precisely, we consider F(b, u) such that T(0, 0) = 0. With indices m and SQ such
(1.33) m , s
77i+ 3 ,
consider 6 G i7 s °([-T
, T0] x fi) such that b = 0 for t 0. Assuming that the state
u = 0 belongs the domain of hyperbolicity (9 in Assumption 1.1 and shrinking
To if necessary, the mixed Cauchy problem (1.1) (1.8) has a unique solution u0 G
Hs°([—To, TQ] x Q) which vanishes for t 0. In this case, UQ given by (1.23) vanishes
for t 0 and is an exact solution of (1.4) there. We show that this solution can be
continued to [0, To] x £1 and that UQ is a good approximation.
T H E O R E M 1.11. There is e0 0 such that for all s G]0,£0] the problem
(1.4) (1.5) has a unique solution ue which vanishes for t 0. Moreover,
(1.34) ||«e-«§||«- + \\U-UUL^=0(S).
This theorem is proved in section 6. Indeed, we first construct a first corrector
u\ such that u\ = 0 for t 0, u\ = 0 on [—To, TQ] X dQ and u£a UQ + £wf satisfies
equation (1.4) up to an error e = O(s). Indeed, when one substitutes u£0 in (1.4),
the 0{e~l) term is killed by the choice (1.23) and because W satisfies (1.6) when
the boundary condition is satisfied. However, it remains an 0 ( e ~ ^ / £ ) term. The
corrector u\, given by a formula analogous to (1.23), can be chosen to cancel this
term (see the general discussion of BKW solutions in [Gr-Gu]). Then the solution
is constructed as u£a -f ev£, where solves
(1.35) Vueav£ + sQ(v£) = f:=e'1e
and Q is at least quadratic in v. Denoting by || \\xm [resp. || \\y™] the norm given
by adding the left [resp right] hand sides of (1.31) and (1.32) one proves that
\eQ(ve)\\y™ £i/4 C(M)
\e(Q(v{) - Q(vl))\\yT e^C{M) | h - v2
provided that
e\\vi\\L~ 1 , e|HU~ 1
( ' } e\\vi\\xp M e l M U r ^ M '
Together with Theorem 1.10, this shows that the equation (1.35) can be solved in
A^m, provided that e is small enough.
The main result in [Gr-Gu] is analogous to Theorem 1.11, but does not give
the existence up to TQ. They proved the linear and nonlinear stability as long as
UQ remains smaller than some small constant in a suitable norm. Here, we get the
stability under the more geometric condition that (T/, uo(t, y)) remains in a domain
where the uniform stability condition Assumption 1.4 holds, knowing that when
this condition fails, strong instabilities can occur. Note also that our estimates in
Theorem 1.10 are stronger than the corresponding estimates in [Gr-Gu], since they
proved estimates for derivatives eZ and not for Z. The price they had to pay was
that they needed a very accurate approximate solution u£a1 so that the solution is
constructed as u£a + eMv with M large, so that control of e derivatives for v gives
control of L°° norms for sMv. Moreover, the accurate approximate solutions were
constructed by BK W expansions, which require a lot of smoothness on i^o. Indeed,
they assumed i^o G C°°.
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