An Inverse Problem for the Relativistic Boltzmann Equation

Abstract

We consider an inverse problem for the Boltzmann equation on a globally hyperbolic Lorentzian spacetime (M, g) with an unknown metric g. We consider measurements done in a neighbourhood $V\subset M$ of a timelike path $\mu$ that connects a point $x^-$ to a point $x^+$. The measurements are modelled by a source-to-solution map, which maps a source supported in V to the restriction of the solution to the Boltzmann equation to the set V. We show that the source-to-solution map uniquely determines the Lorentzian spacetime, up to an isometry, in the set $I^+(x^-)\cap I^-(x^+)\subset M$. The set $I^+(x^-)\cap I^-(x^+)$ is the intersection of the future of the point $x^-$ and the past of the point $x^+$, and hence is the maximal set to where causal signals sent from $x^-$ can propagate and return to the point $x^+$. The proof of the result is based on using the nonlinearity of the Boltzmann equation as a beneficial feature for solving the inverse problem.

Appendices

Appendix A: Auxiliary lemmas

1.1 A.1: Lemmas used in the proof of Proposition 4.3

Lemma 4.2

Let $Y_1$ and $Y_2$ and U be as in Definition 4.1 and adopt also the associated notation. Define

$$\begin{aligned} \Lambda _R = N^* ( \bigcup _{x\in U} \{ x\} \times \mathcal {P}_xU \times \mathcal {P}_xU ). \end{aligned}$$

The submanifold $\Lambda _R$ of $T^*( U \times \mathcal {P}U \times \mathcal {P}U) {\setminus } \{0 \}$ equals the set

$$\begin{aligned} \Lambda _R&= \{ \big (x, y, z,p, q \, ; \, \xi ^x, \xi ^y, \xi ^z,\xi ^p,\xi ^{q} \big ) \in \ T^*(U \times \mathcal {P}U \times \mathcal {P}U) {\setminus } \{0\} : \\&\quad \xi ^x + \xi ^y +\xi ^z = 0 , \ \xi ^p = \xi ^{q} = 0, \ x=y=z \}. \end{aligned}$$

Then we have

$$\begin{aligned} \Lambda _R'&=\{ \big ((x ; \xi ^x), (y, z,p, q \, ; \, \xi ^y, \xi ^z,\xi ^p,\xi ^{q} )\big ) \in \ T^*U \times T^* ( \mathcal {P}U \times \mathcal {P}U) {\setminus } \{0\} : \nonumber \\&\quad \xi ^x =\xi ^y+ \xi ^z \ne 0, \ \xi ^p = \xi ^{q} =0 , \ x=y=z \}. \end{aligned}$$

(A.101)

The spaces $\Lambda '_R \times ( N^*[Y_1 \times Y_2] )$ and $T^*U \times \text {diag}\, T^*( \mathcal {P}M \times \mathcal {P}M)$ intersect transversally in $T^*U \times T^*(\mathcal {P}M \times \mathcal {P}M) \times T^*(\mathcal {P}M \times \mathcal {P}M)$.

Proof

First notice that the manifold $T\Big ( \bigcup _{x\in U} \{ x\} \times \mathcal {P}_xU \times \mathcal {P}_x U\Big )$ can be written as

$$\begin{aligned} \mathcal {V}_1:=\{ (x,y,z, p,q \, ; \, \dot{x},\dot{y}, \dot{z} , \dot{p},\dot{q} ) : \dot{x} = \dot{y}=\dot{z}, \ x = y=z \} . \end{aligned}$$

(A.102)

Note the $\mathcal {V}_1$ is a space of dimension 6n. Let us consider the subspace of $T^*(U\times \mathcal {P}U \times \mathcal {P}U)$

$$\begin{aligned} \mathcal {V}_2:=\{ (x,y,z, p,q\ ; \ \xi ^x, \xi ^y , \xi ^z, \xi ^p,\xi ^{q} ) : x=y=z, \ \xi ^x + \xi ^y +\xi ^z = 0 , \ \xi ^p = \xi ^{q} = 0 \}. \end{aligned}$$

Note that a vector in $\mathcal {V}_1$ paired with a covector in $\mathcal {V}_2$ yields zero since

$$\begin{aligned} (\xi ^x,\xi ^y,\xi ^z,\xi ^p,\xi ^q)\cdot (\dot{x},\dot{y},\dot{z},\dot{p},\dot{q})=\dot{x}\,\xi ^x+\dot{y}\,\xi ^y+\dot{z}\,\xi ^z=\dot{x}(\xi ^x + \xi ^y +\xi ^z)=0. \end{aligned}$$

Thus we have that $\mathcal {V}_2\subset \Lambda _R$. Note that the fibers of $\mathcal {V}_1$ are of dimension $5n-2n=3n$. Note also that the fibers of $\mathcal {V}_2$ have dimension $5n-3n=2n$. We thus have $\mathcal {V}_2=\Lambda _R$, since dimensions of a fiber of the conormal bundle of $\cup _{x\in U} \{ x\} \times \mathcal {P}_xU \times \mathcal {P}_x U$ is the same as the codimension of a fiber of $\mathcal {V}_1$. In conclusion, the coordinate expressions for $\Lambda _R$ and $\Lambda _R'$ hold.

Next we prove the transversality claim. First, fix the notation

$$\begin{aligned} X&= T^*U \times T^*( \mathcal {P}M \times \mathcal {P}M)\times T^*( \mathcal {P}M \times \mathcal {P}M)\\ L_1&= \Lambda '_R \times ( N^*[Y_1 \times Y_2])\\ L_2&= T^*U \times \text {diag}\,T^*( \mathcal {P}M \times \mathcal {P}M). \end{aligned}$$

We want to show that the linear spaces $L_1$ and $L_2$ intersect transversally in X; that is for all $\lambda \in L_1\cap L_2$,

$$\begin{aligned} \text {dim}(T_\lambda L_1 + T_\lambda L_2)&= \text {dim}T_\lambda X. \end{aligned}$$

(A.103)

Let $\lambda \in L_1 \cap L_2$. Since $Y_1$ and $Y_2$ satisfy the intersection property, there exists local parametrisation $(\tilde{x}',\tilde{x}'',\tilde{p}',\tilde{p}'') \mapsto (\tilde{x}',\tilde{x}'',\theta _2(x') + \tilde{p}', \theta _1(x'') + \tilde{p}'')$ (see (4.8)) on TU such that

$$\begin{aligned} \begin{aligned} Y_1 \cap TU&= \{ (x,p) \in TU : \tilde{x}' = 0 , \ \tilde{p}' = 0, \ \tilde{p}'' =0 \}, \\ Y_2 \cap TU&= \{ (x,p) \in TU : \tilde{x}'' = 0, \ \tilde{p}''= 0, \ \tilde{p}' =0 \}. \end{aligned} \end{aligned}$$

