ ∈
λ
D(0, r) and y bounded, the proposition is proved for G . Note that the Green’s
λ
function decreases exponentially when λ is on the right-hand side of all the parabo-
las P , but that it may grow as soon as λ crosses one of the parabolas. We get the
j
result for ∂ G by differentiating (4.15) and (4.16) with respect to x.
ꢀ
x
λ

STABILITY OF BOUNDARY LAYERS
1357
Next we use general results on ordinary differential equations to build and esti-
mate G in  . We will show the following:
λ
2
PROPOSITION 4.3 There exist C > 0 and γ > 0 such that
ꢈꢂ
ꢃꢈ
ꢈ
ꢈ
G (y, x)
C
1/2
λ
−γ |λ| |x−y|
e
1/2
∀x,y ∈ R+, λ ∈ 0 .
ꢈ
ꢈ
1
λ|
≤
ꢈ
ꢈ
∂
G (y, x)
λ
1
/2
x
|λ|
|
PROOF: To solve the nonhomogeneous problem (4.5), we put it into a first-
order system
4.18)
0
(
w = A(x, λ)w + F ,
0
f
where F = ( ). When λ is in  , λ is not on the parabolas. Hence A (λ) does
2
∞
not have eigenvalues on the imaginary axis. Consequently, the system
0
w = A (λ)w
∞
has an exponential dichotomy on R according to [5]. Moreover, using [15], we
+
can choose a projection P (λ) on the eigenspace associated to eigenvalues of neg-
∞
ative real part of A depending analytically on λ in  . Hence thanks to (4.7), the
theorem of [5] about roughness of exponential dichotomy says that system (4.6)
∞
2
also has an exponential dichotomy on R . Moreover, the projection P(λ) given by
+
the theorem of [5] also depends analytically on λ.
0
Note that every bounded solution of w = A(x, λ)w for λ in  is such that
2
w(0) ∈ RP(λ). Hence, since L does not have eigenvalues in  , we get
2
n
RP(λ) ∩ {0} × C = {0} .
n
˜
˜
We then modify analytically P(λ) into P(λ) such that N P(λ) = {0} × C . Using
[
15], we choose a basis (r (λ), . . . , r (λ)) of RP(λ) such that the r (λ) are analytic
1
n
i
in  . We then set
2
˜
P(λ) =
ꢂ
ꢃ
In 0n
−1
,
(
r (λ), . . . , r (λ), e , . . . , e )
(r (λ), . . . , r (λ), e , . . . , e )
1
n
1
n
1
n
1
n
0n 0n
n
˜
where (e , . . . , e ) is the canonical basis of {0} × C . Then as RP(λ) = RP(λ),
1
n
˜
system (4.6) has an exponential dichotomy with projection P(λ).
0
Call X(λ, x) the solution of w = A(x, λ)w such that w(0) = I . Then X
2
n
depends analytically on λ in  . The unique bounded solution of the nonhomoge-
2
I
neous problem (4.6) such that w (0) = 0 is
w(x) =
Z
Z
x
+∞
−1
−1
˜
˜
(I − P)(λ)X(λ, y) F(y)dy .
X(λ, x)P(λ)X(λ, y) F(y)dy −
0
x

1
358
E. GRENIER AND F. ROUSSET
Consequently, the Green’s function for the first-order system is
(
−
1
˜
X(λ, x)P(λ)X(λ, y)
,
x > y ,
x < y .
M(λ, y, x) =
−1
˜
−
X(λ, x)(I − P(λ))X(λ, y)
,
Set
ꢂ
ꢃ
M (λ, y, x) M (λ, y, x)
1
2
M(λ, y, x) =
.
M (λ(y, x) M (λ, y, x)
3
4
Coming back to the original formulation (4.5), we get G (y, x) = M (λ, y, x),
λ
1
∂
G (y, x) = M (λ, y, x). Hence we have constructed a Green’s function for the
x
λ
3
operator L − λ analytic in  .
2
We now have to find estimates for this function. Restricting λ to a bounded
0
subset of  , we get using the definition of exponential dichotomy in [5]
0
(
4.19)
kM(λ, y, x)k ≤ Ce−β|x−y| ∀x ≥ 0 , y ≥ 0 , λ ∈  ∩ {|λ| ≤ R} ,
where C and β > 0 depend only on R, which will be chosen big enough later.
0
It remains to estimate M(λ, y, x) when λ ∈  , |λ| ≥ R. We use the same
method as in [25]. We come back to the original problem (L − λ)u = f , and we
set
λ
x = |λ|1/2x , λ =
,
|
λ|
ꢂ
ꢃ
ꢂ
x
ꢃ
ꢂ
ꢃ
x
x
u(x) = u
,
A (x) = A
,
f (x) = f
.
1
/2
1/2
1/2
|
λ|
|λ|
|λ|
Then reducing the second-order equation to a first-order system, we get
!
!
!
ꢂꢂ
ꢃ
ꢂ
ꢃꢃ
0n
In
0
1
0
0n
In
0
0
w =
w +
=
+ O
w +
,
A
|
A
f
f
1
/2
+
λ
λIn 0n
|λ|
|λ|1
/2
λ|
|λ|
|λ|
0
where is now the derivative with respect to x. The O is uniform in x ∈ R , λ ∈ C.
+
The homogeneous system can be seen as a perturbation of
ꢂ
ꢃ
0n
In
0
w =
w .
λIn 0n
Eigenvalues of this system are ꢁ(λ)1/2, which are not on the imaginary axis when
0
λ ∈  . Hence this system has an exponential dichotomy. Choosing R such that
ꢈ ꢂ
ꢃꢈ
ꢈ
ꢈ
1
ꢈ
ꢈ
O
< 1 ,
ꢈ
1/2 ꢈ
|
R|
we get an exponential dichotomy on R for the perturbed system using again the
theorem of [5]. With the same trick as before, we can choose the analytic projection
+
n
P(λ) given by the theorem of [5] such that N (P(λ)) = {0} × C . Finally, we have
a Green’s function M(λ, y, x) such that
ꢄ
ꢅ
λ
0
(
4.20) kM(λ, y, x)k ≤ Ce−β1|x−y| ∀x, y ∈ R , λ ∈
, λ ∈  , |λ| ≥ R ,
+
|
λ|

