Before working out the core of the proof of Theorem 2.6 in§10, we shall need a short-time estimate, which will act as an “initial regularity layer” for the Newton scheme. (This will give us room later to allow the regularity index to depend on t.) So we run the whole scheme once in this section, and another time in the next section.
Short-time estimates in the analytic class are not new for the non-linear Vlasov equation: see in particular the work of Benachour [8] on the Vlasov–Poisson equation.
His arguments can probably be adapted for our purpose; also the Cauchy–Kovalevskaya method could certainly be applied. We shall provide here an alternative method, based on the analytic function spaces from §4, but not needing the apparatus from §§5–7.
Unlike the more sophisticated estimates which will be performed in§10, these ones are
“almost” Eulerian (the only characteristics are those of free transport). The main tool is given by the following lemma.
Lemma9.1. Let f be an analytic function,λ(t)=λ−Ktand μ(t)=μ−Kt; let T >0 be so small that λ(t), μ(t)>0 for 0tT. Then for any τ∈[0, T]and any p1,
d+ dt
t=τfZλ(t),μ(t);p
τ − K
1+τ∇fZλ(τ),μ(τ);p
τ , (9.1)
where d+/dtstands for the upper right derivative.
Remark 9.2. Time-differentiating Lebesgue integrability exponents is common prac-tice in certain areas of analysis; see e.g. [33]. Time-differentiation with respect to regu-larity exponents is less common; however, as pointed out to us by Strain, Lemma 9.1 is strongly reminiscent of a method recently used by Chemin [17] to derive local analytic regularity bounds for the Navier–Stokes equation. We expect that similar ideas can be applied to more general situations of Cauchy–Kovalevskaya type, especially for first-order equations, and maybe this has already been done.
Proof. For notational simplicity, let us assumed=1. The left-hand side of (9.1) is
k∈Zd n∈Nd0
e2πμ(τ)|k|2πμ(τ)|k|˙ λn(τ)
n! (∇v+2iπkτ)nfˆ(k, v)Lp(dv)
+
k∈Zd n∈N0d
e2πμ(τ)|k|λ(τ˙ )λn−1(τ)
(n−1)! (∇v+2iπkτnfˆ(k, v)Lp(dv)
−K
k∈Zd n∈Nd0
e2πμ(τ)|k|2π|k|λn(τ)
n! (∇v+2iπkτ)nfˆ(k, v)Lp(dv)
−K
k∈Zd n∈Nd0
e2πμ(τ)|k|λn(τ) n!
∇v+2iπkτn+1fˆ(k, v)Lp(dv)
−K
k∈Zd n∈Nd0
e2πμ(τ)|k|λn(τ)
n! (∇v+2iπkτ)n∇xf(k, v)Lp(dv)
+Kτ 1+τ
k∈Zd n∈Nd0
e2πμ(τ)|k|λn(τ)
n! (∇v+2iπkτn∇xf(k, v)
Lp(dv)
− K 1+τ
k∈Zd n∈Nd0
e2πμ(τ)|k|λn(τ)
n! (∇v+2iπkτ)n∇vf(k, v)Lp(dv),
where in the last step we used that
(∇v+2iπkτ)h∇vh−τ2iπkh
1+τ .
The conclusion follows.
Now let us see how to propagate estimates through the Newton scheme described in
§10. The first stage of the iteration (h1in the notation of (8.3)) was considered in§4.12, so we only need to care about higher orders. For anyk1, we solve
∂thk+1+v·∇xhk+1=Σ!k+1, where
Σ!k+1=−(F[hk+1]·∇vfk+F[fk]·∇vhk+1+F[hk]·∇vhk)
(note the difference with (8.7)–(8.8)). Recall thatfk=f0+h1+...+hk. We define λk(t) =λk−2Kt and μk(t) =μk−Kt,
where (λk)∞k=1and (μk)∞k=1 are decreasing sequences of positive numbers.
We assume inductively that at stagenof the iteration, we have constructed (λk)nk=1, (μk)nk=1 and (δk)nk=1 such that
0tTsup hk(t,·)Zλk(t),μk(t);1
t δk for all 1kn
for some fixedT >0. The issue is to constructλn+1,μn+1andδn+1so that the induction hypothesis is satisfied at stagen+1.