(A.104)

We redefine $(x',x'',p',p'')$ as these coordinates. Again, we denote $(x,\xi )=(x',x'')$, $p= (p',p'')$ and similarly $\xi = (\xi ',\xi '')$ for the associated covectors etc. In these coordinates, the elements of $\Lambda _R'= \mathcal {V}_2'$ are of the similar form as above in the sense that each element is of the form

$$\begin{aligned} \begin{aligned} \big ((x;\xi ^y+\xi ^z) ,(y,z,p,q,\xi ^y,\xi ^z,\xi ^p,\xi ^q) \big ), \quad x=y=z, \quad \xi ^x = \xi ^y + \xi ^z, \quad \xi ^p = 0 = \xi ^q \end{aligned} \end{aligned}$$

in the canonical coordinates. We next compute the induced local expressions of $L_1$, $L_2$ and $L_1\cap L_2$ with respect to the coordinate system in (A.104).

If $\lambda \in L_1$, $\lambda$ has the local coordinate form

$$\begin{aligned} \lambda = ((x,\xi ^x, y, p, \xi ^y,0, z,q,\xi ^z, 0),\alpha ,\beta ), \end{aligned}$$

(A.105)

where $(\alpha ,\beta ) \in N^*[Y_1 \times Y_2]$, $(x,\xi ^x)\in T^*U$, and $(y,p,\xi ^y,0),(z,q,\xi ^z,0)\in T^*(\mathcal {P}U)$ are such that $y=z=x$ and $\xi ^y+\xi ^z=\xi ^x$.

If $(\alpha ,\beta )\in N^*[Y_1\times Y_2]$, then $(\alpha ,\beta )$ is nonzero and must satisfy in coordinates

$$\begin{aligned} \alpha&=(0,x'',0,0 \ ; (\xi ^x)',0, (\xi ^p)',(\xi ^p)'') \\ \beta&= (y',0,0,0 \ ; 0,(\xi ^y)'', (\xi ^q)',(\xi ^q)''), \end{aligned}$$

(A.106)

where one (but not both) of the components is allowed to be zero.

The expression for $\lambda \in L_2$ is

$$\begin{aligned} \lambda = (x,\xi ^x, y, p, z, q,\xi ^y, \xi ^p, \xi ^z,\xi ^q, y, p,z, q, \xi ^y, \xi ^p,\xi ^z,\xi ^q). \end{aligned}$$

(A.107)

where $x,y,z \in U$.

Using (A.105), (A.106), and (A.108), we obtain that points $\lambda \in L_1\cap L_2$ are described by

$$\begin{aligned} \lambda&= (0,\xi ^x, \gamma ,\gamma ), \end{aligned}$$

(A.108)

where

$$\begin{aligned} \xi ^x&= ((\xi ^x)',(\xi ^x)''),\\ \gamma&= (\underbrace{0, 0}_{x},\underbrace{0,0}_{p} , \underbrace{0, 0}_{y},\underbrace{0,0}_{q};\underbrace{(\xi ^x)',0}_{\xi ^x}, \underbrace{0,(\xi ^x)''}_{\xi ^y},\underbrace{0,0}_{\xi ^p}, \underbrace{0,0}_{\xi ^q}),\\ \end{aligned}$$

Indeed, if $\lambda \in L_1\cap L_2$, we must have $(x',x'') = (y',0)=(0,z'')$ which is only satisfied if $x=y=z=0$. From (A.105), we have $\xi ^p=0=\xi ^q$ in (A.108). Since $\xi ^x=\xi ^y+\xi ^z$ in (A.105), from (A.106) and (A.108) we find $((\xi ^x)',(\xi ^x)'') = ((\xi ^z)',(\xi ^y)'')$. From (A.104), $(p',p'') ={ (0,0) = (q',q'')}$.

From the above coordinate expressions, we now compute the dimensions of $L_1$, $L_2$, and $L_1\cap L_2$. First note that

$$\begin{aligned} \text {dim}(X)= \text {dim}(T^*U) + \text {dim}( T^*( \mathcal {P}M \times \mathcal {P}M) )+ \text {dim}( T^*( \mathcal {P}M \times \mathcal {P}M)) = 18n \end{aligned}$$

Similarly one computes $\text {dim}(L_2) = 10n$. The expression (A.109) shows that $\text {dim}(L_1\cap L_2)=n$.

Therefore, for $\lambda \in L_1\cap L_2$,

$$\begin{aligned} \text {dim}(T_\lambda L_1) + \text {dim}( T_\lambda L_2) - \text {dim}T_\lambda (L_1 \cap L_2) = 9n +10n -n = 18n = \text {dim}(T_\lambda X). \end{aligned}$$

This shows that $L_1$ is transverse to $L_2$ in X. $\square$

Appendix B: Existence Theorems for Vlasov and Boltzmann Cauchy Problems

In this section, the space (M, g) is assumed to be globally hyperbolic $C^\infty$-Lorentzian manifold. By $\gamma _{(x,p)}:(-T_1,T_2)\rightarrow M$ we denote the inextendible geodesic which satisfies

$$\begin{aligned} \gamma _{(x,p)}(0)=x\quad \text {and}\quad \dot{\gamma }_{(x,p)}(0)=p. \end{aligned}$$

(A.109)

We do not assume that (M, g) is necessarily geodesically complete. Therefore, we might have that $T_1<\infty$ or $T_2<\infty$. We will repeatedly use the fact that if f(x, p) is a smooth function on ${\overline{\mathcal {P}}^+ }M$ whose support on the base variable $x\in M$ is compact, then the map

$$\begin{aligned} (x,p) \mapsto \int _{-\infty }^0 f( \gamma _{(x,p)}(t) , \dot{\gamma }_{(x,p)}(t) ) dt\quad \text {on} \quad (x,p) \in {\overline{\mathcal {P}}^+ }M \end{aligned}$$

(B.110)

is well defined. This is because on a globally hyperbolic Lorentzian manifold any causal geodesic $\gamma$ exits a given compact set $K_\pi$ permanently after finite parameter times. That is, there are parameter times $t_1,t_2$ such that $\gamma (\{t<t_1\}),\gamma (\{t>t_2\})\subset M{\setminus } K_\pi$. Thus the integral above is actually an integral of a smooth function over a finite interval. Further, since f and the geodesic flow on (M, g) are smooth the map in (B.111) is smooth. If (M, g) is not geodesically complete and if $\gamma _{(x,p)}:(-T_1,T_2)\rightarrow M$, we interpret the integral above to be over $(-T_1,0]$. We interpret similarly for all similar integrals in this section without further notice.

