where the constant $C$ does not depend on $k$, and

$\sum_{|I|\leq
N}\big{\|}(t\partial_{x})^{I}P_{[k]}(t,\cdot)\big{\|}_{L^{2}_{x}}\leq
C_{k}\mathcal{F}_{\infty}^{N+2k+3}\frac{(\log t)^{1+k}}{t^{\frac{5}{2}+k}}.$
(5.13)

###### Proof of Proposition 5.2.

The estimate (5.13) follows immediately from Theorem 4.1 (see (4.2)).

For (5.12), consider first the first term on the right hand side of (5.11).
Note that, since $N\geq 3$,

$\displaystyle\sum_{|I|+|J|\leq
N}\big{\|}L^{I}(t^{-1}\partial_{p})^{J}\big{(}\partial_{x^{i}}\check{\phi}_{[k]}\partial_{p^{i}}f_{[k]}\big{)}(t,\cdot,\cdot)\big{\|}_{L^{2}_{x}L^{2}_{p}}\lesssim$
$\displaystyle\qquad\qquad\qquad\sum_{|I|=1}^{N-2}\sup_{x\in\mathbb{R}^{3}}\big{|}(t\partial_{x})^{I}\check{\phi}_{[k]}(t,\cdot)\big{|}\sum_{\begin{subarray}{c}|I|+|J|\leq
N+1\\\ |J|\geq
1\end{subarray}}\big{\|}L^{I}(t^{-1}\partial_{p})^{J}f_{[k]}(t,\cdot,\cdot)\big{\|}_{L^{2}_{x}L^{2}_{p}}$
$\displaystyle\qquad\qquad\qquad+\sum_{|I|=1}^{N+1}\big{\|}(t\partial_{x})^{I}\check{\phi}_{[k]}(t,\cdot)\big{\|}_{L^{2}_{x}}\sum_{\begin{subarray}{c}|I|+|J|\leq
N-2\\\ |J|\geq
1\end{subarray}}\sup_{x\in\mathbb{R}^{3}}\big{\|}L^{I}(t^{-1}\partial_{p})^{J}f_{[k]}(t,x,\cdot)\big{\|}_{L^{2}_{p}},$

It follows from the Sobolev inequality — see Remark 2.2 — that

$\sup_{x\in\mathbb{R}^{3}}\sum_{|I|=1}^{N-2}|(t\partial_{x})^{I}\check{\phi}_{[k]}(t,x)|\leq
Ct^{-\frac{3}{2}}\sum_{|I|=1}^{N}\|(t\partial_{x})^{I}\check{\phi}_{[k]}(t,\cdot)\|_{L^{2}_{x}},$

and similarly, by Theorem 4.1 (see (4.3)) and the Sobolev inequality of
Proposition 2.14,

$\sum_{|I|+|J|\leq
N-2}\sup_{x\in\mathbb{R}^{3}}t^{\frac{3}{2}}\big{\|}L^{I}(t^{-1}\partial_{p})^{J}f_{[k]}(t,x,\cdot)\big{\|}_{L^{2}_{p}}+\sum_{|I|+|J|\leq
N+1}\big{\|}L^{I}(t^{-1}\partial_{p})^{J}f_{[k]}(t,\cdot,\cdot)\big{\|}_{L^{2}_{x}L^{2}_{p}}\\\
\leq C\mathcal{F}_{\infty}^{N+1}+\frac{C_{k}\mathcal{F}_{\infty}^{N+k}\log
t}{t}\leq C\mathcal{F}_{\infty}^{N+1},$

where the constant $C$ does not depend on $k$, and the latter inequality
follows after taking $T_{0}$ suitably large. The second term on the right hand
side of (5.11) is estimated in Theorem 4.1 (see (4.1)). The proof of (5.12)
then follows. ∎

In the following proposition the size of the support of $\check{f}_{[k]}$ is
estimated.

######  Proposition 5.3 (The support of $\check{f}_{[k]}$).

If $T_{0}=T_{0}(\mathcal{F}_{\infty}^{N+2k+4})$ is sufficiently large, then

$\displaystyle\mathrm{supp}(\check{f}_{[k]}(t,\cdot,\cdot))$
$\displaystyle\subset\\{(x,p)\subset\mathbb{R}^{3}\times\mathbb{R}^{3}\mid|x-tp+\log
t\nabla\phi_{\infty}(p)|\leq 2\mathcal{F}_{\infty}^{3}+B,|p|\leq 2B\\}$
$\displaystyle\subset\\{(x,p)\subset\mathbb{R}^{3}\times\mathbb{R}^{3}\mid|x|\leq
2\mathcal{F}_{\infty}^{3}+3Bt,|p|\leq 2B\\}.$

###### Proof.

Theorem 4.1, in particular the property (4.5), implies that

$\mathrm{supp}(f_{[k]})\subset\\{(x,p)\subset\mathbb{R}^{3}\times\mathbb{R}^{3}\mid|x-tp+\log
t\nabla\phi_{\infty}(p)|\leq B,|p|\leq B\\}.$

Consider, therefore the support of $f$. Note that, for
$f_{T_{f}}(x,p):=f(T_{f},x,p)$,

$|x-T_{f}p+\log T_{f}\,\nabla\phi_{\infty}(p)|\leq B,\qquad\text{for all
}(x,p)\in\mathrm{supp}(f_{T_{f}}),$

which follows from the definition (5.2) of $f_{T_{f}}$ and the fact that each
$f_{k,l}$ satisfies (4.5). Define the characteristic curves of $\mathbb{X}$,
denoted
$X,P\colon[T,T_{f}]\times[T,T_{f}]\times\mathbb{R}^{3}\times\mathbb{R}^{3}\to\mathbb{R}^{3}$,
as solutions of

$\frac{dX(s,t,x,p)}{ds}=P(s,t,x,p),\qquad\frac{dP(s,t,x,p)}{ds}=(\nabla\phi)(s,X(s,t,x,p)),$

along with the initial conditions

$X(t,t,x,p)=x,\qquad P(t,t,x,p)=p.$

Clearly, for any $x,p\in\mathbb{R}^{3}$, $T\leq s\leq T_{f}$,

$f(s,X(s,T_{f},x,p),P(s,T_{f},x,p))=f(T_{f},x,p),$

and so

$\mathrm{supp}(f)=\\{(s,X(s,T_{f},x,p),P(s,T_{f},x,p))\mid(x,p)\in\mathrm{supp}(f_{T_{f}}),T\leq
s\leq T_{f}\\}.$ (5.14)

We proceed by subtracting from $X$ and $P$ suitable approximate solutions of
the characteristic equations and estimating the difference (cf. [16], where
similar approximate solutions of the characteristic equations are introduced
in the context of the Einstein–Vlasov system). Define