STABILITY OF BOUNDARY LAYERS
1359
since λ varies in a bounded set.
Coming back into the original coordinates x and y, we get
ꢈꢂ
ꢃꢈ
ꢈ
ꢈ
G (y, x)
λ
ꢈ
ꢈ
(4.21)
1
λ|
≤
ꢈ
ꢈ
∂
G (y, x)
λ
1
/2
x
|
C
1/2
−
γ |λ| |x−y|
0
e
/2
∀x,y ∈ R , λ ∈  , |λ| ≥ R .
+
1
|
λ|
Finally, combining (4.19) and (4.21), the proposition is proved.
ꢀ
4.3 Proof of Theorem 4.1
The proof of Theorem 4.1 is based on the following inversion formula for the
Laplace transform:
Z
λt
(4.22)
G(t, y, x) = e G (y, x)dλ .
λ
∂

This formula is rigorously justified in [25]. In the following, we will use the ana-
lyticity of G and the Cauchy formula to choose optimal paths of integration. As
λ
the proof of the theorem is quite technical, we begin with a lemma.
LEMMA 4.4 For a sufficiently small β > 0, there exists C, σ > 0, and σ > 0
1
2
such that
ꢂ
ꢃ
ꢃ
−
σ1t
2
e
(x − y)
kG(t, y, x)k ≤ C √ exp −σ
,
,
2
t
t
ꢂ
−σ1t
2
e
(x − y)
k∂ G(t, y, x)k ≤ C
exp −σ
x
2
t
t
2
2
2
for any x, y, t ≥ 0 such that β |x − y| ≥ rt .
Remark 4.5. It suffices to find bounds of the form
2
2
1
(x−y)
1
(x−y)
−σ
−σ
e
t
t
√
e
for G and
for ∂ G
x
t
t
since
2
2
σr
(
x−y)
(x−y)
2t
− 2β
t
−
σ
−σ
2
t
e
≤ e
e
.
2
2
2
PROOF: Setting λ = β |x − y| /t , we slightly change the path of integra-
m
tion; we integrate on 0 = ∂({λ ∈ C : −θ ≤ arg λ ≤ θ} \ D(0, λ )). We get (see
m
Figure 4.1)
Z
Z
Z
G(t, y, x) =
+
+
= I + I + I ,
1
2
3
0
−
0+
C
and similarly we write
G(t, y, x) =
Z
Z
Z
∂
+
+
= J + J + J .
x
1
2
3
0
−
0+
C

1
360
E. GRENIER AND F. ROUSSET
ꢂλ
0
+
∂

C
θ
S
1
Pα
−η
<λ
S
2
0
−
FIGURE 4.1. Paths
iθ
Let us bound I . On 0 , we have λ = ρe and |λ| ≥ λ ≥ r. Hence we can
2
+
m
use Proposition 4.3. We get
Z
1/2
Z
+
∞
−γρ |x−y|
+∞ −ρ|cos θ|t
e
1/2
e
−ρ|cos θ|t
dρ ≤ Ce−γ λm
|x−y|
|
I2| ≤ C
e
dρ
1
/2
1/2
ρ
ρ
λm
λm
ꢂ
ꢃ
2
˜
C
1
(x − y)
≤
√ exp −γβ
,
1
/2
|
cos θ|
t
t
which is the expected bound using Remark 4.5. The bound for I is similar. The
1
same calculus gives the bound
Z
+
∞
1/2
−
ρ|cos θ|t −γρ |x−y|
C
e
e
dρ ,
λm
that is to say
ꢂ
ꢃ
2
˜
C
1
(x − y)
exp −γβ
,
1
/2
|
cos θ|
t
t
for J and J .
1
3
Let us bound I . On C we can still use estimates of Proposition 4.3; we get
3
ꢂ
ꢃ
2
1
/2
1
|x − y|
|x−y|
1
−γ λm |x−y|
eλmt
e
−γβ
t
|
I3| ≤ 2θCλ /2eλmt
e
= 2θβC √
√
.
m
t
t
X2
−X /2
2
We then use the classical estimate |X|e− ≤ C e
to get
1
2
1
γ
2
|x−y|
β(β−
)
t
|
I | ≤ 2θβCC √ e
.
3
1
t
We choose 0 < β < γ/2 to get an appropriate bound using Remark 4.5. Similarly,
we have
2
1/2
1
γ
2
|x−y|
λmt −γ λm |x−y|
β(β−
)
t
|
J | ≤ 2θCλ e
e
≤ 2θβCC1 t
e
.
3
m
This ends the proof of the lemma by again using Remark 4.5.
ꢀ
We now give the proof of the theorem.

