Existence time for solutions of semilinear different speed
Klein-Gordon system with weak decay data
Daoyuan Fang ∗ Ru-ying Xue
Center for Mathematical Sciences, Zhejiang University,
Hangzhou 310027, P. R. China.
Abstract
A ²−2 | log ²|−α (α = 2 if d ≥ 3, α = 3 if d = 2) result is obtained for the existence time
of solutions of semilinear different speed Klein-Gordon system with weakly decaying Cauchy
data, of size ², in certain circumstances of nonlinearity.
keywords: Weakly decaying Cauchy data, Existence time, Klein-Gordon equations with
different speeds.
Subject class: 35L70
1
Statement of the Main Result
We know that there are a lot of results on the lower bound problem of the life-spain for solutions
to the following semilinear Klein-Gordon equation
½
¤u + u = F (u, ∂t u, ∂x u),
(1)
u|t=0 = ²u0 , ∂t u|t=0 = ²u1
with small, weak decay Cauchy data. In [1] Delort studied the problem (1) with periodic Cauchy
data , and got a lower bound for the time of existence, of magnitude c²−2 , for a general nonlinearity,
and there are examples showing the optimality of that result for specific nonlinearities. Another
type of Cauchy data with weak decay properties at infinity has been considered in [2], in dimension
d = 1: it was shown that if u0 , u1 are in Sobolev spaces H N (R), H N −1 (R), without any other
assumption at infinity, and if the right hand side satisfies a “null condition” of the type of those
considered by Kosecki [4], the solution to (1) exists over an interval of length c²−4 | log ²|−6 . Very
recently, Delort and Fang in [3] investigated the higher dimensional case. We show that if the
nonlinearity F is a combination of a polynomial, depending only on u, vanishing at order at
least 2 at 0, and of [(∂t u)2 − (∂x u)2 ]G(u), where G is also a polynomial, the solution to (1) with
u0 ∈ H N (Rd ), u1 ∈ H N −1 (Rd ), N > (d + 3)/2, exists on an interval of time of length exp[c²−µ ]
with µ = 2/3 if d = 2, µ = 1 if d ≥ 3.
A natural problem is: can we get a lower bound for the time of existence for the different
speed Klein-Gordon system with weak decay data? In one space dimension this problem has been
∗ The
author was supported by NNSF of China 19671072 and partially by 19971077
1
considered by Fang [6]. In this paper, we devote to study the problem in dimension d ≥ 2. We
consider the following semilinear Klein-Gordon system

¤µ u + µ2 u = F1 (u, v, ∂t u, ∂t v, ∂x u, ∂x v),



¤v + v = F2 (u, v, ∂t u, ∂x u, ∂t v, ∂x v),
(2)
u|t=0 = ²u0 , ∂t u|t=0 = ²u1 ,



v|t=0 = ²v0 , ∂t v|t=0 = ²v1
with different propagation speeds in multi-space dimension, where (t, x) are coordinates on R1+d
Pd
Pd
with d ≥ 2, ¤ = ∂ 2 /∂t2 − j=1 ∂ 2 /∂x2j , ¤µ = ∂ 2 /∂t2 − µ2 j=1 ∂ 2 /∂x2j , 0 < µ 6= 1, U (t, x) =
(u(t, x), v(t, x)), u(t, x) = (u1 , · · · , um1 ), v(t, x) = (v1 , · · · , vm2 ), m = m1 + m2 . We assume that
functions Fi satisfy:
X
X
F1 (U, ∂t U, ∂x U ) = G10 (u, v) +
G1i,j (u)ui ∂t vj +
G2i,j (u)ui ∂x vj
+
1≤i≤m1 ,1≤j≤m2
X
1≤i≤m1 ,1≤j≤m2
G3i,j (u)q1 (∂t ui , ∂x ui ; ∂t uj , ∂x uj ),
(3)
1≤i<j≤m1
and
F2 (U, ∂t U, ∂x U )
= G20 (u, v) +
+
X
1≤i≤m1 ,1≤j≤m2
X
X
G̃1i,j (v)vj ∂t ui +
G̃2i,j (v)vj ∂x uI
1≤i≤m1 ,1≤j≤m2
G̃3i,j (v)q2 (∂t vi , ∂x vi ; ∂t vj , ∂x vj ),
(4)
1≤i<j≤m2
where Gk0 , G`ij and G̃`ij , with k = 1, 2, ` = 1, 2, 3, are polynomials vanishing at least at order 2 at
0, qk , k = 1, 2, is the bilinear form
qk (T, X; T 0 , X 0 ) = T T 0 − XX 0 .
The main theorem is the following:
Theorem 1.1 Let N ∈ N, N ≥ (d + 3)/2. There is a constant c > 0, such that for any pair
(U0 , U1 ) in the unit ball of H N (Rd ) × H N −1 (Rd ), and any ² ∈]0, 1[, problem (2) has a unique
solution U ∈ C 0 (] − T² , T² [, H N ) ∩ C 1 (] − T² , T² [, H N −1 ) with the existence time T² > c²−2 | log ²|−α
with α = 2 if d ≥ 3 and α = 3 if d = 2.
In the whole of the paper, we shall denote by k · k L2 -norms, and call “universal constant” a
constant which does not depend on the different parameters.
2
Reduction to a Local Problem
We shall use rescaling technique to reduce the proof of the theorem to a local existence problem
with Cauchy data of limited smoothness. Let us denote by ∆0 (resp. ∆j , j ∈ N∗ ) the Fourier
multiplier on L2 (Rd ) with symbol φ0 =1l {|ξ|<1} (resp. φj =1l {2j−1 <|ξ|<2j } ).
s
s
(Rd ) (or HN
( Rd , Rm )) the space of
Definition 2.1 If s ∈ R, N ∈ N, we shall denote by HN
h
2
d
families of functions (u )h∈]0,1/2[ , which are in L (R ) for fixed h, such that there is a sequence
(cj (h))j in the unit ball of `2 (N) and C > 0 satisfying
k∆j uh k ≤ Ccj (h)hs | log h|−ν (1 + 2j h)−N .
2
(5)
where we put ν = 1 if d ≥ 3 and ν =
k(uh )h kHNs .
3
2
if d = 2. The best constant C in (5) defines the norm
It is not difficult to know that Theorem 1.1 will be a consequence of the following result:
Theorem 2.2 Let N be an integer, N ≥ (d + 3)/2. There exists δ > 0 such that for any families
d+1
d−1
(V h )h∈]0,1/2[ , (resp. (W h )h∈]0,1/2[ ) in HN2 (resp. in HN2−1 ), of norms in these spaces smaller
than δ, the system

 ¤µ uh + µ2 h12 uh = h12 F1 (U h , h∂t U h , h∂x U h ),
¤v h + h12 v h = h12 F2 (U h , h∂t U h , h∂x U h ),
(6)
 h
U |t=0 = V h , ∂t U h |t=0 = W h
d+1
d−1
has a solution (U h )h ∈ C 0 (] − 1, 1[, HN2 ) ∩ C 1 (] − 1, 1[, HN2−1 ).