$\overline{X}(s,t,x,p)=X(s,t,x,p)-\Big{(}x-(t-s)p-\log\Big{(}\frac{s}{t}\Big{)}\nabla\phi_{\infty}(p)\Big{)},\qquad\overline{P}(s,t,x,p)=P(s,t,x,p)-\Big{(}p-\frac{1}{s}\nabla\phi_{\infty}(p)\Big{)},$

so that

$\frac{d\overline{X}(s,t,x,p)}{ds}=\overline{P}(s,t,x,p),\qquad\frac{d\overline{P}(s,t,x,p)}{ds}=(\nabla\phi)(s,X(s,t,x,p))-\frac{1}{s^{2}}\nabla\phi_{\infty}(p),$
(5.15)

along with the initial conditions

$X(t,t,x,p)=0,\qquad P(t,t,x,p)=\frac{1}{t}\nabla\phi_{\infty}(p).$

Consider some $(x,p)\in\mathrm{supp}(f_{T_{f}})$, $T\leq t\leq s\leq T_{f}$.
Using the fact that

$\sup_{x\in\mathbb{R}^{3}}\left|\nabla\phi(t,x)-t^{-2}\nabla\phi_{\infty}\left(\frac{x}{t}\right)\right|\leq\frac{\mathcal{F}_{\infty}^{N+2k+4}}{t^{\frac{5}{2}}},\qquad\sup_{p\in\mathbb{R}^{3}}\Big{(}|\nabla\phi_{\infty}(p)|+|\nabla^{2}\phi_{\infty}(p)|\Big{)}\lesssim\mathcal{F}_{\infty}^{3},$

(see the bootstrap assumption (5.4) and the fact (3.2)) and,

$\displaystyle\left|\frac{X(s,T_{f},x,p)}{s}-p\right|$
$\displaystyle=s^{-1}\left|\overline{X}(s,T_{f},x,p)-\log
s\,\nabla\phi_{\infty}(p)+x-T_{f}p+\log T_{f}\nabla\phi_{\infty}(p)\right|$
$\displaystyle\lesssim\frac{|\overline{X}(s,T_{f},x,p)|}{s}+\frac{\log
s}{s}\mathcal{F}_{\infty}^{3}+\frac{B}{s},$

it follows that

$\displaystyle\left|\frac{d\overline{P}(s,T_{f},x,p)}{ds}\right|$
$\displaystyle\lesssim\left|(\nabla\phi)(s,X(s,T_{f},x,p))-\frac{1}{s^{2}}(\nabla\phi_{\infty})\left(\frac{X(s,T_{f},x,p)}{s}\right)\right|$
$\displaystyle\quad+\frac{1}{s^{2}}\left|(\nabla\phi_{\infty})\left(\frac{X(s,T_{f},x,p)}{s}\right)-(\nabla\phi_{\infty})(p)\right|$
$\displaystyle\lesssim\mathcal{F}_{\infty}^{3}\frac{|\overline{X}(s,T_{f},x,p)|}{s^{3}}+\frac{\mathcal{F}_{\infty}^{N+2k+4}}{s^{\frac{5}{2}}}+\frac{\mathcal{F}_{\infty}^{3}(B+\log
s\mathcal{F}_{\infty}^{3})}{s^{3}}.$

Integrating the equations (5.15) backwards from time $T_{f}$, the fact that

$\int_{t}^{T_{f}}\int_{s}^{T_{f}}\frac{|\overline{X}(s^{\prime},T_{f},x,p)|}{(s^{\prime})^{3}}ds^{\prime}ds=\int_{t}^{T_{f}}(s-t)\frac{|\overline{X}(s,T_{f},x,p)|}{s^{3}}ds,$

then implies that

$\displaystyle|\overline{X}(t,T_{f},x,p)|$
$\displaystyle\lesssim|\nabla\phi_{\infty}(p)|+\frac{\mathcal{F}_{\infty}^{N+2k+4}}{t^{\frac{1}{2}}}+\frac{\mathcal{F}_{\infty}^{3}(B+\log
t\mathcal{F}_{\infty}^{3})}{t}+\mathcal{F}_{\infty}^{3}\int_{t}^{T_{f}}\frac{|\overline{X}(s,T_{f},x,p)|}{s^{2}}ds$
$\displaystyle\leq\frac{3}{2}\mathcal{F}_{\infty}^{3}+C\mathcal{F}_{\infty}^{3}\int_{t}^{T_{f}}\frac{|\overline{X}(s,T_{f},x,p)|}{s^{2}}ds,$

if $T_{0}$ is suitably large. The Grönwall inequality, Proposition 2.6, then
gives

$|\overline{X}(t,T_{f},x,p)|+t|\overline{P}(t,T_{f},x,p)|\leq\frac{3}{2}\mathcal{F}_{\infty}^{3}+C(\mathcal{F}_{\infty}^{3})^{2}\int_{t}^{T_{f}}\frac{1}{s^{2}}e^{\int_{t}^{s}\frac{C\mathcal{F}_{\infty}^{3}}{(s^{\prime})^{2}}ds^{\prime}}ds\leq\frac{3}{2}\mathcal{F}_{\infty}^{3}+\frac{C(\mathcal{F}_{\infty}^{3})^{2}e^{\frac{C\mathcal{F}_{\infty}^{3}}{t}}}{t}.$

It then follows that

$\big{|}X(t,T_{f},x,p)-tp+\log
t\nabla\phi_{\infty}(p)\big{)}\big{|}+t|P(t,T_{f},x,p)-p|\leq
2\mathcal{F}_{\infty}^{3}+B,$

which, by (5.14), completes the proof. ∎

###  5.2 Estimates for remainders

In this section the remainder quantities $\check{f}_{[k]}$,
$\check{\varrho}_{[k]}$, $\check{\phi}_{[k]}$ are estimated. The main result
of this section is the following estimate for $\check{f}_{[k]}$.

######  Proposition 5.4 (Estimates for
$L^{I}(t^{-1}\partial_{p})^{J}\check{f}_{[k]}$).

Under the assumptions of Theorem 5.1, if $k$ is sufficiently large
($k>(C+1)\mathcal{F}_{\infty}^{N+1}-1$ suffices, where $C$ is the constant
from (5.12)), and $T_{0}$ is sufficiently large (depending on $k$ and
$\mathcal{F}_{\infty}^{N+2k+4}$) then, for all $T\leq t\leq T_{f}$,

$\sum_{|I|+|J|\leq
N}\|L^{I}(t^{-1}\partial_{p})^{J}\check{f}_{[k]}(t,\cdot,\cdot)\|_{L^{2}_{x}L^{2}_{p}}\leq
C_{k}(\mathcal{F}_{\infty}^{N+2k+4})^{2}\frac{(\log t)^{1+k}}{t^{1+k}}.$
(5.16)

The reader is again referred to Section 1.3.2 for an overview of the proof of
Proposition 5.4. The proof requires the following preliminary results.