STABILITY OF BOUNDARY LAYERS
1361
2
2
2
˜
PROOF: Only the case β (x − y) /t ≤ r remains. We take a new path 0. We
integrate along ∂(\D(0, r)). We still have
Z
Z
Z
C
Z
C
G(t, y, x) =
+
+
= I + I + I
1
2
3
˜
0
˜
−
0+
and
Z
Z
˜
0+
∂
G(t, y, x) =
+
+
= J + J + J .
x
1
2
3
˜
0
−
˜
Let us bound I . We can still use estimates of Proposition 4.3. On 0 , we have
2
+
iθ
λ = −η + ρe . We get
Z
+∞
1
1/2
−
ηt −ρ| cos θ|t
−γ˜ρ |x−y|
dρ ,
|
I2| ≤
e
e
e
/2
1
ρ
r˜
where r˜ is defined on Figure 4.1 and γ˜ = γ | sin θ| since |λ| ≥ |ꢂλ| = ρ sin θ.
Moreover, again using Figure 4.1, we see that r˜ = cr where c is a geometrical
2
2
2
constant; hence r˜ ≥ cβ (x − y) /t . This gives
Z
+
∞
2
˜
2
(
x−y)
1
C
(x−y)
I | ≤ Ce−
γ˜cβ
e
−ηt
e
/2
−ρ|cos θ|t
dρ ≤ √ e
−ηt −α˜cβ
e
,
t
t
|
2
1
ρ
t
cr
which is an appropriate bound. The integral I is similar. Moreover, we can per-
1
form the same calculus to get the bound
˜
2
C
(x−y)
−
ηt −α˜cβ
t
e
e
for J and J .
1 2
t
It remains to bound I and J . In D(0, r), we use the estimates of G given in
3
3
λ
Proposition 4.2. Assume y ≤ M. In the following C(M) will be a generic constant
that is independent of x and t.
There are two cases: x < y and x > y.
•
Case x < y.
This is the easier case. We have
ꢉꢂ
ꢃꢉ
ꢉ
ꢉ
G (y, x)
λ
ꢉ
ꢉ
≤
C(M) ∀λ ∈ D(0, r), 0 ≤ x < y ≤ M .
ꢉ
ꢉ
∂
G (y, x)
x
λ
We do not integrate on the circle; we keep on integrating along ∂. We get
Z
+
r˜
C(M)
−
ηt −|ρ||cos θ|t
dρ ≤ C(M)e−ηt
−ηt/2
e ,
|
I3| ≤ C(M)
e
e
≤
√
−r˜
t
and similarly
C(M)
J | ≤ C(M)e−
ηt
≤
−ηt/2
e .
|
3
t
Hence we get the right estimate using the next remark.

1
362
E. GRENIER AND F. ROUSSET
√
−ηt
2
2
2
Remark 4.6. When β (x − y) /t ≤ r, it suffices to find O(e / t) estimates for
−ηt
G and O(e /t) estimates for ∂ G, since
x
2
ηt
2
ηβ2
(x−y)
−ηt
≤ e−
− 2r
t
e
e
.
•
Case x > y.
Using Proposition 4.2, we get
ꢂ ꢃ
Z
N
X
I
J
3
λt
=
Kj where K = e d (y, λ)8 (x, λ)dλ .
j
j
j
3
j=1
C
It suffices to bound each K . Using (4.11), we get
j
<
e(µj−(λ))x
kd (y, λ)8 (x, λ)k ≤ C(M)e
∀x > y, 0 ≤ y ≤ M, λ ∈ D(0, r) .
j
j
Suppose λ < 0; then we have
j
−γ x
kd (y, λ)8 (x, λ)k ≤ C(M)e
≤ C(M) ∀x > y, 0 ≤ y ≤ M, λ ∈ D(0, r) ,
j
j
for some γ > 0, since r can be chosen so small that <(µ (λ)) < 0 for any λ in
j−
D(0, r). Hence using analyticity of d and 8 , we do not integrate on the circle;
j
j
we integrate on ∂. It is the same computation as in the case x < y. We find the
right estimate, again using Remark 4.6.
There remains the case λ > 0. In this case <µ (λ) becomes positive when λ
j
j−
crosses the parabola P .
j
Let us set α = (x − λ t)/2t. We want to integrate on the parabola P defined
j
α
by <µ (λ) = −α, which is the standard path for constant-coefficient parabolic
j−
equations in Laplace transform theory. On this parabola we have
2
λ = −(k + iα) − iλ (k + iα) , k ∈ R .
j
Fix ε > 0 sufficiently small.
If |α| ≤ ε, P intersects ∂ inside C. We then replace the integral on C by an
α
integral on P and integrals on the two segments S and S .
α
1
2
Let us bound the integral on P . We have
α
2
2
<
λt + <µ (λ)x = −k t − α t , |dλ| ≤ C ,
j−
since we integrate on a bounded set. Moreover, using (4.11), we have
!
Z
Z
Z
λt µj (λ)x
λt −ωx µj (λ)
=
e e
R (λ)d (y, λ)dλ + O
e e
e
dλ .
j
j
Pα
Pα
Pα
We first bound the second term
ꢉ
ꢉ
ꢉ
ꢉ
ꢉ
ꢉ
Z
Z
+∞
−α2t −ωx
−k2t
C(M)
2
−ωx µ (λ)x
j
−α t −ωx
ꢉ
ꢉ
e
e
dλꢉ≤ C(M)e
e
e
dk ≤
√
e
e
.
ꢉ
t
−∞
Pα