We shall define some spaces . If j, k ∈ N, and tµ ∈ {µ, 1}, we define
±,t
Φjk µ (τ, ξ) =1l {±τ >0}1l {2j−1 <|ξ|<2j }1l {2k−1 <|τ ∓t √h−2 +ξ0 2 |<2k }
µ
if j > 0, k > 0,
±,t
Φ0k µ (τ, ξ) =1l {±τ >0}1l {|ξ|<1}1l {2k−1 <|τ ∓t √h−2 +ξ02 |<2k }
µ
if k > 0,
±,t
Φj0 µ (τ, ξ) =1l {±τ >0}1l {2j−1 <|ξ|<2j }1l {|τ ∓t √h−2 +ξ02 |<1}
µ
if j > 0,
±,t
Φ00 µ (τ, ξ) =1l {±τ >0}1l {|ξ|<1}1l {|τ ∓t √h−2 +ξ0 2 |<1} .
µ
Let us define the corresponding Fourier multipliers as
±,t
±,t
4jk µ u = F −1 (Φjk µ (τ, ξ)û(τ, ξ))
for u ∈ L2 (R1+d ), where we used Fourier transform and inverse Fourier transform in space-time
variables.
0
s,s
Definition 2.3 Let s, s0 be real numbers, N ∈ N. We shall denote by HN,t
(R1+d ) the space of
µ
1+d
families (uh )h∈]0,1/2[ of L2 -functions
values in R (or in Rm ), such that there is a
PonPR , with
2
sequence (cjk (h))(j,k)∈N2 satisfying j ( k |cjk |) ≤ 1 and a constant C > 0 with
0
±t
k 4jk µ uh k ≤ Ccjk (h)hs | log h|−ν 2−ks (1 + 2j h + 2k h)−N
for any j, k ∈ N, h ∈]0, 1/2[, where tµ ∈ {µ, 1} ,ν =
h
constant C in (7) defines the norm of (u )h in
0
s,s
HN,t
.
µ
3
3
2
(7)
if d = 2 and ν = 1 if d ≥ 3. The best
In the rest of this paper, we shall use the following notation: if b(τ, ξ; τ 0 , ξ 0 ) is a locally bounded
function on R1+d × R1+d , and if f and g are L2 functions with compactly supported Fourier
transforms, we shall define B(f, g) by
Z
\
B(f, g)(τ, ξ) = fˆ(τ − τ 0 , ξ − ξ 0 )ĝ(τ 0 , ξ 0 )b(τ, ξ; τ 0 , ξ 0 ) dτ 0 dξ 0 ;
(8)
If j, j 0 and j 00 are nonnegative integers, we shall set j ¿ j 0 if j ≤ j 0 − 2 and j ∼ j 0 if |j − j 0 | ≤ 3.
00
±r
±r 0
0 00
3
Then, if u, v ∈ L2 (R1+d ) and 4±r
j 00 k00 (4jk u 4j 0 k0 v) 6≡ 0 for (r, r , r ) ∈ {µ, 1} , one has
(j ¿ j 0 and j 0 ∼ j 00 ) or (j 0 ¿ j and j ∼ j 00 ) or (j 00 − 5 ≤ j and j ∼ j 0 ).
d+1 1
(9)
d+1 1
,
,
2
2
2
2
× HN,r
, we shall decompose
To study B(u, v) for (u, v) ∈ HN,r
0
X X
X X
0 0
u=
4ejkr u, v =
4ej 0 kr 0 v,
(10)
e0 ∈{+,−} j 0 ,k0
e∈{+,−} j,k
00 00
and estimate 4ej 00 kr 00 (B(u, v)). Decompose it from (10) as
X X X 00 00
00 00
e0 r 0
4ej 00 kr 00 (B(u, v)) =
4ej 00 kr 00 (B(4er
jk u, 4j 0 k0 v))
(11)
e,e0 j,j 0 k,k0
for fixed e00 ∈ {+, −}, j 00 , k 00 ∈ N, T = (r, r0 , r00 ) ∈ {µ, 1}3 . Using the same proof as that in [2], we
have
Lemma 2.4 There is a positive constant C such that for any J = (j, j 0 , j 00 ) ∈ N3 , K = (k, k 0 , k 00 ) ∈
N3 , E = (e, e0 , e00 ) ∈ {+, −}3 and T = (r, r0 , r00 ) ∈ {µ, 1}3 , we have
00 00
j̃
0 0
0 0
k̃
er
er
er
2 + 2 kbk ∞
k 4ej 00 kr 00 (B(4er
L (AT (J,K,E)) k 4jk uk 4j 0 k0 vk,
jk u, 4j 0 k0 v))k ≤ C2
where j̃ = min(j, j 0 , j 00 ), k̃ = min(k, k 0 , k 00 ).
3
Microlocal estimates
Theorem 3.1 Let M0 be a fixed positive integer. There is a constant C > 0 such that for any
(j, j 0 , j 00 ) with |j − j 0 | ≤ M0 , any (k, k 0 , k 00 ) with k 0 ≤ k ≤ k 00 , any e00 ∈ {+, −},
00 00
0
−r
k 4ej 00 kr 00 (B(4+r
jk u, 4j 0 k0 v))k ≤
C2̃
d−1
k0
2 + 2
+k
2
1
(2̃ h)− 2 (1 + 2̃ h)
0
−r
× kbkL∞ (AT (J,K,E)) k 4+r
jk uk k 4j 0 k0 vk
(12)
for r 6= r0 .
00 00
0
−r
k 4ej 00 kr 00 (B(4+r
jk u, 4j 0 k0 v))k ≤
C2̃
d−1
k0
2 + 2
+k
2
1
(2̃ h)− 2 (1 + 2̂ h)
0
−r
× kbkL∞ (AT (J,K,E)) k 4+r
jk uk k 4j 0 k0 vk
(13)
for r = r0 .
00 00
0
+r
k 4ej 00 kr 00 (B(4+r
jk u, 4j 0 k0 v))k ≤
C2̃
d−1
k0
2 + 2
+k
2
1
(2̃ h)− 2 (1 + 2̃ h)
0
+r
× kbkL∞ (AT (J,K,E)) k 4+r
jk uk k 4j 0 k0 vk
for all T, where j̃ = min(j, j 0 , j 00 ), ĵ = max(j, j 0 , j 00 ), T = (r, r0 , r00 ) ∈ {µ, 1}3 .
4
(14)
Remark that if ̃ = 0, estimates (12), (13) and (14) are implied by inequality (12). As a
consequence, we will be free in the sequel to assume j > 0, j 0 > 0, j 00 > 0. We put f = 4+r
jk u,
0
2
g = 4±r
j 0 k0 v and study the L norm of
00
Z
00
I± (τ, ξ) = Φej 00 k,r00 (τ, ξ)
fˆ(τ − τ 0 , ξ − ξ 0 )ĝ(τ 0 , ξ 0 )b(τ, ξ; τ 0 , ξ 0 ) dτ 0 dξ 0 .