######  Proposition 5.5 (Estimates for
$(t\partial_{x})^{I}\check{\phi}_{[k]}$).

For any multi-index $I$ such that $|I|\leq N$, for all $k\geq 0$ and all
$T\leq t\leq T_{f}$,

$\|\nabla(t\partial_{x})^{I}\check{\phi}_{[k]}(t,\cdot)\|_{L^{2}}\lesssim
t\|(t\partial_{x})^{I}\check{\varrho}_{[k]}(t,\cdot)\|_{L^{2}}.$

###### Proof.

Recall that

$\Delta_{\mathbb{R}^{3}}\check{\phi}_{[k]}=\check{\varrho}_{[k]}.$

The proof follows from Proposition 2.7 and the fact that $|x|\leq 3Bt$ in
$\mathrm{supp}(\check{\varrho}_{[k]})$ (see Proposition 5.3, and the property
(4.5)), since $t\partial_{x^{i}}$ commutes with $\Delta_{\mathbb{R}^{3}}$. ∎

######  Proposition 5.6 (Estimates for
$(t\partial_{x})^{I}\check{\varrho}_{[k]}$).

For any multi-index $I$ such that $|I|\leq N$, for all $k\geq 0$ and all
$T\leq t\leq T_{f}$,

$\|(t\partial_{x})^{I}\check{\varrho}_{[k]}(t,\cdot)\|_{L^{2}_{x}}\lesssim
t^{-\frac{3}{2}}\|L^{I}\check{f}_{[k]}(t,\cdot,\cdot)\|_{L^{2}_{x}L^{2}_{p}}+C_{k}\mathcal{F}_{\infty}^{N+2k+3}\frac{(\log
t)^{1+k}}{t^{\frac{5}{2}+k}}.$

###### Proof.

Recall (see Proposition 5.2) that

$\check{\varrho}_{[k]}(t,x)=\int_{\mathbb{R}^{3}}\check{f}_{[k]}(t,x,p)dp+P_{[k]},$

where $P_{[k]}$ satisfies (5.13). Since

$\int_{\mathbb{R}^{3}}\partial_{p^{i}}L^{J}\check{f}_{[k]}(t,x,p)dp=0,$

for $i=1,2,3$ and each multi-index $J$, it follows that, for any multi-index
$I$,

$(t\partial_{x})^{I}\check{\varrho}_{[k]}(t,x)=\int_{\mathbb{R}^{3}}L^{I}\check{f}_{[k]}(t,x,p)dp+(t\partial_{x})^{I}P_{[k]}(t,x).$

Hence

$\|(t\partial_{x})^{I}\check{\varrho}_{[k]}(t,\cdot)\|_{L^{2}_{x}}\lesssim\Big{(}\int_{\mathbb{R}^{3}}\Big{(}\int_{\mathbb{R}^{3}}L^{I}\check{f}_{[k]}(t,x,p)dp\Big{)}^{2}dx\Big{)}^{\frac{1}{2}}+\|(t\partial_{x})^{I}P_{[k]}(t,\cdot)\|_{L^{2}_{x}}\\\
\lesssim\Big{(}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\mathds{1}_{\check{f}_{[k]}}(t,x,p)dp\int_{\mathbb{R}^{3}}|L^{I}\check{f}_{[k]}(t,x,p)|^{2}dpdx\Big{)}^{\frac{1}{2}}+\|(t\partial_{x})^{I}P_{[k]}(t,\cdot)\|_{L^{2}_{x}}.$

Note now that

$\int_{\mathbb{R}^{3}}\mathds{1}_{\mathrm{supp}(\check{f}_{[k]})}(t,x,p)dp\lesssim\frac{1+(\mathcal{F}_{\infty}^{3})^{3}}{t^{3}},$

which follows from the fact that

$\Big{|}\frac{x}{t}+\frac{\log
t}{t}\nabla\phi_{\infty}\Big{(}\frac{x}{t}\Big{)}-p\Big{|}\leq\frac{3\mathcal{F}_{\infty}^{3}+2B}{t}\quad\text{in}\quad\mathrm{supp}(\check{f}_{[k]}).$

See Proposition 5.3 (and cf. Proposition 4.5). The proof then follows from the
estimate (5.13). ∎

In the following proposition, the terms arising from commuting the equation
(5.10) for $\check{f}_{[k]}$ with the vector fields of Section 2.3 are
estimated, using the bootstrap assumption (5.3).

######  Proposition 5.7 (Commuted equation for $\check{f}_{[k]}$).

If the bootstrap assumption (5.3) holds, then

$\sum_{|I|+|J|\leq
N}\|\mathbb{X}_{\phi}(L^{I}(t^{-1}\partial_{p})^{J}\check{f}_{[k]})\|_{L^{2}_{x}L^{2}_{p}}\lesssim\sum_{|I|+|J|\leq
N}\|L^{I}(t^{-1}\partial_{p})^{J}F_{[k]}\|_{L^{2}_{x}L^{2}_{p}}\\\
+\frac{C_{k}\mathcal{F}_{\infty}^{N+2k+3}(\log
t)^{2}}{t^{2}}\sum_{|I|+|J|=1}^{N}\|L^{I}(t^{-1}\partial_{p})^{J}\check{f}_{[k]}\|_{L^{2}_{x}L^{2}_{p}}+(\mathcal{F}_{\infty}^{N+2k+4})^{2}\frac{\log
t}{t^{\frac{5}{2}}}\sum_{|K|=1}^{N+1}\|(t\partial_{x})^{K}\check{\phi}_{[k]}\|_{L^{2}_{x}}.$
(5.17)

###### Proof.

Recall Proposition 2.10, which implies that, for $i=1,2,3$,

$\displaystyle\Big{[}\mathbb{X}_{\phi},L_{i}\Big{]}=\ $
$\displaystyle-\frac{\log
t}{t^{2}}\partial_{i}\partial_{j}\phi_{\infty}(p)\,L_{j}+\Big{[}t^{-1}\Big{(}\partial_{i}\partial_{k}\phi_{\infty}(p)-\partial_{i}\partial_{k}\phi_{\infty}\Big{(}\frac{x}{t}\Big{)}\Big{)}+\frac{\log
t}{t^{2}}\partial_{l}\phi_{\infty}(p)\,\partial_{i}\partial_{k}\partial_{l}\phi_{\infty}(p)$
$\displaystyle+\frac{(\log
t)^{2}}{t^{2}}\partial_{i}\partial_{j}\phi_{\infty}(p)\partial_{j}\partial_{k}\phi_{\infty}(p)-t\partial_{x^{i}}\partial_{x^{k}}\check{\phi}_{[0]}(t,x)+\log
t\,\partial_{i}\partial_{k}\partial_{l}\phi_{\infty}(p)\,\partial_{x^{l}}\check{\phi}_{[0]}(t,x)\Big{]}t^{-1}\partial_{p^{k}},$

and

$\displaystyle\Big{[}\mathbb{X}_{\phi},t^{-1}\partial_{p^{i}}\Big{]}=-\frac{1}{t^{2}}L_{i}+\frac{\log
t}{t^{2}}\partial_{i}\partial_{k}\phi_{\infty}(p)\,t^{-1}\partial_{p^{k}}.$

Note that, for any smooth $h\colon\mathbb{R}^{3}\to\mathbb{R}$,

$L_{i}\Big{(}h(p)-h\Big{(}\frac{x}{t}\Big{)}\Big{)}=(\partial_{i}h)(p)-(\partial_{i}h)\Big{(}\frac{x}{t}\Big{)}+\frac{\log
t}{t}\partial_{i}\partial_{k}\phi_{\infty}(p)(\partial_{k}h)(p),$

and so

$\sup_{|y(t,x,p)|\leq
B}\Big{|}L_{i}\Big{(}h(p)-h\Big{(}\frac{x}{t}\Big{)}\Big{)}\Big{|}\lesssim\|\nabla^{2}h\|_{L^{\infty}}\frac{2B+\mathcal{F}_{\infty}^{3}\log
t}{t}+\|\nabla h\|_{L^{\infty}}\frac{\mathcal{F}_{\infty}^{3}\log t}{t}.$