STABILITY OF BOUNDARY LAYERS
1363
Next we use the fact that x − a t ≥ −2εt; hence
j
−ωx
−ω(aj −2ε)t
−σt
,
e
≤ e
= e
since we can assume that ε is so small that a − 2ε > 0. Hence we have a bound
j
1
t
−σt
√
e
, which is the right bound for G using Remark 4.6. For the bound of ∂ G,
x
we use
1
1
≤ e
t
σt
−σt
− 2
√
e
.
t
It remains to bound
Z
eµj (λ)x R (λ)d (y, λ)dλ .
j
j
Pα
We use (4.8); hence we get for G a bound
Z
+
∞
2
2
C(M)
2
−
α t
−k t
−α t
C(M)e
e
dk ≤
√
e
−∞
t
along the vector r , which is the correct bound thanks to the definition of α. For
j
∂
G, we again use (4.8). The integrated function is along the vector µ (λ)r . Since
j j
x
|
µ (λ)| ≤ C|λ| ≤ C(|k| + |α|) ,
j
we get an estimate
Z
+
∞
2
2
C(M)
2
C(M)
+ √ |α|e
2
−
α t
−k t
−α t
−α t
.
C(M)e
(|k| + |α|)e
dk ≤
e
t
t
−∞
This is the expected bound thanks to the definition of α, and
2
2
2
X
Xe X ≤ Ce−
−
.
Let us bound the integrals on S and S . We are outside of the parabola P ;
1
2
α
hence <µ (λ) ≤ −α. Consequently,
j−
ꢉ
ꢉ
Z
Z
ꢉ
ꢉ
ꢉ
ꢉ
−ηt −αx
e .
ꢉ
ꢉ
+
ꢉ≤ 2r˜C(M)e
ꢉ
S1
S2
Since
ꢂ
ꢃ
ꢂ
ꢃ
x2
x
2
λ
j
−ηt − αx = −t
− λj + η ≤ −
− η t ,
2
t2
2t
8
−
σt
2
j
we get a bound of the form e , σ > 0, since we can always suppose λ /8−η > 0.
Since
C
≤ √ e
t
σ
2
C
σ
4
−
σt
−
t
−
t
e
≤ t
e
,
it gives the expected bound for G and ∂ G, again using Remark 4.6.
x

1
364
E. GRENIER AND F. ROUSSET
It remains to bound K when |α| > ε. Suppose α > ε; then we integrate on
j
the parabola P and on the two remaining segments S and S . Let us bound the
ε
1
2
integral on P . We have
ε
2
2
2
2
<
λt − (<µ (λ))x = ε t + ελ t − k t − εx ≤ −ε t − k t
j−
j
since (x − λ t)/2t ≥ ε. Consequently,
j
ꢉ
ꢉ
Z
Z
ꢉ
ꢉ
+∞
ꢉ
ꢉ
−tε2
−k2t
C(M)
−ε t
2
ꢉ
ꢉ≤ C(M)e
e
dk ≤
√
e
.
ꢉ
ꢉ
t
−∞
Pε
Hence we have a bound exponentially decreasing in time, and we get the expected
estimate using Remark 4.6 again.
Let us bound the integrals on S and S . On these segments, we have
1
2
<
µ (λ) ≤ −ε ;
j−
hence
ꢉ
ꢉ
Z
Z
ꢉ
ꢉ
ꢉ
ꢉ
−ηt
.
ꢉ
+
ꢉ≤ 2Cr˜e
ꢉ
ꢉ
S1
S2
Thanks to Remark 4.6, it is again the right bound.
In case α < −ε, we integrate on P and on the two remaining segments. For
−ε
the integral on P , we get
−
ε
2
2
2
2
<
λt + <µ (λ) = −k t + tε − tελ + εx ≤ −k t − ε t ,
j−
j
since (x − λ t)/2t ≤ −ε. Hence we finish as for the integral on P .
j
ε
For the integrals on the segments, we use <µ (λ) ≤ ε and
j−
−
ηt + εx ≤ (ελ − η)t
j
since
x − λ t
j
≤
−ε .
2
t
Because we can assume ελ − η < 0 by choosing ε sufficiently small, we have
j
a bound that decreases exponentially in time. Hence, again with the use of Re-
mark 4.6, the theorem is proved.
ꢀ