(15)
By the preceding remark, we will always have |ξ| > 1, |ξ 0 | > 1, |ξ − ξ 0 | > 1. Let us remark that it
− r0
r
is equivalent to prove (2.1.1) or the corresponding inequality in which 4+
jk u, 4j 0 k0 v is replaced
0
0
+ r
−r
+r
− r0
+ r
r
by 4−
jk u, 4j 0 k0 v, for 4jk u = 4jk ū, 4j 0 k0 v = 4j 0 k0 v̄.
To prove theorem 2.1.1, we define two functions depending on h on R1+d × R1+d , for k ∈ N,
` ∈ Z, as
p
p
F±µ (ξ, ξ 0 ) = µ h−2 + (ξ − ξ 0 )2 ± h−2 + ξ 02 ,
(16)
0
µ
χ`,µ
± (ξ, ξ ) = 1l {`2k <F± (ξ,ξ 0 )<(`+1)2k } .
Put
00
(17)
0
θ = 2−j ξ, θ0 = 2−j ξ 0 .
We have 1/2 ≤ |θ| ≤ 1, 1/2 ≤ |θ0 | ≤ 1 since in the integrand of (15) we have ξ ∈ Supp φj 00 and
ξ 0 ∈ Supp φj 0 . Let us define also
ρ = 2j
00
−j 0
0
0
0
, σ = (1 + 2j h)−1 , m = 1 − σ = 2j h(1 + 2j h)−1 ,
(18)
and for a given positive number M ,
©
ª
KM = (θ, θ0 , σ, ρ) ∈ R1+d × R1+d × [0, 1] × [0, M ]; 1/2 ≤ |θ| ≤ 1, 1/2 ≤ |θ0 | ≤ 1 .
(19)
We rescale F±µ defining the following functions on KM :
Gµ± (θ, θ0 , σ, ρ) =
0
00
0
h(1 + 2j h)−1 F±µ (2j θ, 2j θ0 )
= µ(σ 2 + m2 (ρθ − θ0 )2 )1/2 ± (σ 2 + m2 θ02 )1/2 .
Lemma 3.2 i). The function θ0 → Gµ± (θ, θ0 , σ, ρ) has no critical point if σ ≤
√
(20)
1−µ2
.
3
ii). The critical point,θ± , of the function θ0 → Gµ± (θ, θ0 , σ, ρ), if it exists, is unique on KM and
ρθ
is collinear to the parameter θ. Let θ± = 1+λ
, we have µ1 ≤ |λ± | ≤ C(M ) with some positive
±
number C(M ) deponding only on M .
proof: From
∇θ0 Gµ± (θ, θ0 , σ, ρ) = p
−µm2 (ρθ − θ0 )
σ 2 + m2 (ρθ − θ0 )2
±√
m2 θ0
=0
σ 2 + m2 θ02
(21)
we know that ρθ−θ0 and θ0 are collinear, and hence there is some constant λ± so that ρθ−θ0 = λ± θ0 .
Obviously λ± 6= 0 and 1 + λ± 6= 0. It follows from (21) that
±λ± > 0
and
1
σ 2 + m2 θ02
= 2
,
−2
2
µ
−1
(λ± − 1)σ
5
(22)
which means |λ± | > 1 because of 0 < µ < 1. Denote by
2 2 2
m ρ θ
1 + (1+λ)
2 σ2
σ 2 + m2 θ02
K(λ) = 1
=
.
1
2
( λ2 − 1)σ
1 − λ2
It is obvious that K(λ) is a strictly decreasing function when λ > 1 and is a strictly increasing
function when λ < −1. Moreover we have
K(±µ−1 ) >
1
and
1 − µ2
lim K(λ) = 1 <
λ→±∞
Then the system
±λ > 0, K(λ) =
has a unique solution λ± with |λ± | >
√
1−µ2
3
1
µ,
1
.
1 − µ2
1
1 − µ2
and hence Gµ± (θ, θ0 , σ, ρ) has a unique critical point.
1
2
≤ |θ0 | ≤ 1. What
√
1−µ2
1
remains is to prove that |λ± | < C(M ) for some positive constant. By K(λ± ) = 1−µ
,
σ
≥
2
3
and |λ± | ≥ µ1 we know
That σ ≥
follows from (2.1.13), the assumption m + σ = 1 and
1 + M 2 (1 + λ± )−2 σ −2
1
1 + 9M 2 (1 + λ± )−2 (1 − µ2 )−1
≤
≤
.
−2
2
1−µ
1 − λ±
1 − λ−2
±
(23)
Notice that
0 < µ < 1,
1 + 9M 2 (1 + λ± )−2 (1 − µ2 )−1
→ 1 as
1 − λ−2
±
λ± → ±∞.
We deduce from (23) that there exists a positive constant C(M ), deponeding only on M , such that
|λ± | ≤ C(M ).
¤
In the sequel, we will write θ0 = (θ0 1 , θ00 ) ∈ R × Rd−1 and θ = |θ|(1, 0, · · · , 0). It follows from
Lemma 3.2 that the critical point, θ± , of the function θ0 → Gµ± (θ, θ0 , σ, ρ), if it exists, is unique on
KM . Let us define
(
)
p
2
1
−
µ
D± = (θ, θ0 , σ, ρ) ∈ KM : θ0 = θ± , σ ≥
.
(24)
3
Lemma 3.3 For any M > 0 and any neighborhood V± of D± in KM , there is a positive constant
C and a real valued C 1 function H± , defined on a neighborhood of the set KM − V± , such that
H±
is bounded on KM − V± ,
C −1 ≤ |∇θ0 H± (θ, θ0 , σ, ρ)| ≤ C on KM − V± ,
G± (θ, θ0 , σ, ρ) = (µ ± 1)σ + m2 H± (θ, θ0 , σ, ρ) on KM − V± .
proof: Denote by
H± =
G± − (µ ± 1)σ
.
m2
6
(25)
We first prove the boundedness of H± . When m ≥ 1/2, it is obvious that |H± | ≤ C for some
positive number C. When 0 ≤ m ≤ 1/2, we have σ = 1 − m ≥ 1/2 and
H± =
(σ 2
µ(ρθ − θ0 )2
θ02
+ 2
2
0
2
1/2
+ m ζ(ρθ − θ ) )
(σ + m2 ζθ02 )1/2
for some ζ ∈ (0, 1), and hence |H± | ≤ C since (σ 2 + m2 (ρθ − θ0 )2 ) ≥ 1/4, |ρ − θ0 | ≤ M + 1 and
1/2 ≤ |θ0 | ≤ 1. Notice that
∇ θ 0 H± =
−µ(ρθ − θ0 )
θ0
±
.