It follows from a simple induction argument, along with (3.2) and the fact
that, for any $n\geq 0$ and any multi-indices $I$, $J$ such that $|I|+|J|\leq
n$,

$\|\mathbb{X}_{\phi}(L^{I}(t^{-1}\partial_{p})^{J}\check{f}_{[k]})\|_{L^{2}_{x}L^{2}_{p}}\leq\|L^{I}(t^{-1}\partial_{p})^{J}F_{[k]}\|_{L^{2}_{x}L^{2}_{p}}+C\frac{\mathcal{F}_{\infty}^{n+3}(\log
t)^{2}}{t^{2}}\sum_{|I^{\prime}|+|J^{\prime}|=1}^{n}\|L^{I^{\prime}}(t^{-1}\partial_{p})^{J^{\prime}}\check{f}_{[k]}\|_{L^{2}_{x}L^{2}_{p}}\\\
+C(1+\mathcal{F}_{\infty}^{n+3})\,\frac{\log
t}{t}\Big{(}\sum_{|K|=1}^{n+1}\|(t\partial_{x})^{K}\check{\phi}_{[0]}\|_{L^{2}_{x}}\sum_{|I^{\prime}|+|J^{\prime}|=1}^{\lfloor\frac{n+2}{2}\rfloor}\|L^{I^{\prime}}(t^{-1}\partial_{p})^{J^{\prime}}\check{f}_{[k]}\|_{L^{\infty}_{x}L^{2}_{p}}\\\
+\sum_{|K|=1}^{\lfloor\frac{n+4}{2}\rfloor}\|(t\partial_{x})^{K}\check{\phi}_{[0]}\|_{L^{\infty}_{x}}\sum_{|I^{\prime}|+|J^{\prime}|=1}^{n}\|L^{I^{\prime}}(t^{-1}\partial_{p})^{J^{\prime}}\check{f}_{[k]}\|_{L^{2}_{x}L^{2}_{p}}\Big{)},$
(5.18)

The Sobolev inequality of Proposition 2.14 and the fact (3.2) (since $N\geq
3$), implies that the right hand side of (5.18) is controlled by

$\sum_{|I|+|J|\leq
N}\|L^{I}(t^{-1}\partial_{p})^{J}F_{[k]}\|_{L^{2}_{x}L^{2}_{p}}\\\
+\Big{(}\frac{\mathcal{F}_{\infty}^{N+3}(\log
t)^{2}}{t^{2}}+(1+\mathcal{F}_{\infty}^{N+3})\,\frac{\log
t}{t^{\frac{5}{2}}}\sum_{|K|=1}^{N+1}\|(t\partial_{x})^{K}\check{\phi}_{[0]}\|_{L^{2}_{x}}\Big{)}\sum_{|I|+|J|=1}^{N}\|L^{I}(t^{-1}\partial_{p})^{J}\check{f}_{[k]}\|_{L^{2}_{x}L^{2}_{p}}.$

Note now that $\check{\phi}_{[0]}=\check{\phi}_{[k]}+(\phi_{[k]}-\phi_{[0]})$,
and that Proposition 5.5 and Proposition 5.6 imply that

$\sum_{|I|=1}^{N+1}\|(t\partial_{x})^{I}(\phi_{[k]}-\phi_{[0]})(t,\cdot)\|_{L^{2}}\lesssim
t^{\frac{1}{2}}\sum_{|I|\leq
N}\|L^{I}(f_{[k]}-f_{[0]})(t,\cdot,\cdot)\|_{L^{2}_{x}L^{2}_{p}}\leq\frac{C_{k}\mathcal{F}_{\infty}^{N+2k+3}\log
t}{t^{\frac{1}{2}}},$

where (4.4) has been used. It follows that

$\sum_{|I|+|J|\leq
N}\|\mathbb{X}_{\phi}(L^{I}(t^{-1}\partial_{p})^{J}\check{f}_{[k]})\|_{L^{2}_{x}L^{2}_{p}}\lesssim\sum_{|I|+|J|\leq
N}\|L^{I}(t^{-1}\partial_{p})^{J}F_{[k]}\|_{L^{2}_{x}L^{2}_{p}}\\\
+\Big{(}\frac{C_{k}\mathcal{F}_{\infty}^{N+2k+3}(\log
t)^{2}}{t^{2}}+(1+\mathcal{F}_{\infty}^{N+3})\,\frac{\log
t}{t^{\frac{5}{2}}}\sum_{|K|=1}^{N+1}\|(t\partial_{x})^{K}\check{\phi}_{[k]}\|_{L^{2}_{x}}\Big{)}\sum_{|I|+|J|=1}^{N}\|L^{I}(t^{-1}\partial_{p})^{J}\check{f}_{[k]}\|_{L^{2}_{x}L^{2}_{p}}.$

The proof of (5.17) follows from inserting the bootstrap assumption (5.3).

∎

The proof of Proposition 5.4 can now be given.

###### Proof of Proposition 5.4.

If $T_{0}=T_{0}(\mathcal{F}_{\infty}^{N+2k+4})$ is suitably large so that the
coefficient of the final term in Proposition 5.7 satisfies

$(\mathcal{F}_{\infty}^{N+2k+4})^{2}\frac{\log
t}{t}\leq\mathcal{F}_{\infty}^{N+1},$

then Proposition 5.2 and Proposition 5.7 imply that,

$\displaystyle\sum_{|I|+|J|\leq
N}\|\mathbb{X}_{\phi}(L^{I}(t^{-1}\partial_{p})^{J}\check{f}_{[k]})\|_{L^{2}_{x}L^{2}_{p}}\leq\
$ $\displaystyle C_{k}(\mathcal{F}_{\infty}^{N+2k+3})^{2}\frac{(\log
t)^{1+k}}{t^{2+k}}+\frac{C\mathcal{F}_{\infty}^{N+1}}{t^{\frac{3}{2}}}\sum_{|I|=1}^{N+1}\|(t\partial_{x})^{I}\check{\phi}_{[k]}(t,\cdot)\|_{L^{2}}$
$\displaystyle+\frac{C_{k}\mathcal{F}_{\infty}^{N+2k+3}(\log
t)^{2}}{t^{2}}\sum_{|I|+|J|=1}^{N}\|L^{I}(t^{-1}\partial_{p})^{J}\check{f}_{[k]}\|_{L^{2}_{x}L^{2}_{p}}.$

Now Proposition 5.5 and Proposition 5.6 imply that

$\sum_{|I|=1}^{N+1}\|(t\partial_{x})^{I}\check{\phi}_{[k]}(t,\cdot)\|_{L^{2}}\lesssim
t^{\frac{1}{2}}\sum_{|I|\leq
N}\|L^{I}\check{f}_{[k]}(t,\cdot,\cdot)\|_{L^{2}_{x}L^{2}_{p}}+C_{k}\mathcal{F}_{\infty}^{N+2k+3}\frac{(\log
t)^{1+k}}{t^{\frac{1}{2}+k}}.$

Thus, if $T_{0}=T_{0}(\mathcal{F}_{\infty}^{N+2k+3},k)$ is suitably large so
that