Att=0, hn+1=0. Then we estimate the time-derivative ofhn+1Zλn+1(t),μn+1(t);1 t
. Let us first pretend that the regularity indicesλn+1 andμn+1 do not depend ont; then
hn+1(t) = t
0
Σ!n+1 S−(t0 −τ)dτ,
so, by Proposition 4.19,
hn+1Zλn+1,μn+1;1
t
t
0 Σ!n+1τ S−(t−τ)0 Zλn+1,μn+1;1 t
dτ
t
0 Σ!n+1τ Zλn+1,μn+1;1
τ dτ,
and thus
d+
dthn+1Zλn+1,μn+1;1
t Σ!n+1t Zλn+1,μn+1;1 t
.
Finally, according to Lemma 9.1, to this estimate we should add a negative multiple of the norm of∇hn+1 to take into account the time-dependence ofλn+1 andμn+1.
All in all, after application of Proposition 4.24, we get d+
dthn+1(t,·)Zλn+1(t),μn+1(t);1 t
F[hn+1t ]Fλn+1t+μn+1∇vftnZλn+1,μn+1;1 t
+F[ftn]Fλn+1t+μn+1∇vhn+1t Zλn+1,μn+1;1 t
+F[hnt]Fλn+1t+μn+1∇vhntZλn+1,μn+1;1 t
−K∇xhn+1t Zλn+1,μn+1;1
t −K∇vhn+1t Zλn+1,μn+1;1
t ,
where K >0,t is sufficiently small, and all exponents λn+1 andμn+1 in the right-hand side actually depend ont.
From Proposition 4.15 (iv) we easily get F[h]Fλt+μC∇hZλ,μ;1
t . Moreover, by Proposition 4.10,
∇fnZλn+1,μn+1;1
t
n k=1
∇hkZλn+1,μn+1;1
t C
n k=1
hkZλk+1,μk+1;1
min{λk−λn+1t , μk−μn+1}.
We end up with the bound d+
dthn+1(t,·)Zλn+1(t),μn+1(t);1 t
C n k=1
δk
min{λk−λn+1, μk−μn+1}−K
∇hn+1Zλn+1(t),μn+1(t);1 t
+ δ2n
min{λn−λn+1, μn−μn+1}. We conclude that if
n k=1
δk
min{λk−λn+1, μk−μn+1}K
C, (9.2)
then we may choose
δn+1= δn2
min{λn−λn+1, μn−μn+1}. (9.3) This is our first encounter with the principle of “stratification” of errors, which will be crucial in the next section: to control the error at stagen+1, we use not only the smallness of the error from stagen, but also an information about all previous errors; namely the fact that the convergence of the size of the error is much faster than the convergence of the regularity loss. Let us see how this works. We chooseλk−λk+1=μk−μk+1=Λ/k2, where Λ>0 is arbitrarily small. Then forkn,λk−λn+1Λ/k2, and thereforeδn+1δn2n2/Λ.
The problem is to check that
∞ n=1
n2δn<∞. (9.4)
Indeed, then we can chooseK large enough for (9.2) to be satisfied, and thenT small enough that, say, λ∗−2KTλ and μ∗−KTμ, where λ<λ∗ and μ<μ∗ have been fixed in advance.
Ifδ1=δ, the general term in the series of (9.4) is n2δ2n
Λn(22)2n−1(32)2n−2(42)2n−2...((n−1)2)2n2.
To prove the convergence forδsmall enough, we assume by induction thatδnzan, where ais fixed in the interval (1,2) (saya=1.5); and we claim that this condition propagates ifz >0 is small enough. Indeed,
δn+1z2an
Λ n2zan+1z(2−a)ann2
Λ ,
and this is bounded above byzan+1 ifz is so small that z(2−a)an Λ
n2 for alln∈N.
This concludes the iteration argument. Note that the convergence is still extremely fast—like O(zan) for any a<2. (Of course, when a approaches 2, the constants be-come huge, and the restriction on the size of the perturbation bebe-comes more and more stringent.)
Remark 9.3. The method used in this section can certainly be applied to more gen-eral situations of Cauchy–Kovalevskaya type. Actually, as pointed out to us by Bony and G´erard, the use of a regularity index which decays linearly in time, combined with a Newton iteration, was used by Nirenberg [73] to prove an abstract Cauchy–Kovalevskaya theorem. Nirenberg uses a time-integral formulation, so there is nothing in [73] compa-rable to Lemma 9.1, and the details of the proof of convergence differ from ours; but the general strategy is similar. Nirenberg’s proof was later simplified by Nishida [74] with a clever fixed-point argument; in the present section anyway, our final goal is to provide short-term estimates for the successive corrections arising from the Newton scheme.