(σ 2 + m2 (ρθ − θ0 )2 )1/2
(σ 2 + m2 θ02 )1/2
Since |∇θ0 H± | |{σ=0} = |µ ± 1| > 0, we can choose σ0 > 0 small so that
|∇θ0 H± | ≤ 2|µ ± 1| on the set KM ∩ {σ ≤ σ0 }.
T
On the compact set KM {σ ≥ σ0 }, the norm |∇θ0 H± | is a continuous function and vanishes only
if θ0 = θ± . The boundedness of |∇θ0 H± | now follows from the definition of V± .
¤
Let us introduce, for δ small positive number, the following open subsets of KM :
(
)
p
2
1
−
µ
0
V±δ = (θ, θ0 , σ, ρ) ∈ KM ; |θ0 − θ±
| < δ, σ >
6
(26)
Lemma 3.4 Let M be a fixed positive number. There is C > 0, δ > 0, and two real valued C 1
1
2
functions H±
and H±
, defined on an open neighborhood of V±δ ∩ {ρ > 0}, bounded with bounded
0
first θ -derivative on V±δ ∩ {ρ > 0}, such that
n
C −1 ≤ |H±
(θ, θ0 , σ, ρ)| ≤ C, n = 1, 2,
(27)
1
Gµ± (|θ|, θ0 , σ, ρ) = Gµ± (|θ|, θ± , σ, ρ) + m2 σ 2 (θ10 − θ± )2 H±
(θ, θ0 , σ, ρ)
2
+ m2 θ002 H±
(θ, θ0 , σ, ρ)
on V±δ ∩ {ρ > 0}.
proof: Denote by
θ = |θ|(1, 0, · · · , 0), θ0 − θ± = η = (η1 , η 00 )
£
¤1
£
¤1
2 2
A = σ 2 + m2 (ρθ − θ± )2 2 , B = σ 2 + m2 θ±
.
Then η1 = θ10 − θ± , η 00 = θ00 , θ± =
1
1+λ± |θ|(1, 0, · · ·
, 0) and
h
i 21
h
i1
2
2 2
2m2 θ± η1 +m2 η 2 2
± )η1 +m η
Gµ± (|θ|, θ0 , σ, ρ) = µA 1 + −2m (ρθ−θ
±
B
1
+
A2 i
B2
h
£ µ
¤ 2 2 2
−µ(ρθ−θ± )
θ±
µ
1
= G± (|θ|, θ± , σ, ρ) +
± B η1 + 2A3 ± 2B 3 m σ η1
A
£µ
¤ 2 002
1
4 3
+ 2A ± 2B m η + g5 m η1 + g6 m4 η 003 ,
where
gi (i = 1, 2, ·√
· · , 6) are bounded continuous functions on KM . Notice that ±λ± > 1/µ,
√
1−µ2
1−µ2
σ ≤ A ≤ C,
σ ≤ B ≤ C. Using 5θ0 Gµ± (|θ|, θ± , σ, ρ) = 0 and Lemma (3.2) we have
3
3
−µ(ρθ − θ± ) θ±
±
= 0,
A
B
7
µ 2
µ
1
µρ|θ|
µ(1 + λ± )
µ(1 − µ)(1 + C(M ))
( − 1) ≤
±
=
=
≤
,
2 µ
2A 2B
2θ± A
2A
2µ
and
C≤
2
(1 + λ± )3
µ
1
µ (1 + λ± )σ 2 + m2 θ±
1
±
=
≥
> 0.
2A3
2B 3
2A
A2 B 2
C
Let
1
H±
=
µ
1
g5 m2 η1
±
+
,
3
3
2A
2B
σ2
2
H±
=
√
The conclusion now follows from (28) and the fact σ ≥
4
µ
1
±
+ g6 m2 η 00 .
2A 2B
1−µ2
6
and |η| < δ small.
(28)
¤
Proof of Theorem 3.1
We first prove the case r 6= r0 . Without loss of generality, let us consider the case r = µ, r0 = 1,
± 1
µ
that is, we want to estimate the integral (15) with f = 4+
jk u, g = 4j 0 k0 v. Remember that we
00
0
0
assumed |j − j | ≤ M0 . Together with condition (9), this implies that ρ = 2j −j stays smaller
j0
than a fixed positive constant M . For δ > 0 and ξ± = 2 θ± we denote by
(
)
p
1 − µ2
δ
0
0
0
j0
W± = (ξ, ξ ) : ξ ∈ Suppφj 00 , ξ ∈ Suppφj 0 , |ξ − ξ± | < δ2 , σ ≥
,
6
00
0
where
+ I±
and split I± = I±
0
I±
=
00 00
Φej 00 k,r00
Z
fˆ(τ − τ 0 , ξ − ξ 0 )ĝ(τ 0 , ξ 0 )b(τ, ξ; τ 0 , ξ 0 )1l W±δ dτ 0 dξ 0 .
(29)
0
, we introduce a result considered in [3] (Lemma 2.1.3 and Lemma 2.1.4 ). For any
To estimate I±
triple (j, j 0 , j 00 ) and any real number R ∈]0, 2], let us fix a finite partition of Rd − {0} in conical
subdomain, (γa )a∈S , where S is a finite subset of S d−1 , such that there is a universal constant
c > 0 with
½
¾
½
¾
ξ0
ξ0
0 0
d
j 00 −j
0 0
d
j 00 −j
ξ ; ξ ∈ R − {0}, | 0 − a| < c2
R ⊂ γa ⊂ ξ ; ξ ∈ R − {0}, | 0 − a| < 2
R .
|ξ |
|ξ |
Set P = N × S and for p = (q, a) ∈ P define
0
−1
0
00
00
Qp
=
{ξ 0 ; 2j
≤ |ξ 0 | ≤ 2j , q2j ≤ |ξ 0 | ≤ (q + 1)2j , ξ 0 ∈ γa },
Q̃p
=
{η; 2j−1 ≤ |η| ≤ 2j , and ∃(ξ, ξ 0 )with2j
ξ ξ ˙
η = ξ − ξ 0 , |ξ 0 − (ξ 0 ) | ≤ R2j 0 }.
|ξ| |ξ|
00
−1
Then there is a universal constant C > 0 such that
X
X
1=
1l Qp , k
1l Q̃p kL∞ ≤ C.
p∈P
p∈P
00
Estimate of I±
8
00
≤ |ξ| ≤ 2j , ξ 0 ∈ Qp ,
00
We have I±
=
P
00
I±p
with
Z
e00 ,r 00
= Φj 00 k00
fˆp (τ − τ 0 , ξ − ξ 0 )gˆp (τ 0 , ξ 0 )b(τ, ξ; τ 0 , ξ 0 )1l c W±δ dτ 0 dξ 0 ,
p∈P
00
I±p
P 00 `
00
where fˆp = fˆ1l Q̃p , gˆp = ĝ1l Qp . We write also I±p
= ` I±p
with
Z
00 00
00 `
0
0
0
I±p
= Φej 00 k,r00
fˆp (τ − τ 0 , ξ − ξ 0 )gˆp (τ 0 , ξ 0 )b(τ, ξ; τ 0 , ξ 0 )1l c W±δ χ`µ
± (ξ, ξ )dτ dξ ,
00 `
The absolute value of I±p
is smaller than
µZ
k0
2
2 φj 00 (ξ)
0
0
0
|fˆp (τ − τ 0 , ξ − ξ 0 )|2 |gˆp (τ 0 , ξ 0 )|2 |b|21l c W±δ χ`,µ
± (ξ, ξ ) dτ dξ
µZ
×
0
0
χ`,µ
δ dξ
± (ξ, ξ )1l Qp (ξ 0 )1l c W±
¶1/2
¶1/2
.