$\frac{C_{k}\mathcal{F}_{\infty}^{N+2k+3}(\log
t)^{2}}{t}\leq\mathcal{F}_{\infty}^{N+1},$

then

$\displaystyle\sum_{|I|+|J|\leq
N}\|\mathbb{X}_{\phi}(L^{I}(t^{-1}\partial_{p})^{J}\check{f}_{[k]})\|_{L^{2}_{x}L^{2}_{p}}\leq\frac{C\mathcal{F}_{\infty}^{N+1}}{t}\sum_{|I|+|J|\leq
N}\|L^{I}(t^{-1}\partial_{p})^{J}\check{f}_{[k]}\|_{L^{2}_{x}L^{2}_{p}}+C_{k}(\mathcal{F}_{\infty}^{N+2k+4})^{2}\frac{(\log
t)^{1+k}}{t^{2+k}}.$

Defining

$\check{\mathcal{E}}_{[k]}(t):=\sum_{|I|+|J|=0}^{N}\|L^{I}(t^{-1}\partial_{p})^{J}\check{f}_{[k]}(t,\cdot,\cdot)\|_{L^{2}_{x}L^{2}_{p}},$

and noting that the vanishing of $\check{f}_{[k]}$ at time $T_{f}$ (see (5.9))
implies that $\check{\mathcal{E}}_{[k]}(T_{f})=0$, Proposition 2.9 then
implies that, for all $T\leq t\leq T_{f}$,

$\check{\mathcal{E}}_{[k]}(t)\leq
C_{k}(\mathcal{F}_{\infty}^{N+2k+4})^{2}\frac{(\log
t)^{1+k}}{t^{1+k}}+C\mathcal{F}_{\infty}^{N+1}\int_{t}^{T_{f}}\frac{\check{\mathcal{E}}_{[k]}(s)}{s}ds.$
(5.19)

Defining

$a(t)=\frac{C\mathcal{F}_{\infty}^{N+1}}{t},\qquad
b(t)=C_{k}(\mathcal{F}_{\infty}^{N+2k+4})^{2}\frac{(\log t)^{1+k}}{t^{1+k}},$

it follows that

$e^{\int_{t}^{s}a(s^{\prime})ds^{\prime}}=\left(\frac{s}{t}\right)^{C\mathcal{F}_{\infty}^{N+1}}.$

Thus, if $k>C\mathcal{F}_{\infty}^{N+1}-1$,

$\displaystyle\int_{t}^{T_{f}}a(s)b(s)e^{\int_{t}^{s}a(s^{\prime})ds^{\prime}}ds$
$\displaystyle\leq\frac{C_{k}\mathcal{F}_{\infty}^{N+1}(\mathcal{F}_{\infty}^{N+2k+4})^{2}}{t^{C\mathcal{F}_{\infty}^{3}}}\int_{t}^{T_{f}}\frac{(\log
s)^{1+k}}{s^{2+k-C\mathcal{F}_{\infty}^{N+1}}}ds\leq\frac{C_{k}\mathcal{F}_{\infty}^{N+1}(\mathcal{F}_{\infty}^{N+2k+4})^{2}(\log
t)^{1+k}}{t^{1+k}},$

and the Grönwall inequality, Proposition 2.6, implies that

$\check{\mathcal{E}}_{[k]}(t)\leq
C_{k}(\mathcal{F}_{\infty}^{N+2k+4})^{2}(1+\mathcal{F}_{\infty}^{N+1})\frac{(\log
t)^{1+k}}{t^{1+k}},$

which completes the proof of the estimate (5.16) (after increasing the
elements of the sequence $\\{\mathfrak{n}(n)\\}$ appropriately).

∎

This section is ended with the following proposition, which will be used to
recover the bootstrap assumption (5.4).

######  Proposition 5.8 (Estimate for $\nabla\check{\phi}_{[0]}$).

Under the assumptions of Proposition 5.4, for all $T\leq t\leq T_{f}$,

$\sum_{|K|=1}^{N+1}\|(t\partial_{x})^{K}\check{\phi}_{[0]}\|_{L^{2}_{x}}\lesssim\frac{C_{k}\mathcal{F}_{\infty}^{N+2k+4}\log
t}{t^{\frac{1}{2}}}.$ (5.20)

###### Proof.

Proposition 5.5 and Proposition 5.6 imply that

$\sum_{|I|=1}^{N+1}\|(t\partial_{x})^{I}(\check{\phi}_{[0]})(t,\cdot)\|_{L^{2}}\lesssim
t^{\frac{1}{2}}\sum_{|I|\leq
N}\|L^{I}\check{f}_{[0]}(t,\cdot,\cdot)\|_{L^{2}_{x}L^{2}_{p}}\\\ \leq
t^{\frac{1}{2}}\sum_{|I|\leq
N}\big{(}\|L^{I}\check{f}_{[k]}(t,\cdot,\cdot)\|_{L^{2}_{x}L^{2}_{p}}+\|L^{I}(f_{[k]}-f_{[0]})(t,\cdot,\cdot)\|_{L^{2}_{x}L^{2}_{p}}\big{)}\leq\frac{C_{k}\mathcal{F}_{\infty}^{N+2k+4}\log
t}{t^{\frac{1}{2}}},$

where the final inequality follows from Proposition 5.4 and Theorem 4.1 (see
(4.4)). ∎

###  5.3 Estimates for the finite problems: the proof Theorem 5.1

The proof of Theorem 5.1 can now be given.

###### Proof of Theorem 5.1.

The estimate (5.5) follows from Proposition 5.4. The estimate (5.6) follows
from Proposition 5.6 and the estimate (5.5), and the estimate (5.7) follows
from Proposition 5.5 and the estimate (5.6).

The bootstrap assumption (5.3) clearly holds with the right hand side replaced
by $\frac{1}{2}(\mathcal{F}_{\infty}^{N+2k+4})^{2}$ if $T_{0}$ is sufficiently
large. For the improvement of (5.4), note that Proposition 5.8 and the Sobolev
inequality of Remark 2.2 imply that, for all $T\leq t\leq T_{f}$,

$\sup_{x\in\mathbb{R}^{3}}|\nabla\check{\phi}_{[0]}(t,x)|\leq\frac{C}{t^{\frac{5}{2}}}\sum_{|I|=1}^{N+1}\|(t\partial_{x})^{I}\check{\phi}_{[0]}\|_{L^{2}_{x}}\leq\frac{C_{k}\mathcal{F}_{\infty}^{N+2k+4}\log
t}{t^{3}},$ (5.21)

and so, if $T_{0}$ is sufficiently large, (5.4) holds with the right hand side
replaced by $\frac{\mathcal{F}_{\infty}^{N+2k+4}}{2t^{5/2}}$.

Finally, the property (5.8) follows from Proposition 5.3. ∎

##  6 The proof of the main results

In this section the proofs of Theorem 3.1, Theorem 3.3, and Theorem 3.4 are
given. The following (non-optimal) local well posedness theorem will be used.