STABILITY OF BOUNDARY LAYERS
1365
ε
5
The General Case: Construction of G
5.1 Principle of the Construction
Let us now prove Theorem 1.3. We split f (uapp(t, x)) into an interior term, a
0
boundary layer term, and a remainder.Let
int
0
int
A (t, x) = f (u (t, x)) ,
ꢊ
ꢋ
ꢊ
ꢊ
ꢋꢋ
x
x
b
0
int
b,0
0
int
A t,
= f u (t, 0) + u
t,
− f (u (t, 0)) ,
ε
ε
ꢊ
ꢋ
x
int
b
A(t, x) = A (t, x) + A t,
,
ε
ꢊ
ꢋ
x
0
app
int
b
˜
A(t, x) = f (u (t, x)) − A (t, x) − A t,
.
ε
The aim of this section is to use Theorem 2.1 to derive estimates on the Green’s
function of the operator
ꢊꢊ
ꢊ
ꢋ
ꢋ ꢋ
x
int
b
˜
L u = −∂x A (t, x) + A t,
+ A(t, x) u + ε∂ u ,
ε
xx
ε
with Dirichlet condition at {x = 0}, which is a first step in the proof of Theo-
rem 1.3.
By assumption we can write
int
int
int
int
−1
,
A (t, x) = P (t, x)D (t, x)(P (t, x))
where Pint and Dint depend smoothly on (t, x). Moreover, A , P , and (P )
int −1
int
int
are uniformly bounded on [0, T ] × R as well as their first and second derivatives.
+
int
i
int
int
i
Let λ (t, x) be the coefficients of D (t, x). For x < 0 we define λ (t, x) =
int
i
int
i
int
i
λ (t, 0). Let X (τ, y; t) be the characteristics of λ (t, x) defined by
int
int
int
i
int
i
X (τ, y; τ) = y, ∂ X (τ, y; t) = λ (t, X (τ, y; t)) .
i
t
i
To begin, we will assume more than a noncharacteristic boundary. At first we
0
assume that there exist δ > 0 and 0 ≤ N ≤ N + 1 such that
(
int
−1
0
0
< δ < λ (t, x) < δ
for 1 ≤ i ≤ N
i
int
(5.1)
∀x ≥ 0
−
1
0
−
δ
< λ (t, x) < −δ < 0 for N + 1 ≤ i ≤ N
i
+
for t ∈ R . We make this assumption until Section 5.7. Let P be the projection
+
0
−
0
on the first N eigenvectors, and P be the projection on the eigenvectors N +
1
, . . . , N defined by
ꢁ
int
ꢁ
int
−1
P (t, x) = P (t, x)D (P (t, x))
+
0
where D = diag(1, . . . , 1, 0, . . . , 0) is a diagonal matrix with N unit coefficients
+
−
0
and D = diag(0, . . . , 0, 1, . . . , 1) has N + 1 − N unit coefficients such that
−
D + D = Id.

1
366
E. GRENIER AND F. ROUSSET
Note that
ꢊ
ꢊ
ꢋꢋ
x
0
int
0
int
0
int
b,0
˜
A(t, x) = − f (u (t, x)) + f (u (t, 0)) − f u (t, 0) + u
t,
ε
ꢊ
ꢊ
ꢋꢋ
ꢊ
ꢊ
ꢋꢋ
x
x
0
int
b,0
0
int
b,0
+
+
f u (t, x) + u
t,
− f u (t, x) + u
t,
ε
ε
0
app
f (u (t, x))
Z
Z
1
ꢀ
ꢁꢀ
ꢁ
0
0
int
int
int
int
int
=
−
f
f
u (t, 0) + µ(u (t, x) − u (t, 0)) u (t, x) − u (t, 0)
0
0
0
1
ꢊ
ꢊ
ꢋ
ꢋ
x
ꢀ
ꢁ
00
int
b,0
int
int
+
u (t, 0) + u
t,
+ µ u (t, x) − u (t, 0)
ε
int
int
(
u (t, x) − u (t, 0))
ꢊ
ꢊ
ꢋꢋ
x
int
b,0
+ f (uapp(t, x)) .
0
−
˜
f u (t, x) + u
t,
ε
Hence
ꢉ
ꢊ
ꢋꢉ
ꢉ
x
ꢉ b,0
int
int
(
5.2)
|A(t, x)| ≤ Cꢉu
t, ꢉ|u (t, x) − u (t, 0)| + C ε ≤ Cε
0
ε
for some constant C > 0, and similarly
−ωx/ε
˜
(
5.3)
|∂ A(t, x)| ≤ C ε + C e
.
x
0
0
5
.2 Construction of Gapp
We need to approximate the exact Green’s function by the Green’s function built
in Section 3 away from the boundary, and by the Green’s function constructed in
Section 4 near the boundary. This construction involves three cutoff parameters,
M , M , and M , with M > 4M and M > 4M > 0 and leads to numerous error
1
2
3
3
2
2
1
1
1
terms that have to be estimated in L ([0, T ], L ) norms in terms of T , ε, and the
three cutoff parameters.
In all the proof, C denotes constants that do not depend on M , M , or M . Let
1
2
3
χ(x) be a smooth function that equals 0 for x ≤ 1 and 1 for x ≥ 2. We define three
in
out
bl
approximate Green functions G (ingoing waves), G (outgoing waves), and G
(
boundary layer) as follows:
ꢂ x
ꢃ
in
Gint,in(τ, y, t, x)
G (τ, y, t, x) = χ
M ε
1
with
Gint,in(τ, y, t, x) = P (t, x)D G (τ, y, t, x)(P (τ, y))
int
+
int
−1
,
T
where G is defined as in (3.4),
T
ꢂ x
ꢃ
out
Gint,out(τ, y, t, x)
G (τ, y, t, x) = χ
M ε
1
with
Gint,out(τ, y, t, x) = P (t, x)D G (τ, y, t, x)(P (τ, y))
int
−
int
−1
.
T