We record the following lemma.

Lemma B.1

Let (M, g) be a globally hyperbolic Lorentzian manifold, let X be a compact subset of ${\overline{\mathcal {P}}^+ }M$ and let $K_{\pi }$ be a compact subset of M. Then the function $\ell :{\overline{\mathcal {P}}^+ }M \rightarrow \mathbb {R}$

$$\begin{aligned} \ell (x,p)= {\max } \{s\ge 0: \gamma _{(x,p)}(-s)\in K_{\pi } \}. \end{aligned}$$

is well defined, upper semi-continuous and there is the maximum

$$\begin{aligned} l_0 = \max \big \{ \ell (y,q): (y,q )\in X \}<\infty . \end{aligned}$$

(B.111)

In addition, if $\lambda >0$ then $\ell (x,\lambda p ) = \lambda ^{-1} \ell (x,p)$.

Proof

Since the globally hyperbolic manifold (M, g) is causally disprisoning and causally pseudoconvex, any of its causal geodesics exits the compact set $K_{\pi }$ permanently after finite parameter times in the corresponding inextendible domain, see e.g. [5, Proposition 1] and [4, Lemma 11.19]. Hence, $\ell (x,p)<\infty$ is well defined for all $(x,p)\in {\overline{\mathcal {P}}^+ }M$.

The upper semi-continuity of $\ell$ follows from the global hyperbolicity of (M, g) and the compactness of $K_\pi$.

The maximum in (B.112) exists since $\ell$ is upper semi-continuous and X is compact. Since $\gamma _{(x,\lambda p)} (s) = \gamma _{(x, p)} (\lambda s)$, for all $\lambda \in \mathbb {R}$, we have that $\ell (x,\lambda p ) = \lambda ^{-1} \ell (x,p)$, for $\lambda >0$. $\square$

Theorem 3.2

Assume that (M, g) is a globally hyperbolic $C^\infty$-Lorentzian manifold. Let $\mathcal {C}$ be a Cauchy surface of (M, g), $K\subset {\overline{\mathcal {P}}^+ }\mathcal {C}^+$ be compact and $k\ge 0$. Let also $f\in C_K^k({\overline{\mathcal {P}}^+ }M)$. Then, the problem

$$\begin{aligned} \mathcal {X}u(x,p)&= f(x,p)&\text {on} \quad {\overline{\mathcal {P}}^+ }M \nonumber \\ u(x,p)&= 0&\text {on} \quad {\overline{\mathcal {P}}^+ }\mathcal {C}^- \end{aligned}$$

(B.112)

has a unique solution u in $C^k ( {\overline{\mathcal {P}}^+ }M )$. In particular, if $Z\subset {\overline{\mathcal {P}}^+ }M$ is compact, there is a constant $c_{k,K,Z}>0$ such that

$$\begin{aligned} \Vert u|_Z \Vert _{C^k ( Z)}\le c_{k,K,Z}||f||_{C^{k} ( {\overline{\mathcal {P}}^+ }M )}. \end{aligned}$$

(B.113)

If $k=0$, the estimate above is independent of Z:

$$\begin{aligned} \Vert u \Vert _{C ( {\overline{\mathcal {P}}^+ }M)}\le c_{K}||f||_{C( {\overline{\mathcal {P}}^+ }M )}. \end{aligned}$$

Proof

Let us denote by $K_{\pi }=\pi (K)$ the compact set containing $\pi (\text {supp}\,(f))$. Let $(x,p)\in {\overline{\mathcal {P}}^+ }M$ and $f\in C_c^k({\overline{\mathcal {P}}^+ }\mathcal {C}^+)$. Evaluating (B.113) at $(\gamma _{(x,p)}(s),\dot{\gamma }_{(x,p)}(s))$ reads

$$\begin{aligned} (\mathcal {X}u)(\gamma _{(x,p)}(s),\dot{\gamma }_{(x,p)}(s))=f(\gamma _{(x,p)}(s),\dot{\gamma }_{(x,p)}(s)). \end{aligned}$$

Since $\mathcal {X}$ is the geodesic vector field we have for all s that

$$\begin{aligned} (\mathcal {X}u)(\gamma _{(x,p)}(s),\dot{\gamma }_{(x,p)}(s))=\frac{d}{ds}u(\gamma _{(x,p)}(s),\dot{\gamma }_{(x,p)}(s)). \end{aligned}$$

By integrating in s, we obtain

$$\begin{aligned} u(x,p) = \int _{-\infty }^0 f(\gamma _{(x,p)} (s), \dot{\gamma }_{(x,p)} (s) )ds. \end{aligned}$$

(B.114)

Here we used that $f(\gamma _{(x,p)} (s), \dot{\gamma }_{(x,p)} (s) )$ vanishes for $s<-\ell (x,p)$ by Lemma B.1, where

$$\begin{aligned} \ell (x,p)=\max \{s\ge 0: \gamma _{(x,p)}(-s)\in K_{\pi } \}. \end{aligned}$$

Indeed, any inextendible causal geodesic in a globally hyperbolic (M, g) leaves permanently the compact set $\pi \text {supp}(f)$ (see page 48). This holds even without assumptions on completeness.

We verify that u is a solution to (B.113). Note that if $(y,q)=\big (\gamma _{(x,p)} (s), \dot{\gamma }_{(x,p)} (s)\big )$, then

$$\begin{aligned} \gamma _{(y,q)}(z)=\gamma _{(x,p)}(z+s) \text { and } \dot{\gamma }_{(y,q)} (z)=\dot{\gamma }_{(x,p)}(z+s). \end{aligned}$$

It follows that

$$\begin{aligned} u(\gamma _{(x,p)}(s),\dot{\gamma }_{(x,p)}(s))&=\int _{-\infty }^0f\big (\gamma _{(x,p)}(z+s),\dot{\gamma }_{(x,p)}(z+s)\big )dz\\&=\int _{-\infty }^sf\big (\gamma _{(x,p)}(z),\dot{\gamma }_{(x,p)}(z)\big )dz \end{aligned}$$

and consequently

$$\begin{aligned} \mathcal {X}u(x,p)=\frac{d}{ds}\Big |_{s=0}u(\gamma _{(x,p)}(s),\dot{\gamma }_{(x,p)}(s))=f\big (\gamma _{(x,p)}(0),\dot{\gamma }_{(x,p)}(0)\big )=f(x,p). \end{aligned}$$

If $(x,p)\in {\overline{\mathcal {P}}^+ }\mathcal {C}^-$, then $u(x,p)=0$ by the integral formula (B.115) and the fact that $f\in C_c^k({\overline{\mathcal {P}}^+ }\mathcal {C}^+)$. We have now shown that a solution u to (B.113) exists. The solution u is unique since it was obtained by integrating the Eq. (B.113).