######  Theorem 6.1 (Local existence for Vlasov–Poisson).

There is $k$ such that, for any $t_{0}\in\mathbb{R}$ and any smooth compactly
supported function
$f_{0}\colon\mathbb{R}^{3}_{x}\times\mathbb{R}^{3}_{p}\to[0,\infty)$, there
exists $T=T(\|f_{0}\|_{H^{k}_{x}H^{k}_{p}})>0$ and a unique smooth solution
$(f,\varrho,\phi)$ of the Vlasov–Poisson system (1.1)–(1.2) on
$(t_{0}-T,t_{0}+T)\times\mathbb{R}^{3}_{x}\times\mathbb{R}^{3}_{p}$ such that

$f(t_{0},x,p)=f_{0}(x,p),$

for all $(x,p)\in\mathbb{R}^{3}\times\mathbb{R}^{3}$.

Let $T_{0}\gg 1$ be sufficiently large so as to satisfy the assumptions of
Theorem 4.1 and Theorem 5.1. Before the proofs of Theorem 3.1, Theorem 3.3,
and Theorem 3.4 are given, Theorem 5.1 is used to establish, for each
$T_{f}>T_{0}$, the existence of a solution of Vlasov–Poisson on the interval
$[T_{0},T_{f}]$ with trivial remainder $\check{f}_{[k]}$ at $t=T_{f}$.

######  Theorem 6.2 (Existence of solutions of the finite problems on
$[T_{0},T_{f}]$).

Let $k>k_{*}$ be as in Theorem 5.1. For any $T_{f}>T_{0}$, there exists a
solution
$f\colon[T_{0},T_{f}]\times\mathbb{R}^{3}\times\mathbb{R}^{3}\to[0,\infty)$ of
Vlasov–Poisson (1.1)–(1.2) such that

$f|_{t=T_{f}}=f_{[k]}(T_{f},\cdot,\cdot),$ (6.1)

which satisfies the estimates (5.5)–(5.7) and the support property (5.8) for
all $t\in[T_{0},T_{f}]$.

###### Proof.

Let $\mathcal{B}\subset[T_{0},T_{f}]$ be the set of times $T\in[T_{0},T_{f}]$
such that a solution
$f\colon[T,T_{f}]\times\mathbb{R}^{3}\times\mathbb{R}^{3}\to[0,\infty)$ of
Vlasov–Poisson (1.1)–(1.2) exists on $[T,T_{f}]$ satisfying the final
condition (6.1) and the estimates (5.3)–(5.4).

The set $\mathcal{B}$ is a non-empty, closed subset of $[T_{0},T_{f}]$, by
Theorem 6.1 and the fact that $\check{f}_{[k]}(T_{f},\cdot,\cdot)\equiv 0$ and

$\sup_{x\in\mathbb{R}^{3}}|\nabla\check{\phi}_{[0]}(T_{f},x)|\leq\frac{C_{k}\mathcal{F}_{\infty}^{N+2k+4}\log
T_{f}}{(T_{f})^{3}}\leq\frac{\mathcal{F}_{\infty}^{N+2k+4}}{2(T_{f})^{\frac{5}{2}}},$

for $T_{0}$ as in Theorem 5.1 (see, for example, the estimate (5.21)). If
$T\in\mathcal{B}$ satisfies $T>T_{0}$, Theorem 6.1 implies that there is
$\delta>0$ such that the solution extends to the interval $[T-\delta,T_{f}]$.
Theorem 5.1 implies that the inequalities (5.3)–(5.4) in fact hold with better
constants and thus, by continuity, continue to hold for all
$t\in[T-\delta,T_{f}]$ if $\delta$ is sufficiently small. The set
$\mathcal{B}$ is therefore open, and hence equal to $[T_{0},T_{f}]$, and the
proof follows from Theorem 5.1. ∎

The proofs of Theorem 3.1 and Theorem 3.3 can now be given. It is convenient
to give both together.

###### Proof of Theorem 3.1 and Theorem 3.3.

The proof of Theorem 3.1 and Theorem 3.3 is divided into several steps. First,
the main part of the proof — namely the existence of the solution
$(f,\varrho,\phi)$ — is given. Then properties of the solution, including the
expansions of Theorem 3.3 and the support property (3.8), are established.
Finally, the part of Theorem 3.1 concerning the fact that the solution attains
the scattering data $f_{\infty}$ — in the sense that the estimate (3.6) holds
— is shown.

The existence of the solution: Consider some $T_{f}^{\prime}>T_{f}>T_{0}$. By
Theorem 6.2, there exist corresponding solutions

$f^{(T_{f}^{\prime})}\colon[T_{0},T_{f}^{\prime}]\times\mathbb{R}^{3}\times\mathbb{R}^{3}\to[0,\infty),\qquad
f^{(T_{f})}\colon[T_{0},T_{f}]\times\mathbb{R}^{3}\times\mathbb{R}^{3}\to[0,\infty),$

of the Vlasov–Poisson system, which each satisfy the estimates (5.5)–(5.7) and
the support property (5.8). The differences

$\widetilde{f}:=f^{(T_{f}^{\prime})}-f^{(T_{f})},\qquad\widetilde{\phi}:=\phi^{(T_{f}^{\prime})}-\phi^{(T_{f})},\qquad\widetilde{\varrho}:=\varrho^{(T_{f}^{\prime})}-\varrho^{(T_{f})},$

satisfy the system

$\mathbb{X}_{\phi^{\prime}}\widetilde{f}=\partial_{x^{i}}\widetilde{\phi}\,\partial_{p^{i}}f^{(T_{f})},\qquad\Delta_{\mathbb{R}^{3}}\widetilde{\phi}=\widetilde{\varrho},$
(6.2)

on $[T_{0},T_{f}]$, where $\phi^{\prime}=\phi^{(T_{f}^{\prime})}$, along with
the final condition

$\widetilde{f}(T_{f},x,p)=\check{f}_{[k]}^{(T_{f}^{\prime})}(T_{f},x,p),$
(6.3)

for all $x,p\in\mathbb{R}^{3}$.

The proof proceeds by estimating this difference on $[T_{0},T_{f}]$, in a
manner similar to the proof of Theorem 5.1. Indeed, using the latter of (6.2)
and the fact that $\widetilde{f}$ satisfies the support property (5.8), minor
adaptations of Proposition 5.5 and Proposition 5.6 give, for any multi-index
$I$, and any $T_{0}\leq t\leq T_{f}$,

$\|\nabla(t\partial_{x})^{I}\widetilde{\phi}(t,\cdot)\|_{L^{2}}\lesssim
t\|(t\partial_{x})^{I}\widetilde{\varrho}(t,\cdot)\|_{L^{2}}\lesssim
t^{-\frac{1}{2}}\|L^{I}\widetilde{f}(t,\cdot,\cdot)\|_{L^{2}_{x}L^{2}_{p}}.$
(6.4)

Thus, minor adaptations of the proofs of Proposition 5.2 and Proposition 5.7,
using the fact that $f^{(T_{f}^{\prime})}$ and $f^{(T_{f})}$ satisfy the
estimates (5.5)–(5.7), give