STABILITY OF BOUNDARY LAYERS
1367
Note the truncation in x through multiplications by χ(x/M ε) to insure Dirichlet
1
boundary conditions.
bl
b
−1
Next we define G . Let G (ε y; t, x) be the Green’s function constructed in
Section 4 with
b
int
A(x) := A (τ, x) + A (τ, 0) .
We define
ꢂ
ꢂ
ꢃꢃ
x
bl
2
G (τ, y; t, x) = 1 − χ
G (τ, y; t, x)
M3ε
where
2
−1
b
−1
−1
−1
G (τ, y; t, x) = ε G (ε (t − τ), ε y, ε x) .
Basically we freeze the profile at time τ, look only at the boundary layer, and
−1
completely ignore what happens for x ꢃ ε. Note the ε factors that appear in the
2
scaling. Using Theorem 4.1, we get for G the following estimates:
ꢂ
ꢃ
X
2
C(y)
(x − λ (t − τ))
j
2
(
5.4)
|G (τ, y, t, x)| ≤
√
exp −
ꢂ
ε(t − τ)
C(y)
Cε(t − τ)
j,λj >0
ꢃ
ꢂ
ꢃ
2
t − τ
(x − y)
+
√
exp −σ
ꢂ
exp −
,
ε(t − τ)
C(y)
ε
Cε(t − τ)
ꢃ
X
2
(x − λ (t − τ))
j
2
(
5.5) |∂ G (τ, y, t, x)| ≤
exp −
x
ε(t − τ)
Cε(t − τ)
j,λj >0
ꢂ
ꢃ
ꢂ
ꢃ
2
C(y)
t − τ
(x − y)
+
exp −σ
exp −
.
ε(t − τ)
ε
Cε(t − τ)
Let us now define the projectors 5 and the kernels S . We take
k
k
in
out
bl
S = S = G , S3 = G
,
S = G ,
1
2
4
and
ꢂ
ꢂ
ꢃꢂ
ꢂ
ꢃꢃ
ꢂ
ꢃ
y
2y
2y
+
+
P ,
5
5
= χ
= χ
1 − χ
P , 5 = χ
1
2
M2ε
y
M3ε
M3ε
ꢃ
ꢂ
ꢂ
ꢃꢃ
y
−
P ,
5 = 1 − χ
.
3
4
M2ε
M2ε
5
.3 Bounds on the Error Terms of Gin
Let us compute the error term R defined by
in
ε
in
R = (∂ − L ) G .
in
t
int
+
+
T
To shorten the expressions, we abbreviate P as P and D G as G . We split
T
in
1
in
R into two terms, E and E , where
in
2
ꢂ
ꢃ
x
in
1
ε
int,in
E = χ
(∂ − L ) G
t
M1ε

1
368
E. GRENIER AND F. ROUSSET
is the evolution error and where
0
00
M12
0 ꢊ ꢋ
χ
χ
χ
x
in
∂x Gint,in − ε−1
Gint,in + ε−1
f 0(uapp)Gint,in
E = −2
2
M1
M
ε
1
is the truncation error. More precisely, we have
ꢂ
ꢃ
ꢂ x
ꢃ
ꢂ x
ꢃ
x
ꢀ
ꢁ
in
int
+
−1
,
E = χ
B1 + χ
B2 + χ
P D − DX ∂ G P
1
x
T
M1ε
M1ε
M1ε
+
where B is the sum of the terms in GT
1
+
−1
int
+
−1
+
−1
B = (∂ P)G P + ∂ (A P)G P − ε(∂ P)G P
1
t
T
x
T
xx
T
b
+
−1
+
−1
,
˜
+
∂ (A P)G P + ∂ (AP)G P
x
T
x
T
+
B is the sum of the terms involving ∂ G
2
x
T
+
−1
b
+
−1
+
−1
,
˜
B = −2ε∂ P∂ G P + A P∂ G P + AP∂ G P
2
x
x
T
x
T
x
T
and
int
i
D = diag(λ (τ, X(τ, y, t))) .
X
Note that E 1i n vanishes for x ≤ M ε, and E vanishes for x ≤ M ε and for
in
2
1
1
x ≥ 2M ε.
1
LEMMA 5.1 Let M ≥ M . We have
2
in
k1
E (τ, y, t, x)k
∞
1
≤ C (T + ε) + C ,
y≥Mε
1
L
,L
1
2
τ,y t,x
where C is independent of M , M , and M , and where C , which depends only
1
1
2
3
2
on M, goes to 0 as M → +∞. Moreover,
in
k1
E (τ, y, t, x)k
∞
1
≤ C (M , M) with lim C (M , M) = 0 .
y≥Mε
2
L
,L
3
1
3
1
τ,y t,x
M→+∞
in
1
PROOF: Let us bound the evolution error E . Using (1.13), (5.2), and (5.3),
ꢊ
ꢋ
1
−αx/ε
+
|
B | ≤ C 1 + e
|G | ;
1
T
ε
consequently, to bound kχ B k
∞
1
, we need to estimate uniformly in τ and y
1
L
,L
τ,y t,x
the following integrals:
2
2
(
x−X
j )
(x−X
j )
Z
Z
Z
Z
T
+∞
−
T
+∞
−
ε(t−τ)
ε(t−τ)
e
e
x
dx dt .
ε
−
α
ε
α
1
=
√
dx dt ,
α
2, j
=
√
e
, j
τ
M1ε
ε(t − τ)
τ
M1ε
ε(t − τ)
x−Xj
The easier one to estimate is α . Setting z = √
, we get
1
, j
ε(t−τ)
Z
Z
T
+∞
−
z2
α
≤
e
dz dt ≤ CT .
1
, j
τ
−∞
Next we estimate α . We set z = x/ε, s = (t − τ)/ε, Y = y/ε, and
2
, j
1
˜
X (τ, Y, s) = X (τ, εY, τ + εs) − Y .
j
j
ε