The last integral is bounded from above by
Z
0
2j d 1l {`2k−j0 m<Gµ (θ,θ0 ,σ,ρ)<(`+1)2k−j0 m}1l c V±δ1l Q̄(2j0 θ0 ) dθ0 ,
(30)
(31)
±
00
where Q̄, obtained from Qp by rotation, is contained in a cube of diameter C2j . Using Lemma 3.3,
(31) can be controlled by
Z
0
(32)
2j d 1l {`L− (µ±1)σ ≤H± ≤(`+1)L− (µ±1)σ }1l Q̄(2j0 θ0 ) dθ0
m2
m2
0
with L = 2k−j /m. Since C1 ≤ | 5θ0 H± | ≤ C, we can cover the domain of integration by a
universally bounded finite number local charts, over which H± can be taken as a local coordinate
satisfying universal bounds. This allows us to bound (32) by
0
C2j
0
d
2k−j (j 00 −j 0 )(d−1)
2
.
m
Plugging into (30), using almost othogonality and summing in ` and p, we get
00
kI±
k ≤ C2j̃
d−1
k0
2 + 2
+k
2
1
1
(1 + 2ĵ h) 2 (2j̃ h)− 2 kbkkf kkgk
(33)
since |j − j 0 | ≤ M0 implies that j 00 can be controlled by j̃.
0
Estimate of I±
P 0`
0
0
0
We set ξ 0 = ξ±
+ 2j η, and let us decompose I±
= ` I±
with
Z
00 00
0 `
0
I±
(τ, ξ) = Φej 00 k,r00
fˆ(τ − τ 0 , ξ − ξ 0 )ĝ(τ 0 , ξ 0 )bχ`,µ
1 W±δ dτ 0 dξ 0
± (ξ, ξ )l
P
0
0 ` 2 21
Using almost othogonality (see [3]), we have kI±
k ≤ C( ` kI±
k ) and
¯ 0 `¯
¯I± ¯
µZ
≤ 2j
× 2
0d
2
k0
2
0
0
0
|fˆ(τ − τ 0 , ξ − ξ 0 )|2 ĝ(τ 0 , ξ 0 )|2 b2 χ`,µ
± (ξ, ξ )dτ dξ
µZ
1l W δ (ξ,ξ± +2j0 η) χ`,µ
± (ξ, ξ±
±
9
j0
+ 2 η)dη
¶ 21
¶ 12
.
(34)
The last integral can be bounded from above by, using Lemma 3.4,
Z
1l {|η00 |<δ,|η1 |<δ}1l {`L−T <σ2 η12 H±1 +η00 2 H±2 <(`+1)L−T } dη,
(35)
0
where L = 2k−j /m, T = Gµ± (θ, θ± , σ, ρ)/m2 . Performing the change of variables inverse to w1 =
1 12
2 12
η1 (H±
) ans w2 = η 00 (H±
) , we get the upper bound of (35)
Z
0
0
0
C 1l {|w1 |+w2 |≤Cδ}1l {`L−T <σ2 w12 +w22 <(`+1)L−T } dw1 dw2 ≤ C2k−j (1 + 2j h)2 (2j h)−1 .
Plugging into (34), using the fact that j 0 can be controlled by ĵ and summing in ` we obtain
0
kI±
k ≤ C2j̃
d−1
k0
2 + 2
+k
2
1
1
(1 + 2j̃ h) 2 (2j̃ h)− 2 kbkkf k kgk.
For the case of r = r0 in Theorem 3.1, we can get the conclusion from Theorem 2.1.1 in [3].
00
00
0
± r
r
0
00
00
We wish to bound k4ej 00 k00 r (B(4+
jk u, 4j 0 k0 v))k when k ≤ k ≤ k , e = ± and j is much
0
00
0
00
smaller than j ∼ j or j much smaller than j ∼ j . We will study only the second case, since the
first one is similar. The result is the following one:
Proposition 4.1 There is a positive integer N0 such that if j 0 ≤ j 00 − N0 , k 0 ≤ k ≤ k 00 and
(r, r0 , r00 ) ∈ {µ, 1}, one has a universal constant C with
° 00 00
°
k0
k
1
° e r
°
± r0
r
j̃ d−1
2 + 2 + 2 (1 + 2j̃ h)(2j̃ h)− 2
°4j 00 k00 (B(4+
jk u, 4j 0 k0 v))° ≤ C2
±
r
× kbkL∞ (AT (J,K,E)) k 4+
jk uk k 4j 0 k0
r0
vk.
(36)
To prove this proposition, we shall use the following notations
00
0
00
00
00
ξ = 2j θ, ξ 0 = 2j θ0 , σ = (1 + 2j h)−1 , m = 2j h(1 + 2j h)−1 , ρ = 2j
0
−j 00
.
We will have 1/2 ≤ |θ| ≤ 1, 1/2 ≤ |θ0 | ≤ 1 and ρ small. As before, we set θ = |θ|(1, 0, · · · , 0),
θ0 = (θ10 , θ00 ). Let us also set
Gµ± (|θ|, θ0 , σ, ρ) =
00
00
0
h(1 + 2j h)−1 F±µ (2j θ, 2j θ0 )
= µ(σ 2 + m2 (θ − ρθ0 )2 )1/2 ± (σ 2 + m2 ρ2 θ02 )1/2
(37)
if r = 1, r0 = µ. We shall study the local behavior of Gµ± . For δ > 0 small, let us define
V±δ
δ
K±
= {(|θ|, θ0 , σ, ρ); 1/2 ≤ |θ| ≤ 1, 1/2 ≤ |θ0 | ≤ 1, ρ ∈]0, 1], σ ∈ [0, 1],
σ/ρ < δ, ±θ10 > 0, |θ00 | < δ},
= {(|θ|, θ0 , σ, ρ); 1/2 ≤ |θ| ≤ 1, 1/2 ≤ |θ0 | ≤ 1, ρ ∈]0, 1], σ ∈ [0, 1]} − V±δ .
(38)
Lemma 4.2 i) There is δ > 0, C > 0, and a real valued C 1 function, H± , defined on an open
neighborhood of V±δ , such that on V±δ ∩ {ρ < δ}
Gµ± (|θ|, θ0 , σ, ρ) = µm|θ| + ρmH± (|θ|, θ0 , σ, ρ),
|H± (|θ|, θ0 , σ, ρ)| ≤ C, C −1 ≤ |
10
∂H±
(|θ|, θ0 , σ, ρ)| ≤ C.