Next we prove the estimate (B.114). We have by the representation formula (B.115) for the solution u that

$$\begin{aligned} \sup _{(x,p)\in {\overline{\mathcal {P}}^+ }M}|u(x,p) |=\sup _{(x,p)\in {\overline{\mathcal {P}}^+ }K_\pi }|u(x,p) |. \end{aligned}$$

(B.115)

The Eq. (B.116) holds since $\pi (\text {supp}\,(f))$ is properly contained in $K_{\pi }$. Let e be some auxiliary smooth Riemannian metric on M and let $SK_{\pi }\subset TM$ be the unit sphere bundle with respect to e over $K_{\pi }$. Let us also denote

$$\begin{aligned} X=SK_{\pi }\cap {\overline{\mathcal {P}}^+ }K_{\pi } \end{aligned}$$

the bundle of future directed causal (with respect to g) vectors that have unit length in the Riemannian metric e. Since X is a closed subset of the compact set $SK_{\pi }$, we have that X is compact. By Lemma B.1 we have that

$$\begin{aligned} l_0 = \max \big \{ \ell (y,q): (y,q )\in X \} \end{aligned}$$

exists.

Let us continue to estimate $|u(x,p) |$ for $(x,p)\in {\overline{\mathcal {P}}^+ }K_{\pi }$. If $(x,p)\in {\overline{\mathcal {P}}^+ }K_{\pi }$, then there is $\lambda >0$ such that $(x,\lambda ^{-1} p)\in X$. Let us denote $q=\lambda ^{-1}\, p\in SK_{\pi }$. We have that

$$\begin{aligned} u(x,p)&= \int _{-\infty }^0 f(\gamma _{(x,\lambda q)} (s), \dot{\gamma }_{(x,\lambda q)} (s) )ds =\int _{-\infty }^0 f(\gamma _{(x,q)} (\lambda s), \lambda \dot{\gamma }_{(x,q)} (\lambda s) )ds \nonumber \\&=\frac{1}{\lambda }\int _{-\infty }^0 f(\gamma _{(x,q)} (s), \lambda \dot{\gamma }_{(x,q)} (s) )ds=\frac{1}{\lambda }\int _{-l_0}^0 f(\gamma _{(x,q)} (s), \lambda \dot{\gamma }_{(x,q)} (s) )ds. \end{aligned}$$

(B.116)

Here we used

$$\begin{aligned} \gamma _{(x,\lambda p)}(z)=\gamma _{(x,p)}(\lambda z) \quad \text { and } \quad \frac{d}{dz} \gamma _{(x,\lambda p)}(z)=\lambda \dot{\gamma }_{(x,p)}(\lambda z). \end{aligned}$$

Let us define two positive real numbers

$$\begin{aligned} C&=\max _{s\in [0,l_0]}\max _{(x,q)\in X}|\dot{\gamma }_{(x,q)}(-s) |_e<\infty \\ R&=\inf \{r>0: f|_{B_e(0,r)\subset T_xM}=0 \text{ for } \text{ all } x\in K_{\pi }\}>0. \end{aligned}$$

Here for $x\in K_{\pi }$ the set $B_e(0,r)$ is the unit ball of radius r with respect to the Riemannian metric e in the tangent space $T_xM$. The constant R is positive since f has compact support in ${\overline{\mathcal {P}}^+ }M$ by assumption. Let us define

$$\begin{aligned} \lambda _{\text {min}}:=\frac{R}{C}>0. \end{aligned}$$

Then if $\lambda < \lambda _{\text {min}}=\frac{R}{C}$, we have for $(x,q)\in X$ that

$$\begin{aligned} \int _{-l_0}^0 f(\gamma _{(x,q)} (s), \lambda \dot{\gamma }_{(x,q)} (s) )ds=0, \end{aligned}$$

since in this case $|\lambda \dot{\gamma }_{(x,q)}(s) |< R$ for all $s\in [-l_0,0]$. It follows that for all $\lambda >0$ we have that

$$\begin{aligned} \frac{1}{\lambda }\left| \int _{-l_0}^0 f(\gamma _{(x,q)} (s), \lambda \dot{\gamma }_{(x,q)} (s) )ds\right| \le \frac{{l_0}}{\lambda _{\text {min}}}\Vert f \Vert _{C({\overline{\mathcal {P}}^+ }M)}. \end{aligned}$$

Finally, combining the above with (B.116) and (B.117) shows that

$$\begin{aligned} \Vert u \Vert _{C({\overline{\mathcal {P}}^+ }M)}&=\sup _{(x,p)\in {\overline{\mathcal {P}}^+ }M}|u(x,p) | \le \sup _{\lambda >0}\sup _{(x,q)\in X}\frac{1}{\lambda }\left| \int _{-l_0}^0 f(\gamma _{(x,q)} (s), \lambda \dot{\gamma }_{(x,q)} (s) )ds\right| \\&\le \frac{l_0}{\lambda _{\text {min}}}\Vert f \Vert _{C({\overline{\mathcal {P}}^+ }M)}. \end{aligned}$$

Let $Z\subset {\overline{\mathcal {P}}^+ }M$ be a compact set. We next show that

$$\begin{aligned} \Vert u \Vert _{C^k (Z)}\le c_{k,K,Z}\Vert f \Vert _{C^{k}( {\overline{\mathcal {P}}^+ }M )} \end{aligned}$$

for $k\ge 1$.