STABILITY OF BOUNDARY LAYERS
1369
We get
Z
Z
+∞
+∞
˜
)2
(
z−Y−X
j
dz ds
−
−αz
s
α
2
≤ C
e
e
√
.
, j
s
0
0
Hence using Lemma 7.1, as all the characteristics are going to the right, α2, j is
uniformly bounded in Y and goes to 0 when M → +∞.
Next, we bound kχ B k
∞
1
. Since
2
L
,L
τ,y t,x
ꢀ
x
ε
ꢁ
−α
+
|
B | ≤ C ε + e
|∂ G | ,
2
x
T
we need to estimate uniformly in (τ, y)
Z
Z
T
+∞
(x−X )2
|
x − X |
j
dx dt
j
−
ε(t−τ)
α
= ε
e
√
,
3
4
, j
, j
ε(t − τ)
x − X |
ε(t − τ)
τ
M1ε
Z
Z
T
+∞
(x−X )2
|
j
x
ε
dx dt
j
−
−α
ε(t−τ)
α
=
e
e
√
.
ε(t − τ)
ε(t − τ)
τ
M1ε
We use the same variable changes as previously and get
Z
Z
T
+∞
√
z2
dz dt
√
α
3
≤
ε
|z|e−
√
≤ C εT
, j
τ
−∞
t − τ
and
Z
Z
+∞
+∞
˜
2
(
z−Y−X
j )
|
z − Y − x˜|
dz ds
−
−αz
s
α
4
≤ C
e
e
√
,
, j
s
s
0
0
which is uniformly bounded in Y ≥ 0 and tends to 0 when M → +∞ by
Lemma 7.1. Next we have to bound the error in the transport
ꢂ
ꢃ
x
ꢀ
ꢁ
int
+
−1
,
χ
P D − DX ∂ G P
x
T
M1ε
√
which is bounded as in Theorem 3.1 by C(T + εT ).
in
It remains to bound the truncation error E . To this end we have to estimate
2
two types of integral,
2
2
(
x−X
j )
(z−Y−X˜
s
j )
Z
Z
Z
T
M1ε
−
+∞
M1
−
e
ε(t−τ)
1
e
1
α
α
=
=
√
dx dt ≤ M
√s
dz ds ,
5
, j
, j
M1ε
ε(t − τ)
1
τ
1
M ε/2
0
1
M /2
Z
T
M1ε
(x−X )2
1
|x − X |
j
dx dt
j
−
ε(t−τ)
e
√
6
M1
ε(t − τ)
ε(t − τ)
2
)
τ
1
Z
+
M1
˜
˜
(x−Y−X
1
|x − Y − X |
j
dz ds
j
−
s
≤ M
e
√
.
s
s
1
0
1
M /2
Lemma 7.1 then ends the proof.
ꢀ

1
370
E. GRENIER AND F. ROUSSET
5
.4 Bounds on the Error Terms of Gout
This section is very similar to the previous one; however, the conclusions are
out
1
out
different. We define the error terms R , E , and E2 analogously to the corre-
out
sponding “in” terms.
LEMMA 5.2 We have
out
k1y≥M2ε E1 (τ, y, t, x)kL
∞
1
≤ C (T + ε) + C ,
,L
4
5
τ,y t,x
where C is independent of M , M , and M and where C → 0 as M → +∞.
4
1
2
3
5
1
Moreover,
out
k1
E2 (τ, y, t, x)kL
∞
1
≤ C6
y≥M2ε
,L
τ,y t,x
with C locally bounded in M .
6
1
PROOF: Let us bound the evolution error E 1o ut. The terms α1, j and α3, j can be
treated as in the previous section, but using Lemma 7.1 we can only bound α2, j
and α4, j by constants that go to 0 as M → +∞, and similarly for the truncation
1
in
error E .
ꢀ
2
5
.5 Bounds on the Error Terms of Gbl
b
bl
Let R be the error term for G . First we split it into an evolution error and a
truncation error,
where
b
ˆ
ˆ
R = E + E ,
1
2
ꢂ
ꢂ
ꢃꢃ
x
ε
2
ˆ
E = 1 − χ
(∂ − L ) G
t
1
M3ε
is the evolution error and
ꢂ
ꢂ x
ꢃ
ꢂ x ꢃꢃ
1 χ0
M3ε
f (u ) +
1
G2
0
app
00
χ
ˆ
E = −
2
2
M ε
M ε
M ε
3
3
3
ꢂ
ꢃ
2
x
0
∂x G2
+ M
χ
3
M3ε
in
ˆ
ˆ
is the truncation error. We split E into two parts: the ingoing part E and the
2
2
out
ˆ
ˆ
outgoing part E2 , which are defined as the projections of E onto the ingoing and
2
outgoing modes
in
2
+
out
2
−
ˆ
ˆ
ˆ
E
ˆ
E = P E ,
= P E .
2
2
ˆ
ˆ
Note that E has its support in [M ε, 2M ε], and E its support in [0, 2M ε].
2
3
3
1
3
We shall show the following:
LEMMA 5.3 We have
ˆ
(
i)
k1y≤2M2ε E (τ, y, t, x)k
∞
1
≤ C (T + ε) ,
≤ C8 ,
1
L
L
,L
7
τ,y t,x
in
ˆ
(ii)
k1y≤2M2ε E (τ, y, t, x)k
∞
1
2
,L
τ,y t,x