$\displaystyle\sum_{|I|+|J|\leq
N}\\!\\!\|\mathbb{X}_{\phi}(L^{I}(t^{-1}\partial_{p})^{J}\widetilde{f})\|_{L^{2}_{x}L^{2}_{p}}\lesssim\
$
$\displaystyle\frac{\mathcal{F}_{\infty}^{N+1}}{t^{\frac{3}{2}}}\sum_{|I|=1}^{N+1}\|(t\partial_{x})^{I}\widetilde{\phi}(t,\cdot)\|_{L^{2}}+\frac{\mathcal{F}_{\infty}^{N+3}(\log
t)^{2}}{t^{2}}\\!\\!\sum_{|I|+|J|\leq
N}\\!\\!\|L^{I}(t^{-1}\partial_{p})^{J}\widetilde{f}\|_{L^{2}_{x}L^{2}_{p}}$
$\displaystyle\lesssim\ $
$\displaystyle\frac{\mathcal{F}_{\infty}^{N+1}}{t}\sum_{|I|+|J|\leq
N}\|L^{I}(t^{-1}\partial_{p})^{J}\widetilde{f}\|_{L^{2}_{x}L^{2}_{p}}.$

Thus, using the estimate (5.5) for the final condition (6.3), Proposition 2.9
gives that

$\widetilde{\mathcal{E}}(t):=\sum_{|I|+|J|\leq
N}\|L^{I}(t^{-1}\partial_{p})^{J}\widetilde{f}(t,\cdot,\cdot)\|_{L^{2}_{x}L^{2}_{p}}$

satisfies, for all $T_{0}\leq t\leq T_{f}$,

$\widetilde{\mathcal{E}}(t)\leq
C_{k}(\mathcal{F}_{\infty}^{N+2k+4})^{2}\frac{(\log
T_{f})^{1+k}}{(T_{f})^{1+k}}+\int_{t}^{T_{f}}\frac{C\mathcal{F}_{\infty}^{N+1}}{s}\widetilde{\mathcal{E}}(s)ds.$

The Grönwall inequality, Proposition 2.6, with

$a(t)=\frac{C\mathcal{F}_{\infty}^{N+1}}{s},\qquad b(t)\equiv
C_{k}(\mathcal{F}_{\infty}^{N+2k+4})^{2}\frac{(\log
T_{f})^{1+k}}{(T_{f})^{1+k}}$

then gives

$\displaystyle\widetilde{\mathcal{E}}(t)\leq
C_{k}(\mathcal{F}_{\infty}^{N+2k+4})^{2}\frac{(\log
T_{f})^{1+k}}{(T_{f})^{1+k}}\bigg{(}1+\frac{C\mathcal{F}_{\infty}^{N+1}}{t^{C\mathcal{F}_{\infty}^{N+1}}}\int_{t}^{T_{f}}s^{C\mathcal{F}_{\infty}^{N+1}-1}ds\bigg{)}.$

Thus, provided $k\geq C\mathcal{F}_{\infty}^{N+1}+1$,

$\displaystyle\sup_{T_{0}\leq t\leq T_{f}}\sum_{|I|+|J|\leq
N}\|L^{I}(t^{-1}\partial_{p})^{J}(f^{(T_{f}^{\prime})}-f^{(T_{f})})(t,\cdot,\cdot)\|_{L^{2}_{x}L^{2}_{p}}\leq\frac{C_{k}(\mathcal{F}_{\infty}^{N+2k+4})^{2}}{T_{f}}.$

Returning to (6.4), one has similar estimates for
$\phi^{(T_{f}^{\prime})}-\phi^{(T_{f})}$ and
$\varrho^{(T_{f}^{\prime})}-\varrho^{(T_{f})}$. It follows that, for any
increasing sequence $T_{f}^{n}\to\infty$, the sequence
$\\{(f^{(T_{f}^{n})},\phi^{(T_{f}^{n})},\varrho^{(T_{f}^{n})})\\}$ is Cauchy
on their common domains of the elements, in the above norms, and thus
converges to a unique limit $(f,\phi,\varrho)$ (independent of the sequence
$\\{T_{f}^{n}\\}$ chosen) which moreover satisfies the estimates (5.5)–(5.7)
and the support property (5.8) on $[T_{0},\infty)$ and solves the
Vlasov–Poisson system (1.1)–(1.2) on $[T_{0},\infty)$.

Properties of the solution: The fact that $f$ satisfies the support property
(3.8) follows from (4.5), along with the fact that each $f^{(T_{f})}$
satisfies (5.8).

Consider now the estimates (3.10)–(3.12) and recall $k_{*}$ from Theorem 5.1.
For $K\geq k_{*}$ the estimates (3.10)–(3.12) follow immediately from the fact
that $(f,\varrho,\phi)$ satisfy (5.5)–(5.7). For $0\leq K<k_{*}$, note that

$\check{f}_{[K]}(t,x,p)=\check{f}_{[k_{*}]}(t,x,p)-\sum_{k=K+1}^{k_{*}}\sum_{l=0}^{k}\frac{(\log
t)^{l}}{t^{k}}f_{k,l}\big{(}x-tp+\log t\,\nabla\phi_{\infty}(p),p\big{)},$

and thus (4.6) and (5.5) imply that

$\sum_{|I|+|J|\leq
N}\|L^{I}(t^{-1}\partial_{p})^{J}\check{f}_{[K]}(t,\cdot,\cdot)\|_{L^{2}_{x}L^{2}_{p}}\leq
C_{k_{*}}\Big{(}(\mathcal{F}_{\infty}^{N+2k_{*}+4})^{2}\frac{(\log
t)^{1+k_{*}}}{t^{1+k_{*}}}+\sum_{k=K+1}^{k_{*}}\frac{(\log
t)^{k}}{t^{k}}(\mathcal{F}_{\infty}^{N+k})^{2}\Big{)},$

and (3.10) follows if $T_{0}$ is suitably large. Similarly for (3.11)–(3.12).

The $L^{\infty}$ estimates (3.13) follow from the $L^{2}$ estimates (3.10) and
the Sobolev inequality of Proposition 2.3, and the $L^{\infty}$ estimates
(3.14)–(3.15) follow from the $L^{2}$ estimates (3.11)–(3.12) and the Sobolev
inequality of Remark 2.2.

The scattering data is attained: In order to see that the data $f_{\infty}$ is
attained — in the sense that the estimate (3.6) holds — first note that, for

$y(t,x,p)=x-tp+\log t\nabla\phi_{\infty}(p),\qquad\tilde{y}(t,x,p)=x+tp-\log
t\nabla\phi_{\infty}(p),$

one has

$\int\int|f(t,\tilde{y}(t,x,p),p)-f_{\infty}(x,p)|^{2}dpdx=\int\int|f(t,x,p)-f_{\infty}(y(t,x,p),p)|^{2}dpdx,$

and so the zeroth order part of the estimate (3.6) follows from setting $K=0$
in (3.10). For the terms involving first order derivatives, recall first
Remark 2.13 (considering, in particular, (2.8) and (2.9) with $h=f_{\infty}$)
and note also that