∂θ10
(39)
ii) For any fixed δ > 0 small, there is ρ0 > 0, C > 0 and a real valued C 1 function, H± , defined
δ
δ
on an open neighborhood of K±
∩ {ρ < ρ0 }, such that on K±
∩ {ρ < ρ0 }
Gµ± (|θ|, θ0 , σ, ρ) = ±σ + µ(σ 2 + m2 θ2 )1/2 + ρm2 H± (|θ|, θ0 , σ, ρ),
(40)
|H± (|θ|, θ0 , σ, ρ)| ≤ C, C −1 ≤ |5θ10 H± (|θ|, θ0 , σ, ρ)| ≤ C.
proof:
i) Set ν =
µσ
ρm
and write
mρ(θ002 + ν 2 )
= µm|θ| − mρ(µθ10 ∓ |θ10 |) +
2
¶
µ
1
µρ
002
2
002
2
+
(θ
+
ν
)g
±
±
(θ
+
ν
)g
×
1
2
|θ| − ρθ10
|θ10 |
Gµ± (θ, θ0 , σ, ρ)
where ||θ| − ρθ10 | ≥ 14 , |θ10 | ≥ 14 , g1 and g2 are two bounded C 1 functions on V±δ for δ small. Denote
by
µ
¶
θ002 + ν 2
µρ
1
0
H± = (1 ± µ)θ1 +
± 0
+ (θ002 + ν 2 )2 (g1 ± g2 ).
2
|θ| − ρθ10
|θ1 |
Obviously we have on V±δ , for δ small,
Gµ± (|θ|, θ0 , σ, ρ) = µm|θ| + ρmH± (|θ|, θ0 , σ, ρ),
¯
¯
¯
1 − µ ¯¯ ∂H±
0
0
|H± (|θ|, θ , σ, ρ)| ≤ C,
≤ ¯ 0 (|θ|, θ , σ, ρ)¯¯ ≤ 2(1 − µ).
2
∂θ1
ii) Let
H± =
i
1 h µ
0
2
2 2 1/2
G
(|θ|,
θ
,
σ,
ρ)
−
µ(σ
+
m
θ
)
∓
σ
.
±
m2 ρ
(41)
This definition implies at once that H± is bounded. For ρ < ρ0 small, we have
5θ10 H± = −
ρθ10
µ|θ|
±
+ ρg1
(σ 2 + m2 θ2 )1/2
(σ 2 + m2 ρ2 θ02 )1/2
δ
and
where g1 is a bounded function. Obviously | 5θ10 H± | ≤ C on K±
| 5θ10 H± ||σ=0 ≥ |µ ± 1| ≥ 1 − µ > 0.
Hence there exists a small positive number σ0 such that | 5θ10 H± | ≥
For σ0 ≤ σ ≤ 1, we can choose ρ0 > 0 small so that
| 5θ10 H± | ≥
Then, we have
1
C
1−µ
2
δ
∩ {0 ≤ σ ≤ σ0 }.
on K±
µ
ρ
µ
− |θ10 | | | − ρ|g1 | ≥
> 0.
8
σ
16
δ
≤ | 5θ10 H± | ≤ C on K±
∩ {ρ ≤ ρ0 } for ρ0 small.
¤
Lemma 4.3 There is a positive integer N0 such that if j 0 ≤ j 00 − N0 , k 0 ≤ k ≤ k 00 , one has a
universal constant C with
° 00 00
°
d−1
1
k0
k
° e r
°
± r0
r
u,
4
v))
°4j 00 k00 (B(4+
° ≤ 2j̃ 2 + 2 + 2 (1 + 2j̃ h)(2j̃ h)− 2
0
0
jk
j k
±
r
× kbkL∞ (AT (J,K,E)) k 4+
jk uk k 4j 0 k0
11
r0
vk.
(42)
proof: If r = r0 , the result comes from Proposition 2.3.1 in [3]. If r 6= r0 , without loss of generality,
± 1
µ
we consider the case r = µ, r0 = 1. To do this, we denote f = 4+
jk u, g = 4j 0 k0 v and
I± =
Z
00 00
Φej 00 k,r00
fˆ(τ − τ 0 , ξ − ξ 0 )ĝ(τ 0 , ξ 0 )b(τ, ξ; τ 0 , ξ 0 ) dτ 0 dξ 0 .
Let us define
W±δ
=
n
00
00
0
0
(ξ, ξ 0 ); 2j −1 ≤ |ξ| ≤ 2j , 2j −1 ≤ |ξ 0 | ≤ 2j ,
o
00
0
00
0
2j −j (1 + 2j h)−1 < δ, ±ξ10 > 0, |ξ 00 | < 2j δ
0
00
and decompose I± = I±
+ I±
with
00
00
Z
fˆ(τ − τ 0 , ξ − ξ 0 )ĝ(τ 0 , ξ 0 )b1l W±δ dτ 0 dξ 0 ,
0
I±
= Φej 00 k,r00
where δ is fixed so that Lemma 4.2 is true. Taking the integer N0 of the statement of proposition 4.1
large enough, we may assume that ρ is smaller than ρ0 of Lemma 4.2 ii).
0
Estimate of I±
0
We write I±
=
P+∞
0 `
with
I±
Z
00 00
0
0
0
= Φej 00 k,r00
fˆ(τ − τ 0 , ξ − ξ 0 )ĝ(τ 0 , ξ 0 )bχ`,µ
δ dτ dξ .
± (ξ, ξ )1l W±
`=0
0 `
I±
and get
µZ
0 `
|I±
|
0
0
0
|fˆ(τ − τ 0 , ξ − ξ 0 )|2 |ĝ(τ 0 , ξ 0 )|2 |b|2 χ`µ
± (ξ, ξ ) dτ dξ
≤ φj 00 (ξ)
µZ
0
×2k /2
0
0
1l W±δ χ`µ
± (ξ, ξ ) dξ
(43)
¶1/2
¶1/2
.
The last integral can be written
Z
j0d
2
1l V±δ (|θ|, θ0 , σ, ρ)1l {|θ00 |<δ}1l {`2k−j00 m<Gµ <(`+1)2k−j00 m} dθ0 ,
(44)
±
By Lemma 4.2, (44) can be controlled from above by
00
0
C2j d δ d−1
0
2k−j
= C2j (d−1)+k .
ρ
Plugging into (43), we can get
0
kI±
k ≤ C2j
0 d−1
k0
2 + 2
+k
2
kf k kgk kbkL∞ (AT (J,K,E)) .
00
Estimate of I±
00
We decompose I±
=
P
`
00 `
I±
and have
µZ
00 `
|I±
| ≤ φj 00 (ξ)
0
0
0
|fˆ(τ − τ 0 , ξ − ξ 0 )|2 |ĝ(τ 0 , ξ 0 )|2 |b|2 χ`,µ
± (ξ, ξ ) dτ dξ
12
¶1/2
µZ
0
0
0
φj 0 (ξ 0 )1l c W±δ χ`µ
± (ξ, ξ ) dξ
×2k /2
¶1/2
.