STABILITY OF BOUNDARY LAYERS
out
1371
ˆ
E
(iii)
k1y≤2M2ε
(τ, y, t, x)kL
∞
1
≤ C + C T ,
2
,L
9
10
τ,y t,x
where C is locally bounded in M and M , C and C are locally bounded in M
7
2
3
8
10
2
(
uniformly in M ), and C , which depends on M and M , goes to 0 as M → +∞.
3
9
2
3
3
ˆ
PROOF: Let us prove (i). We write E under the form
1
ꢂ
ꢂ
ꢃꢃ
x
ꢀ
ꢁ
b
b
ˆ
ˆ
ˆ
E = 1 − χ
B G + B ∂ G
,
1
1
2
x
M3ε
where
ꢊ
ꢊ
ꢋ
ꢊ
ꢋꢋ
1
x
x
int
∂x Ab t,
b
ˆ
˜
B = ∂ A (t, x) +
− ∂ A τ,
+ ∂ A(t, x) ,
1
x
x
x
ε
ε
ε
ꢊ
ꢋ
ꢊ
ꢋ
x
x
int
b
int
b
ˆ
˜
B = A (t, x) + A t,
− A (τ, 0) − A τ,
+ A(t, x) .
2
ε
ε
We will first bound
ꢈꢂ
ꢈ
ꢂ x ꢃꢃ
ꢈ
ꢈ
ꢈ
ꢈ
b
ꢈ
ˆ
1
− χ
B G
.
1
ꢈ
M ε
3
∞
,L
1
L
τ,y t,x
Using (5.4), (5.2), and
ꢂ
ꢃ
ꢉ
ꢉ
t − τ
−
αx/ε
ꢉˆ ꢉ
B ≤ C 1 +
e
1
ε
leads to the bounds
Z
Z
Z
Z
Z
T
T
T
T
2M3ε
(x−λ (t−τ))2
1
j
−
−
ε(t−τ)
γ
1
=
=
=
=
√
e
e
dx dt ,
, j
τ
τ
τ
τ
0
0
0
0
ε(t − τ)
Z
Z
Z
2M3ε
2
1
t−τ
(x−y)
σ
−
ε
ε(t−τ)
γ
γ
√
e
dx dt ,
1
,r
ε(t − τ)
2M3ε
(x−λ (t−τ))2
t − τ
αx
ε
j
−
−
−
ε(t−τ)
√
e
e
e
dx dt ,
2
, j
ε ε(t − τ)
2M3ε
2
t − τ
√
t−τ
αx
ε
(x−y)
σ
−
−
ε
ε(t−τ)
γ
2
e
e
dx dt .
,r
ε ε(t − τ)
Let us begin with γ . We set
1
, j
x − λ (t − τ)
j
z =
√
,
ε(t − τ)
which gives γ1, j ≤ CT . A similar change of variables gives the same estimate for
γ .
1
,r
Next we bound γ . We have
2
, j
Z
Z
T
M3ε
(x−λ (t−τ))2
1
αx
ε
j
−
−
ε(t−τ)
γ
2
≤ T
√
e
e
dx dt .
, j
τ
0
ε ε(t − τ)
We then set
x
t − τ
z =
,
s =
;
ε
ε

1
372
E. GRENIER AND F. ROUSSET
we get
Z
Z
+∞
+∞
2
(
z−λ s)
1
j
√ e αze
−
−
ds dz ≤ CT
s
γ
2
≤ CT
, j
s
0
0
thanks to Lemma 7.1.
Next
Z
T
t − τ e t−τ
−σ
ε
γ
≤ C
− χ
dt ≤ Cε .
2
,r
ε
0
It remains to bound
ꢈꢂ
ꢈ
ꢂ
ꢃꢃ
ꢈ
ꢈ
ꢈ
ꢈ
x
b
ꢈ
ˆ
1
B ∂ G
.
2
x
ꢈ
M3ε
∞
1
L
,L
τ,y t,x
We write
ꢊ
ꢋ
x
int
int
int
b
ˆ
B = A (t, x) − A (t, 0) + A (t, 0) + A t,
2
ε
ꢊ
ꢊ
ꢋꢋ
x
int
b
˜
−
A (τ, 0) + A τ,
+ A(t, x) .
ε
Hence, using (1.13) and (5.2), we get
ꢉ
ꢉ
ꢉˆ ꢉ
B ≤ C (x + (t − τ) + ε) .
2
Using (5.5) leads to the bounds
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
T
T
T
T
T
T
2M3ε
2M3ε
2M3ε
2M3ε
2M3ε
2M3ε
(x−λ (t−τ))2
x
j
−
−
ε(t−τ)
δ
=
=
=
=
=
=
e
e
dx dt ,
1
, j
ε(t − τ)
x
τ
τ
τ
τ
τ
τ
0
0
0
0
0
0
2
t−τ
(x−y)
σ
−
ε
ε(t−τ)
δ
e
dx dt ,
1
,r
ε(t − τ)
(x−λ (t−τ))2
1
ε
1
ε
j
−
ε(t−τ)
δ
2
e
dx dt ,
, j
2
t−τ
(x−y)
−σ
−
ε
ε(t−τ)
δ
δ
e
e
dx dt ,
2
,r
(x−λ (t−τ))2
1
j
−
ε(t−τ)
e
dx dt ,
3
, j
t − τ
2
t−τ
(x−y)
−
σ
−
ε
ε(t−τ)
δ
e
e
dx dt .
3
,r
Let us bound δ . First we have
1
, j
Z
Z
T
+∞
(x−λ (t−τ))2
1
j
−
ε(t−τ)
δ
1
≤ 2M3
e
dx dt .
, j
(
t − τ)
τ
0
Then we set
and we get
x − λ (t − τ)
j
z =
√
,
ε(t − τ)
√
δ
1
≤ 2C(M ) εT .
, j
3