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