(45)
If ρ < ρ0 , by Lemma 4.2 we write the last integral
Z
0
2j d 1l K±δ (θ, θ0 , σ, ρ)1l {`L−T <H± <(`+1)L−T } dθ0
(46)
¡
¢1
00
whereL = 2k−j /ρm, T = ±σ + µ(σ 2 + m2 θ2 ) 2 ρ−1 m−2 . Since C1 ≤ | 5θ10 H± | ≤ C, we can take
H± as a local coordinate satisfying universal bounds, and hence we can bound (46) by
0
00
2j d 2k−j /ρm ≤ 2j
0
(d−1)+k
00
00
(1 + 2j h)(2j h)−1 .
Plugging into (45), summing in `, we get
00
kI±
k ≤ C2j̃
d−1
k0
2 + 2
+k
2
1
(1 + h2j̃ )(2j̃ h)− 2 kf kkgkkbk.
¤
To end this section, let us state a theorem which gathers the results of theorem 3.1 and proposition 4.1.
Theorem 4.4 i) Assume j3 ¿ j1 ∼ j2 r1 = r2 , and either (k3 = k 00 , e1 6= e2 ) or (k3 6= k 00 , e1 =
e2 ). One has, for a universal constant C,
00
00
0
0
k4ej 00 k00 r (B(∆ejk r u, ∆ej 0 k0 r v))k
≤ C2̃
k1
d−1
2 − 2
+
k2
2
0
1
(1 + 2ĵ h)(2j̃ h)− 2
0
× k∆ejk r ukk∆ej 0 k0 r vkkbk,
(47)
ii) In all the other cases,
00
00
0
0
k4ej 00 k00 r (B(∆ejk r u, ∆ej 0 k0 r v))k ≤
C2̃
k1
d−1
2 − 2
+
k2
2
0
1
(1 + 2j̃ h)(2j̃ h)− 2
0
× k∆ejk r ukk∆ej 0 k0 r vkkbk,
(48)
where j̃ = min(j, j 0 , j 00 ), ĵ = max(j, j 0 , j 00 ).
5
The proof of the main theorem
In this section, we first show that the nonlinearities involved in the right hand side of (1.2.2) are
s,s0
bounded on HN,r
for convenient values of s, s0 , N and r ∈ {µ, 1}, then we prove the main result.
Let B(u, v) be one of the following expressions when u and v are functions on Rd+1 .
h−2 uv, or h−1 u∂t v, or h−1 u∂x v, or ∂t u∂t v − ∂x u∂x v.
We have (see [3])
d+1 1
,2
2
Proposition 5.1 Let N ≥ (d + 3)/2, the map (u, v) → B(u, v) is continuous on HN,r
d−1
,− 1
with values in HN2−1,r 2 for r ∈ {µ, 1}.
13
d+1 1
,2
2
× HN,r
Now we prove that the nonlinearity u∂x v in the right hand side of ( 6) is bounded operator
d+1 1
d+1 1
,
d−1
,
d+1 1
,2
,− 1
2
2
2
2
2
from HN,µ
× HN,1
to HN2−1,µ 2 ( resp. from HN,1
2 0
0
1/h , ξ /h, or τ /h. It is obvious that
kbkL∞ ≤
Let
d−1
,− 1
to HN2−1,1 2 ). Denote by b =
0
0
1
(1 + 2j h + 2k h).
h2
1
uv,
h2
B(u, v) =
d+1 1
,2
2
× HN,µ
1
u∂x v,
h
or
or
(49)
1
u∂t v.
h
d+1 1
,2
2
Proposition 5.2 Let N ≥ (d+3)/2. The map (u, v) → B(u, v) is continuous on HN,µ
d−1
d+1 1
,2
2
×HN,1
,− 1
with values in HN2−1,µ 2 .
proof: We shall prove this proposition by writing
X X X 00
00
0
4ej 00 k00 µ (B(u, v)) =
4ej 00 k00 µ (B(∆ejk µ u, ∆ej 0 k0 1 v)),
(50)
j,j 0 k,k0 e,e0
α). Estimate of
X
X
00
0
k4ej 00 k00 µ (B(∆ejk µ u, ∆ej 0 k0 1 v))k.
(51)
j,j 0 {k,k0 ;k≤k00 ,k0 ≤k00 }
The general term of (51) is bounded from above by
0
0
Ch−2 (1 + 2j h + 2k h)2j̃
d−1
k
k0
2 +2+ 2
d+1 1
,2
d+1 1
,2
× HN21
Using the assumption (u, v) ∈ HN2µ
Chd−1 2j̃
d−1
2
0
(1 + 2j̃ h)(2j̃ h)−1/2 k∆ejk µ ukk∆ej 0 k0 1 vk.
, we can bound this expression by
0
0
(1 + 2j̃ h)(2j̃ h)−1/2 (1 + 2j h + 2k h)−N +1 (1 + 2j h + 2k h)−N
| log h|−2ν kukH (d+1)/2,1/2 kvkH (d+1)/2,1/2 cjk c0j 0 k0 ΘT (J, K, E)
N,µ
(52)
N,1
P P
P P
with j ( k |cjk |)2 ≤ 1 and j 0 ( k0 |c0j 0 k0 |)2 ≤ 1, where ΘT (J, K, E) is the characteristic function
of the set E T and
n
00
ET =
(J, K, E) ∈ N3 × N3 × {+, −}3 , ∃(τ, ξ) ∈ Supp Φje00 k,µ00 ,
o
0
∃(τ 0 , ξ 0 ) ∈ Supp Φej 0 k,10 , with (τ − τ 0 , ξ − ξ 0 ) ∈ SuppΦe,µ
.
(53)
jk
Let us set
dj 00 k00 =
X
X
2
−k00
2
h
d−1
2
2j̃
d−1
2
0
0
(1 + 2j̃ h)(2j̃ h)−1/2 (1 + 2j h + 2k h)−N +1
j,j 0 {k,k0 ;k≤k00 ,k0 ≤k00 }
00
00
×(1 + 2j h + 2k h)−N (1 + 2j h + 2k h)N −1 | log h|−ν kuk
14
d+1 1
,
2
2
HN,µ
kvk
d+1 1
,
2
2
HN,1
cjk c0j 0 k0 ΘT (J, K, E)
(54)
then we have
dj 00 ≤ C
X
d
0
d
2j̃( 2 −1) h 2 −1 | log h|−ν (1 + h inf(2j , 2j ))−N +2 cj c0j 0 ΘT (J)
j,j 0
for `2 sequences (cj )j , (c0j 0 )j 0 , and the result follows from Lemma 3.1.3 in [3], since N ≥ (d + 3)/2,
ν ≥ 1 if d ≥ 3 and ν ≥ 3/2 if d = 2.