We have proven that this estimate holds for $k=0$. We prove the claim for $k>0$. Let $\partial$ denote any of the partial differentials $\partial _{x^a}$ or $\partial _{p^a}$ in canonical coordinates of the bundle TM. We apply $\partial$ to the formula (B.115) of the solution u to obtain

$$\begin{aligned} \partial u(x,p)=\int _{-\infty }^0 \left[ \frac{\partial f}{\partial x^\alpha }(\gamma _{(x,p)} (s), \dot{\gamma }_{(x,p)} (s) )\partial \gamma _{(x,p)}^\alpha (s) +\frac{\partial f}{\partial p^\alpha }(\gamma _{(x,p)} (s), \dot{\gamma }_{(x,p)} (s) )\partial \dot{\gamma }_{(x,p)}^\alpha (s)\right] ds. \end{aligned}$$

Since $\frac{\partial f}{\partial x^\alpha }$ and $\frac{\partial f}{\partial p^\alpha }$ have the same properties as f, and the smooth coefficients $\partial \gamma _{(x,p)}^\alpha$ and $\partial \dot{\gamma }_{(x,p)}^\alpha$ are uniformly bounded for $(x,p)\in Z\subset {\overline{\mathcal {P}}^+ }M$, we may apply the proof above to show that

$$\begin{aligned} \Vert u \Vert _{C^{1} (Z)}\le c_{1,K,Z}\Vert f \Vert _{C^{1}( {\overline{\mathcal {P}}^+ }M )}. \end{aligned}$$

The proof for $k\ge 2$ is similar. $\square$

By using the solution formula (B.115),

$$\begin{aligned} u(x,p) = \int _{-\infty }^0 f(\gamma _{(x,p)} (s), \dot{\gamma }_{(x,p)} (s) )ds, \end{aligned}$$

in the proof of Theorem 3.2, we have the following result for Cauchy problems for the equation $\mathcal {X}u =f$ restricted to ${\mathcal {P}}^+ M$ and $L^+ M$. We denote by $L^+\mathcal {C}^{\pm }$ the bundle of future-directed lightlike vectors in the future $\mathcal {C}^+$ or past $\mathcal {C}^-$ of a Cauchy surface $\mathcal {C}$.

Corollary B.2

Assume as in Theorem 3.2 and adopt its notation. Then for the compact set $K\subset {\overline{\mathcal {P}}^+ }\mathcal {C}^+$, the Cauchy problems

$$\begin{aligned} \mathcal {X}u(x,p)&= f(x,p)&\text {on} \quad {\mathcal {P}}^+ M \nonumber \\ u(x,p)&= 0&\text {on} \quad {\mathcal {P}}^+ \mathcal {C}^-, \end{aligned}$$

(B.117)

and

$$\begin{aligned} \mathcal {X}u(x,p)&= f(x,p)&\text {on} \quad L^+ M \nonumber \\ u(x,p)&= 0&\text {on} \quad L^+ \mathcal {C}^-, \end{aligned}$$

(B.118)

have continuous solution operators $\mathcal {X}^{-1}: C^k_K({\mathcal {P}}^+ M)\rightarrow C^k({\mathcal {P}}^+ M)$ and $\mathcal {X}_L^{-1}:C^k_K(L^+ M)\rightarrow C^k(L^+ M)$ respectively.

Note that we slightly abused notation by denoting by $\mathcal {X}^{-1}$ the solution operator to both Cauchy problems (B.113) and (B.118). Here $C_K^k({\mathcal {P}}^+ M)$ and $C_K^k(L^+M)$ are defined similarly as $C_K^k({\overline{\mathcal {P}}^+ }M)$. Since ${\mathcal {P}}^+ M$ and $L^+ M$ are manifolds without boundary, we are able to use standard results to extend the problems (B.118) and (B.119) for a class of distributional sources f.

Lemma B.3

Assume that (M, g) is a globally hyperbolic $C^\infty$-Lorentzian manifold. Let $\mathcal {C}$ be a Cauchy surface of (M, g)

• (1) The solution operator $\mathcal {X}^{-1}$ to the Cauchy problem (B.118) on ${\mathcal {P}}^+ M$ has a unique continuous extension to $f\in \{h\in \mathcal {D}'({\mathcal {P}}^+ M): WF(h)\cap N^*({\mathcal {P}}^+ \mathcal {C})=\emptyset , \ h=0 \text { in } {\mathcal {P}}^+ \mathcal {C}^-\}$. If S is a submanifold of ${\mathcal {P}}^+ \mathcal {C}^+$, $f\in I^l({\mathcal {P}}^+ M; N^*S)$, $l\in \mathbb {R}$, then we have that $u={\mathcal {X}^{-1}}f$ satisfies $\chi u\in I^{l-1/4}({\mathcal {P}}^+ M; N^*K_S)$ for any $\chi \in C_c^\infty ({\mathcal {P}}^+ M)$ with $\text {supp}(\chi )\subset \subset {\mathcal {P}}^+ M{\setminus }{S}$.

• (2) The solution operator $\mathcal {X}_L^{-1}$ to the Cauchy problem (B.119) on $L^+M$ has a unique continuous extension to $\{h\in \mathcal {D}'(L^+ M): WF(h)\cap N^*({\mathcal {P}}^+ \mathcal {C})=\emptyset , \ h=0 \text { in } L^+\mathcal {C}^-\}$. If S is a submanifold of $L^+ \mathcal {C}^+$, $f\in I^l(L^+M; N^*S)$, $l\in \mathbb {R}$, then we have that $u={\mathcal {X}^{-1}}f$ satisfies $\chi u \in I^{l-1/4}(L^+M; N^*K_S)$ for any $\chi \in C_c^\infty (L^+ M)$ with $\text {supp}(\chi )\subset \subset L^+ M{\setminus }{S}$.

Proof

Let us first consider the solution operator ${\mathcal {X}^{-1}}$ to (B.118). We will refer to [14, Theorem 5.1.6]. To do that, we consider ${\mathcal {P}}^+ M$ as $\mathbb {R}\times {\mathcal {P}}^+ \mathcal {C}$ by using the flowout parametrization $\mathbb {R}\times {\mathcal {P}}^+ \mathcal {C} \rightarrow {\mathcal {P}}^+ M$ given by

$$\begin{aligned} (s, (x,p))\mapsto \dot{\gamma }_{(x,p)}(s), \quad s\in \mathbb {R}, \ (x,p) \in {\mathcal {P}}^+ \mathcal {C}. \end{aligned}$$

Also, by reviewing the proof of Lemma 3.1, we conclude that $\mathcal {X}$ is strictly hyperbolic with respect to ${\mathcal {P}}^+ \mathcal {C}$. Then, by [14, Theorem 5.1.6], the problem (B.118) has a unique solution for $f\in \{h\in \mathcal {D}'({\mathcal {P}}^+ M): WF(h)\cap N^*({\mathcal {P}}^+ \mathcal {C})=\emptyset , \ h=0 \text { in } {\mathcal {P}}^+ \mathcal {C}^-\}$. By [14, Remarks after Theorem 5.1.6] the solution operator ${\mathcal {X}^{-1}}$ to (B.118) extends continuously to $\{h\in \mathcal {D}'({\mathcal {P}}^+ M): WF(h)\cap N^*({\mathcal {P}}^+ \mathcal {C})=\emptyset , \ h=0 \text { in } {\mathcal {P}}^+ \mathcal {C}^-\}$.