$\displaystyle\partial_{x^{i}}\Big{(}f(t,\tilde{y}(t,x,p),p)\Big{)}=\ $
$\displaystyle(\partial_{x^{i}}f)(t,\tilde{y}(t,x,p),p),$
$\displaystyle\partial_{p^{i}}\Big{(}f(t,\tilde{y}(t,x,p),p)\Big{)}=\ $
$\displaystyle\Big{(}\delta_{ij}-\frac{\log
t}{t}\partial^{2}_{i,j}\phi_{\infty}(p)\Big{)}(L_{i}f)(t,\tilde{y}(t,x,p),p)$
$\displaystyle+\frac{(\log
t)^{2}}{t}\partial_{i,j}^{2}\phi_{\infty}(p)\partial_{j,k}^{2}\phi_{\infty}(p)(t^{-1}\partial_{p^{k}}f)(t,\tilde{y}(t,x,p),p),$

and

$\displaystyle(\partial_{x^{i}}f_{\infty})(y(t,x,p),p)=\ $
$\displaystyle-(t^{-1}\partial_{p^{i}})\big{(}f_{\infty}(y(t,x,p),p)\big{)}$
$\displaystyle+\frac{\log
t}{t}\partial_{k}\partial_{i}\phi_{\infty}(p)(\partial_{x^{k}}f_{\infty})(y(t,x,p),p)+t^{-1}(\partial_{p^{i}}f_{\infty})(y(t,x,p),p)$
$\displaystyle(\partial_{p^{i}}f_{\infty})(y(t,x,p),p)=\ $ $\displaystyle
L_{i}\big{(}f_{\infty}(y(t,x,p),p)\big{)}$ $\displaystyle-\frac{(\log
t)^{2}}{t}\partial_{k}\partial_{i}\phi_{\infty}(p)(\partial_{x^{k}}f_{\infty})(y(t,x,p),p)-\frac{\log
t}{t}\partial_{k}\partial_{i}\phi_{\infty}(p)(\partial_{p^{k}}f_{\infty})(y(t,x,p),p).$

It follows that

$\displaystyle(\partial_{x^{i}}f)(t,x,p)-(\partial_{x^{i}}f_{\infty})(y(t,x,p),p)=-(t^{-1}\partial_{p^{i}})\big{(}f(t,x,p)-f_{\infty}(y(t,x,p),p)\big{)}$
$\displaystyle+\frac{1}{t}\big{(}\partial_{x^{i}}+t\partial_{p^{i}}\big{)}f(t,x,p)-\frac{\log
t}{t}\partial_{k}\partial_{i}\phi_{\infty}(p)(\partial_{x^{k}}f_{\infty})(y(t,x,p),p)-t^{-1}(\partial_{p^{i}}f_{\infty})(y(t,x,p),p).$

Hence

$\displaystyle\int\int\big{|}\partial_{x^{i}}\big{(}f(t,\tilde{y}(t,x,p),p)-f_{\infty}(x,p)\big{)}\big{|}^{2}dpdx=\int\int\big{|}(\partial_{x^{i}}f)(t,x,p)-(\partial_{x^{i}}f_{\infty})(y(t,x,p),p)\big{|}^{2}dpdx$
$\displaystyle\qquad\qquad\lesssim\int\int\big{|}t^{-1}\partial_{p^{i}}\big{(}f(t,x,p)-f_{\infty}(y(t,x,p),p)\big{)}\big{|}^{2}dpdx+\frac{\mathcal{F}_{\infty}^{3}\log
t}{t}\lesssim\frac{\mathcal{F}_{\infty}^{N+4}\log t}{t},$

and similarly,

$\displaystyle\int\int\big{|}\partial_{p^{i}}\big{(}f(t,\tilde{y}(t,x,p),p)-f_{\infty}(x,p)\big{)}\big{|}^{2}dpdx$
$\displaystyle\qquad\qquad\lesssim\int\int\big{|}L_{i}\big{(}f(t,x,p)-f_{\infty}(y(t,x,p),p)\big{)}\big{|}^{2}dpdx+\frac{\mathcal{F}_{\infty}^{3}(\log
t)^{2}}{t}\lesssim\frac{\mathcal{F}_{\infty}^{N+4}(\log t)^{2}}{t},$

by (3.10) with $K=0$. The part of (3.6) concerning higher order derivatives
follows similarly. The $L^{\infty}$ estimate (3.7) follows from (3.6) and the
Sobolev inequality of Proposition 2.3. ∎

Finally, the proof of Theorem 3.4 is given.

###### Proof of Theorem 3.4.

Consider the solution $(f,\varrho,\phi)$ of Theorem 3.1, and suppose that
$(f^{\prime},\varrho^{\prime},\phi^{\prime})$ is another solution which
satisfies (3.16) for some $K=K(\mathcal{F}_{\infty}^{N+1})$ and for all $t$
sufficiently large, and $f^{\prime}$ satisfies the support property (3.8). It
follows that the difference of the two solutions satisfies the system

$\mathbb{X}_{\phi^{\prime}}(f-f^{\prime})=\partial_{x^{i}}(\phi-\phi^{\prime})\,\partial_{p^{i}}f,\qquad\Delta_{\mathbb{R}^{3}}(\phi-\phi^{\prime})=\varrho-\varrho^{\prime}.$

Since

$\sup_{x\in\mathbb{R}^{3}}\|\partial_{p^{i}}f(t,x,\cdot)\|_{L^{2}_{p}}\leq\frac{C\mathcal{F}_{\infty}^{N}}{t^{\frac{1}{2}}},\qquad\|\nabla(\phi-\phi^{\prime})(t,\cdot)\|_{L^{2}}\lesssim
t\|(\varrho-\varrho^{\prime})(t,\cdot)\|_{L^{2}}\lesssim
t^{-\frac{1}{2}}\|(f-f^{\prime})(t,\cdot,\cdot)\|_{L^{2}_{x}L^{2}_{p}}.$

where the latter follows from the support property (3.8) (see Proposition 5.5
and Proposition 5.6), it follows that

$\|\mathbb{X}_{\phi^{\prime}}(f-f^{\prime})(t,\cdot,\cdot)\|_{L^{2}_{x}L^{2}_{p}}\leq\frac{C\mathcal{F}_{\infty}^{N}}{t}\|(f-f^{\prime})(t,\cdot,\cdot)\|_{L^{2}_{x}L^{2}_{p}}.$

Proposition 2.9 and the Grönwall inequality, Proposition 2.6, with
$a(t)=t^{-1}C\mathcal{F}_{\infty}^{N}$,
$b(t)\equiv\|(f-f^{\prime})(T,\cdot,\cdot)\|_{L^{2}_{x}L^{2}_{p}}$ then imply
that, for any $T$ large and $t\leq T$,

$\displaystyle\|(f-f^{\prime})(t,\cdot,\cdot)\|_{L^{2}_{x}L^{2}_{p}}$
$\displaystyle\leq\|(f-f^{\prime})(T,\cdot,\cdot)\|_{L^{2}_{x}L^{2}_{p}}\Big{(}1+\frac{C\mathcal{F}_{\infty}^{N}}{t^{C\mathcal{F}_{\infty}^{N}}}\int_{t}^{T}s^{C\mathcal{F}_{\infty}^{N}-1}ds\Big{)}$
$\displaystyle\leq
C_{K}\Big{(}1+(\mathcal{F}_{\infty}^{N+2K+4})^{2}\Big{)}\frac{(\log
T)^{1+K}}{T^{1+K}}T^{C\mathcal{F}_{\infty}^{N}}.$

where the latter follows from (3.10) and the assumption (3.16). If $K\geq
C\mathcal{F}_{\infty}^{N}$, letting $T\to\infty$, it follows that the left
hand side vanishes for all $t$, and thus
$(f^{\prime},\varrho^{\prime},\phi^{\prime})\equiv(f,\varrho,\phi)$. ∎

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