β). Estimate for e00 6= e of
X
X
00
0
k4ej 00 k00 µ (B(∆ejk µ u, ∆ej 0 k0 1 v))k.
(55)
j,j 0 {k,k0 ;k0 ≤k,k00 ≤k}
We bound the general term of (55) by
Chd−1 2j̃
d−1
k00
2 + 2
−k
2
(1 + 2j h + 2k h)−N | log h|−2ν kuk
with
0
1
0
(1 + 2j̃ h)(2j̃ h)− 2 (1 + 2j h + 2k h)−N +1
d+1 1
,
2
2
HN,µ
kvk
d+1 1
,
2
2
HN,1
cjk c0j 0 k0 ΘT (J, K, E)
P P
P P
2
0
2
j(
k |cjk |) ≤ 1 and
j0 (
k0 |cj 0 k0 |) ≤ 1. Set
X
X
d−1
0
0
k0
1
dj 00 k00 =
2j̃ 2 − 2 (1 + 2j̃ h)(2j̃ h)− 2 (1 + 2j h + 2k h)−N +1
j,j 0 {k,k0 ;k0 ≤k,k00 ≤k}
00
00
(1 + 2j h + 2k h)−N (1 + 2j h + 2k h)N −1 |logh|−ν cjk c0j 0 k0 ΘT (J, K, E)
Then
dj 00 ≤ C
X
0
d
d
2j̃( 2 −1) h 2 −1 | log h|−ν (1 + h inf(2j , 2j ))−N +2 cj c0j 0
j,j 0
for `2 sequences (cj )j , (c0j 0 )j 0 . Now the result follows from Lemma 3.1.3 in [3], since N ≥ (d + 3)/2,
ν ≥ 1 if d ≥ 3 and ν ≥ 3/2 if d = 2.
γ) Estimate for e00 = e of
X
X
00
0
k4ej 00 k00 µ (B(∆ejk µ u, ∆ej 0 k0 1 v))k.
(56)
j,j 0 {k,k0 ;k≤k0 ,k00 ≤k0 }
We bound the general term of (56) by, using the fact j 0 << j ∼ j 00 ,
Chd−1 2j̃
d−1
k00
2 + 2
0
− k2
×(1 + 2j h + 2k h)−N | log h|−2ν kuk
d+1 1
,
2
2
HN µ
with
0
1
0
(1 + 2j h)(2j̃ h)− 2 (1 + 2j h + 2k h)−N +1
kvk
d+1 1
,
2
2
HN,1
cjk c0j 0 k0 ΘT (J, K, E)
P P
P P
2
0
2
j(
k |cjk |) ≤ 1 and
j0 (
k0 |cj 0 k0 |) ≤ 1. Set
X
X
d−1
0
0
k0
1
dj 00 k00 =
2j̃ 2 − 2 (1 + 2j h)(2j̃ h)− 2 (1 + 2j h + 2k h)−N +1
j,j 0 {k,k0 ;k≤k0 ,k00 ≤k0 }
00
00
×(1 + 2j h + 2k h)−N (1 + 2j h + 2k h)N −1 h
We get
dj 00 ≤ C
X
d
d−1
2
| log h|−ν cjk c0j 0 k0 ΘT (J, K, E)
(57)
0
d
2j̃( 2 −1) h 2 −1 | log h|−ν (1 + h inf(2j , 2j ))−N +2 cj c0j 0
j,j 0
2
(c0j 0 )j 0 .
for ` sequences (cj )j ,
Now the result follows from Lemma 3.1.3 in [3] as well. ¤ We will
use the following proposition which is similar to proposition 3.1.6 in [3]:
15
d−1
d+1 1
,2
,− 1
2
Proposition 5.3 Let N ≥ (d + 3)/2. The map (u, v) → u · v is continuous on HN2−1,r 2 × HN,r
d−1
,− 1
with values in HN2−1,r 2 , r ∈ {µ, 1}.
The following proposition may be found in [2] proposition 3.2.2:
Proposition 5.4 Let s ∈ R, N ∈ N, N ≥ (d + 3)/2, and tµ ∈ {µ, 1},
s−1,−1/2
s−1
s
d
h
d+1
(uh0 )h ∈ HN
(Rd ), (uh1 )h ∈ HN
).
−1 (R ), (ftµ )h ∈ HN −1,tµ (R
Let (uh )h be the solution of
½
¤tµ uh + h−2 (tµ )2 uh = fthµ ,
uh |t=0 = uh0 , ∂t uh |t=0 = uh1 .
(58)
s,1/2
Then for any χ ∈ C0∞ (R), (χ(t)uh )h is in HN,tµ and the map
((uh0 )h , (uh1 )h , (fthµ )h ) → (χ(t)uh )h
is continuous on the preceding spaces.
Proof of theorem 2.2: Take χ ∈ C0∞ (R), χ ≡ 1 on a neighborhood of [−1, 1]. Let U 0 = (U 0 , V 0 )
be the solution of ¤µ U 0 + h−2 µ2 U 0 = 0, ¤V 0 + h−2 V 0 = 0, U 0 |t=0 = V h , ∂t U 0 |t=0 = W h , and
for n ≥ 0 define U n+1 = (U n+1 , V n+1 ) by
¤µ U n+1 + h−2 µ2 U n+1 = h−2 F1 (χ(t)U n , h∂t (χ(t)U n ), h∂x (χ(t)U n )),
¤V n+1 + h−2 V n+1 = h−2 F2 (χ(t)U n , h∂t (χ(t)U n ), h∂x (χ(t)U n )),
U n+1 |t=0 = V h ,
∂t U n+1 |t=0 = W h .
The assumption ( 3) on F , together with propositions 5.1, 5.2 and 5.3, implies that if χ(t)U n is in
(d+1)/2,1/2
(d+1)/2,1/2
(d−1)/2,−1/2
HN,µ
, χ(t)V n is in HN,1
, the right hand side of the first equation (50) in HN −1,µ
(d−1)/2,−1/2
and the second is in HN −1,1
(d+1)/2,1/2
HN,1
.
n+1
(d+1)2,1/2
. Proposition 5.4 shows that χ(t)U n+1 is then in HN,µ
h
,
h
and χ(t)V
is then in
If (V )h and (W )h are small enough, one deduces from
that the boundedness and convergence of sequence (χ(t)U n )n . The limit provides the solution we
(d+1)/2,1/2
(d+1)/2
are seeking for on ] − 1, 1[. Since χ(t)U ∈ HN,µ
implies U ∈ C 0 (] − 1, 1[, HN
) and
(d−1)/2
∂t U ∈ C 0 (] − 1, 1[, HN −1
(d+1)/2,1/2
), and from χ(t)V ∈ HN,1
(d−1)/2
and ∂t V ∈ C 0 (] − 1, 1[, HN −1
(d+1)/2
implies V ∈ C 0 (] − 1, 1[, HN
), we get the properties of the statement of the theorem.
)
¤
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