If $S\subset \mathcal {P}^+ \mathcal {C}^+$ and $f\in I^l({\mathcal {P}}^+ M; N^*S)$, then we have $f\in \{h\in \mathcal {D}'({\mathcal {P}}^+ M): WF(h)\cap N^*({\mathcal {P}}^+ \mathcal {C})=\emptyset \}$, because

$$\begin{aligned} WF(f) \cap N^* ({\mathcal {P}}^+ \mathcal {C} ) \subset N^*S \cap N^* ({\mathcal {P}}^+ \mathcal {C}) = \emptyset . \end{aligned}$$

By using the definitions of the sets $C_0$ and $R_0$, which appear in [14, Theorem 5.1.6], we obtain

$$\begin{aligned} C_0\circ R_0&=\{(( x,p ;\, \xi ), (z,w ;\, \lambda ) ) \in T^*({\mathcal {P}}^+ M) \times T^*({\mathcal {P}}^+ M) {\setminus } \{ 0\}: (x,p;\, \xi )\\&\quad \text { on the bicharacteristic } \text { strip through } \lambda \in T_{(z,w)}^*{\mathcal {P}}^+ M \\&\quad \text { with } (z,w)\in {\mathcal {P}}^+ \mathcal {C} \ \text {and} \ \sigma _{-i\mathcal {X}} ( z,w ;\, \lambda ) = 0\}. \end{aligned}$$

Here $\sigma _{-i\mathcal {X}}$ is the principal symbol of $\mathcal {X}$, see (3.2). Let $\chi _1\in C_c^\infty ({\mathcal {P}}^+ M)$ be such that $\text {supp}(\chi _1)\subset \subset {\mathcal {P}}^+ M{\setminus } S$. We choose $\chi _2\in C_c^\infty ({\mathcal {P}}^+ M)$ such that $\chi _2$ equals 1 on a neighborhood of S and $\text {supp}\,(\chi _2)\subset {\mathcal {P}}^+ \mathcal {C}^+$ and such that $\text {supp}(\chi _1)\cap \text {supp}(\chi _2)=\emptyset$. Let us denote $A=\chi _1\mathcal {I}$ and $B=\chi _2\mathcal {I}$, where $\mathcal {I}$ is the identity operator. If we consider A and B as pseudodifferential operators of class $\Psi ^0_{1,0}({\mathcal {P}}^+ M)$, we have that $(WF(A) \times WF(B) )\cap [ \text {diag}(T^*(\mathcal {P}^+ M)) \cup (C_0 \circ R_0) ] = \emptyset$. We write

$$\begin{aligned} \chi _1 u = \chi _1{\mathcal {X}^{-1}}\chi _2 f+\chi _1{\mathcal {X}^{-1}}(1-\chi _2) f=A{\mathcal {X}^{-1}}B f, \end{aligned}$$

where we used $(1-\chi _2) f=0$ so that ${\mathcal {X}^{-1}}(1-\chi _2)f=0$. By [14, Theorem 5.1.6], we have that

$$\begin{aligned} A{\mathcal {X}^{-1}}B\in I^{-1/4} (\mathcal {P}^+ M,\mathcal {P}^+ M; \Lambda _\mathcal {X}). \end{aligned}$$

The flowout of the conormal bundle over S under $\mathcal {X}$ is the conormal bundle of the geodesic flowout of S. That is

$$\begin{aligned} \Lambda _\mathcal {X} \circ N^*S = N^* K_{S}. \end{aligned}$$

Finally, by applying [14, Theorem 2.4.1, Theorem 4.2.2], we have

$$\begin{aligned} \chi _1 v\in I^{l - 1/4 }(\mathcal {P}^+ M; N^* K_{S}). \end{aligned}$$

(B.119)

Renaming $\chi _1$ as $\chi$ concludes the proof of (1). The proof of (2) is a similar application of [14, Theorem 5.1.6] by using the flowout parametrization for $L^+M$ given by $(s, (x,p))\mapsto \dot{\gamma }_{(x,p)}(s)$, $(x,p)\in L^+ \mathcal {C}$ and $s\in \mathbb {R}$. $\square$

Next we prove that the Boltzmann equation has unique small solutions for small enough sources. Before that, we give an estimate regarding the collision operator in the following lemma. Following our convention of this section, the integral in the statement of the lemma over a geodesic parameter is interpreted to be over the largest interval of the form $(-T,0]$, $T>0$, where the geodesic exists.

Lemma B.4

Let (M, g) be a globally hyperbolic Lorentzian manifold and let $\mathcal {Q}$ be a collision operator with an admissible collision kernel $A : \Sigma \rightarrow \mathbb {R}$ in the sense of Definition 1.2. Then there exists a constant $C_A>0$ such that

$$\begin{aligned} \left| \int _{-\infty }^0 \mathcal {Q}[v,u]( \gamma _{(x,p)}(s),\dot{\gamma }_{(x,p)}(s)) ds \right| \le C_A \Vert u\Vert _{C({\overline{\mathcal {P}}^+ }M )}\Vert v\Vert _{C({\overline{\mathcal {P}}^+ }M)} \end{aligned}$$

for every $(x,p) \in {\overline{\mathcal {P}}^+ }M$ and $u,v \in C_b({\overline{\mathcal {P}}^+ }M )$.

Proof

Let us define a compact set $K_{\pi }:=\pi (\text {supp}\,(A))$. Let e be some auxiliary Riemannian metric on M. Let us denote

$$\begin{aligned} X=SK_{\pi }\cap {\overline{\mathcal {P}}^+ }K_{\pi } \end{aligned}$$

the bundle of future directed causal (with respect to g) vectors who have unit length in the Riemannian metric e. Since X is a closed subset of the compact set $SK_{\pi }$, we have that X is compact. By Lemma B.1, we have that there exists the maximum

$$\begin{aligned} l_0 = \max \big \{ \ell (y,q): (y,q )\in X \}, \end{aligned}$$

where $\ell (x,p)=\sup \{s\ge 0: \gamma _{(x,p)}(-s)\in K_{\pi } \}$.

