1. INTRODUCTION
9
This theorem is proved in section 4 below together with slight improvements
which are needed in the proof of the estimates for the derivatives. Let us just
mention where the difficulty lies. The coefficients of Vu% depend on ^p{x/e) and thus
are not (uniformly) Lipschitzean. Moreover, the coefficient of u in Vu% has a factor
^ in front of it. Thus the usual energy method using integration by parts yields
singular and apparently uncontrolled terms. This is exactly where the smallness
assumption in [Gr-Gu] comes in. Using it together with a tricky argument, the
authors were able to absorb the singular terms. Our main objective in this paper,
is to replace the smallness argument by a detailed analysis of
Vuea
and to use the
Assumption 1.4 to construct symmetrizers.
The next step is to prove estimates for the derivatives of the solution u. The
classical approach is to differentiate (1.25) with respect to vector fields which are
tangent to R x dfl, in order have natural boundary conditions for the derivatives.
For non characteristic problems, the normal derivatives are deduced from the tan-
gential ones using the equations. Here, we can adapt the first argument, but the
second certainly fails since the coefficients of
Vuea
are singular in the normal direc-
tion and the solution cannot be (uniformly) smooth in this direction. This leads us
to introduce spaces with conormal Sobolev smoothness. Such spaces have already
been widely used in the study of boundary value problems, see e.g. [Ra2], [Gu2].
Let {Zk}okk denote a finite set of generators of vector fields tangent to R x dVt,
with for instance ZQ = dt. For [ / c M x l ] and m G N, define the space
. .
nm(U)
:= {u G
L2(U)
:Zkl... Zkpu G
L2(U),
(1.28)
Vpm,V(fci,...,fc
p
)e{0,...fc}
p
}
This space is equipped with the obvious norm, denoted by ||
\\um{U)-
In order to solve nonlinear problems, we need work in Banach algebras which
means here that we have to supplement the
Hm
estimates with L°° estimates.
Introduce the following norms
(1.29) Nlw*(c/) =
IMIL~
+.]T^ Yl \\Zkl...Zkpu\\L~ .
p=l lfci,...,/cpfc
Reinforcing (1.26), we now assume that on OT0
:— [—Tb,Tb]
x fi,
( u0 e ^
m + 2
' ° ° (CiTo), be ^
m + 2
' ° ° (QTo)
( L 3 0 )
\ sup
||ue||Wm
+ £ | | V ^
£
| |
W
+e
2
||V;y||
w
oo.
THEOREM
1.10. There are C 0 and
SQ
such that all e
E]0,£O]
and all f G
Wm([—To,
To] x Ct) vanishing for t 0, the solution of equation (1.25) satisfies
(1.31) \\u\\Hm + VZ\\dxu\\H~ +
e3/2\\d2xu\\Hm
CWfWn-n
If in addition m 2 -f ^ ^ and f G L°°([—TO,TQ] X Q), then the solution u also
satisfies
(1.32) ||«||
w
, + e\\8xu\W +
z2\\d2xu\\L™
C(||/||«
m
+ e||/||i=o) .
These results can be used to solve the nonlinear equations (1.4). In order to
avoid technical discussions on compatibility conditions for the Cauchy data and the
boundary conditions, we consider here the simple case where the Cauchy data for
(1.1) and (1.2) are zero, but with a non trivial forcing term, see [Gr-Gu]. More
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