Let us define another compact set $\mathcal {K}$ as

$$\begin{aligned} \mathcal {K}:=\big \{ (y,r) \in {\overline{\mathcal {P}}^+ }M: (y,r)=(\gamma _{(x,q)} (s) , \dot{\gamma }_{(x,q)} (s)), \ s\in [-\ell (x,q),0], \ (x,q)\in X \big \}. \end{aligned}$$

To see that $\mathcal {K}$ is compact, note that it is the image of the compact set $\{(s,(x,q)): s\in [-\ell (x,q),0], \ (x,q)\in X \}$ under the geodesic flow. The set $\{(s,(x,q)): s\in [-\ell (x,q),0], \ (x,q)\in X \}$ is compact since it is bounded by Lemma B.1 and closed by the upper semi-continuity of $\ell$. Since the collision kernel is admissible, the function

$$\begin{aligned} \lambda \mapsto F_{x,p}(\lambda ) = \Vert A (x, \lambda p , \,\cdot \,, \,\cdot \,, \,\cdot \,) \Vert _{L^1(\Sigma _{x,\lambda p})}, \end{aligned}$$

is by assumption continuously differentiable in $\lambda$ and attains its minimum value zero at $\lambda =0$. Thus, for any $(x,p)\in {\overline{\mathcal {P}}^+ }M$, we have that

$$\begin{aligned} \lambda ^{-1} F_{x,p}(\lambda ) \longrightarrow \frac{d}{d \lambda }\Big |_{\lambda =0}F_{x,p}(\lambda )=0 \end{aligned}$$

as $\lambda \rightarrow 0$. Since the continuous function $(x,p)\mapsto \frac{d}{d \lambda }\big |_{\lambda =0}F_{x,p}(\lambda )$ on the compact set $\mathcal {K}$ is uniformly continuous, there is a constant $\lambda _0 >0$ such that

$$\begin{aligned} \lambda ^{-1} F_{y,r}(\lambda ) \le 1 \end{aligned}$$

(B.120)

for $0< \lambda < \lambda _0$ and for (y, r) in the compact set $\mathcal {K}$.

Let $(x,p)\in {\overline{\mathcal {P}}^+ }M$ and write $(x,p)=(x,\lambda q)$. Recall from Lemma B.1 that $\lambda \ell (x,p) = \ell (x,\lambda ^{-1}p)=\ell (x,q)$. We have that

$$\begin{aligned} \begin{aligned}&{\bigg |} \int _{-\infty }^0 \mathcal {Q}[ u, v ] \big ( \gamma _{(x,p)} (s) , \dot{\gamma }_{(x,p)} (s) \big ) ds {\bigg |} \\&={\bigg |} \int _{-\ell (x,p)}^0 \mathcal {Q}[ u, v ] \big ( \gamma _{(x,\lambda q)} (s) , \dot{\gamma }_{(x,\lambda q)} (s) \big ) ds {\bigg |} \\&={\bigg |} \int _{-\ell (x,p)}^0 \mathcal {Q}[ u, v ] \big ( \gamma _{(x,q)} (\lambda s) , \lambda \dot{\gamma }_{(x,q)} (\lambda s) \big ) ds {\bigg |} \\&= \frac{1}{\lambda }{\bigg |} \int _{-\lambda \ell (x,p)}^0 \mathcal {Q}[ u, v ] \big ( \gamma _{(x,q)} (s') , \lambda \dot{\gamma }_{(x,q)} (s') \big ) ds'{\bigg |} \\&=\frac{1}{\lambda } {\bigg |} \int _{-\ell (x,q)}^0 \mathcal {Q}[ u, v ] \big ( \gamma _{(x,q)} (s') , \lambda \dot{\gamma }_{(x,q)} (s') \big ) ds' {\bigg |} \\&\le 2 \Vert u\Vert _{C({\overline{\mathcal {P}}^+ }M)}\Vert v\Vert _{C({\overline{\mathcal {P}}^+ }M)} \frac{1}{\lambda } \int _{-l_0}^0 \Vert A(\gamma _{(x,q)} (s') , \lambda \dot{\gamma }_{(x,q)} (s'), \,\cdot \,,\,\cdot \,,\,\cdot \,)\Vert _{L^1(\Sigma _{x,p})} ds' \\&\le {\left\{ \begin{array}{ll} 2 l_0 \Vert u \Vert _{C({\overline{\mathcal {P}}^+ }M)} \Vert v\Vert _{C({\overline{\mathcal {P}}^+ }M)} &{} \quad \text {if} \quad \lambda < \lambda _0 \\ 2\lambda _0^{-1}l_0 \Vert u \Vert _{C({\overline{\mathcal {P}}^+ }M)}\Vert v\Vert _{C({\overline{\mathcal {P}}^+ }M)} &{}\quad \text {if} \quad \lambda \ge \lambda _0 \\ \end{array}\right. } \\ \end{aligned} \end{aligned}$$

(B.121)

Here, for $\lambda \ge \lambda _0$, we used the condition (4) of the assumptions in the definition of an admissible kernel. For $\lambda \le \lambda _0$ we used (B.121). We also did a change of the variable in the integration as $s'=\lambda s$. This proves the claim. $\square$

Theorem 1.3

Let (M, g) be a globally hyperbolic $C^\infty$-Lorentzian manifold of dimension $n\ge 3$. Let also $\mathcal {C}$ be a Cauchy surface of M and $K\subset {\overline{\mathcal {P}}^+ }\mathcal {C}^+$ be compact. Assume that $A: \Sigma \rightarrow \mathbb {R}$ is an admissible collision kernel in the sense of Definition 1.2. Moreover, assume that $\pi (\text {supp}A) \subset \mathcal {C}^+$.

There are open neighbourhoods $B_1 \subset C_K({\overline{\mathcal {P}}^+ }\mathcal {C}^+ )$ and $B_2\subset C_b( {\overline{\mathcal {P}}^+ }M )$ of the respective origins such that if $f \in B_1$, the Cauchy problem

$$\begin{aligned} \mathcal {X}u(x,p)-\mathcal {Q}[u,u](x,p)&= f(x,p) \quad&\text { on } {\overline{\mathcal {P}}^+ }M \nonumber \\ u(x,p)&= 0 \quad&\text { on } {\overline{\mathcal {P}}^+ }\mathcal {C}^- \end{aligned}$$

(B.122)

has a unique solution $u\in B_2$. There is a constant $c_{A,K}>0$ such that

$$\begin{aligned} \Vert u \Vert _{C( {\overline{\mathcal {P}}^+ }M )}\le c_{A,K}\Vert f \Vert _{C({\overline{\mathcal {P}}^+ }M)}.\end{aligned}$$

Proof

We integrate the Eq. B.123 along the flow of $\mathcal {X}$ in TM and then use the implicit function theorem in Banach spaces for the resulting equation. (Integrating the Eq. (B.123) avoids some technicalities regarding Banach spaces, which there would be in the application of the implicit function theorem without the integration.) We define the mapping

$$\begin{aligned} F : C_K({\overline{\mathcal {P}}^+ }\mathcal {C}^+ ) \times C_b( {\overline{\mathcal {P}}^+ }M ) \rightarrow C_b( {\overline{\mathcal {P}}^+ }M ) , \end{aligned}$$

by

$$\begin{aligned}&F ( f, u ) (x,p) = u (x,p) - \int _{-\infty }^0 \mathcal {Q}[ u, u ] \nonumber \\&\big ( \gamma _{(x,p)} (s) , \dot{\gamma }_{(x,p)} (s) \big ) ds - \int _{-\infty }^0 f (\gamma _{(x,p)} (s) , \dot{\gamma }_{(x,p)}(s)) ds. \end{aligned}$$

(B.123)

Let us denote

$$\begin{aligned} Z:=\pi [\text {supp}(A)]\cup \pi [\text {supp}(f)] {\subset \mathcal {C}^+ } \end{aligned}$$

where $\pi$ is the canonical projection. In geodesically incomplete geometry, the line integrals above are interpreted as integrals over the associated lower half $(-T_1,0]$ of the inextendible domain $(-T_1,T_2)$, $T_1,T_2 \in (0,\infty ]$ of the geodesic $\gamma _{(x,p)}$. The tails of the integrals are zero. Indeed, as shown in Lemma B.1, the integrals in the definition of F above contribute only over the bounded interval $[-\ell (x,p),0] \subset (-T_1,0]$, where $\ell (x,p)=\max \{s\ge 0: \gamma _{(x,p)}(-s)\in Z \} < \infty$ is the exit time inside the inextendible domain. In combination with Lemma B.4, we have that F is well-defined.

Using $\gamma _{(\gamma _{x,p}(t),\dot{\gamma }_{x,p}(t))} (s) = \gamma _{x,p} (t+s)$ we compute

$$\begin{aligned} \begin{aligned}&\mathcal {X} \left( \int _{-\infty }^0 \mathcal {Q}[ u, u ] \big ( \gamma _{(\cdot ,\cdot )} (s) , \dot{\gamma }_{(\cdot ,\cdot )} (s) \big ) ds \right) (x,p)\\&=\partial _t\left( \int _{-\infty }^0 \mathcal {Q}[ u, u ] \big ( \gamma _{(x,p)} (t+s) , \dot{\gamma }_{(x,p)} (t+s) \big ) ds \right) \Big |_{t=0} \\&=\partial _t\left( \int _{-\infty }^t \mathcal {Q}[ u, u ] \big ( \gamma _{(x,p)} (s) , \dot{\gamma }_{(x,p)} (s) \big ) ds \right) \Big |_{t=0}\\&= \mathcal {Q}[ u, u ] \big ( \gamma _{(x,p)} (0) , \dot{\gamma }_{(x,p)} (0) \big ) = \mathcal {Q}[ u, u ](x,p) .\\ \end{aligned} \end{aligned}$$

The same argument yields also that

$$\begin{aligned} \mathcal {X} \left( \int _{-\infty }^0 f (\gamma _{(\cdot ,\cdot )} (s) , \dot{\gamma }_{(\cdot ,\cdot )}(s)) ds \right) (x,p) = f(x,p) \end{aligned}$$

Hence, $F(u,f) = 0$ implies that u satisfies the first equation in (B.123).

The second equation in (B.123), that is, the zero initial condition follows from the causality. Indeed, if (x, p) is a causal, future-pointing vector in the lower half $\mathcal {C}^-$, then points in $\{ \gamma _{x,p}(-s) : s\ge 0 \}$ are in the causal past of x and therefore also lie on the lower half. In particular, such points do not belong to $Z \subset \mathcal {C}^+$. As vectors with base-points outside Z do not contribute to the collision term nor f, the initial condition for u with $F(u,f) = 0$ follows by applying this to the integrals above.

We apply the implicit function theorem for Banach spaces (see e.g. [41, Theorem 9.6]) to F to obtain a solution u if the source $f\in C_K({\overline{\mathcal {P}}^+ }\mathcal {C}^+ )$ is small enough. First note that $F(0,0)=0$. Additionally, observe that for $u,\,v\in C_b( {\overline{\mathcal {P}}^+ }M )$ we have

$$\begin{aligned} \mathcal {Q}[ u+v, u +v ] = \mathcal {Q}[ u, u ] + \mathcal {Q}[v, u ] + \mathcal {Q}[ u, v ] + \mathcal {Q}[ v, v ] \end{aligned}$$

(B.124)

since $\mathcal {Q}$ is linear in both of its arguments.

Next, we argue that F is continuously Frechét differentiable (in the sense of [41, Definition 9.2]). Let $u,\,v\in C_b( {\overline{\mathcal {P}}^+ }M )$ and $f,\,h\in C_K({\overline{\mathcal {P}}^+ }\mathcal {C}^+ )$. We have that

$$\begin{aligned} F(f+h,u+v)-F(f,u) - L(f,u)(h,v)=- \int _{-\infty }^0 \mathcal {Q}[ v, v ] \big ( \gamma _{(x,p)} (s) , \dot{\gamma }_{(x,p)} (s) \big ) ds, \end{aligned}$$

where

$$\begin{aligned} L(f,u)(h,v)&:= v (x,p) - \int _{-\infty }^0 h (\gamma _{(x,p)} (s) , \dot{\gamma }_{(x,p)}(s)) ds\\&\quad - \int _{-\infty }^0 \mathcal {Q}[ u, v ] \big ( \gamma _{(x,p)} (s) , \dot{\gamma }_{(x,p)} (s) \big ) ds \\&\quad -\int _{-\infty }^0 \mathcal {Q}[ v, u] \big ( \gamma _{(x,p)} (s) , \dot{\gamma }_{(x,p)} (s) \big ) ds. \end{aligned}$$

It thus follows from Lemma B.4 that

$$\begin{aligned} \Vert F(f+h,u+v)-F(f,u) - L(f,u)(h,v) \Vert _{C_b({\overline{\mathcal {P}}^+ }M )}^2 \le C_A \Vert v\Vert _{C_b({\overline{\mathcal {P}}^+ }M)}^2. \end{aligned}$$

We conclude that the Frechét derivative of F at (f, u) is given by $DF(f,u)(h,v) = L(f,u)(h,v)$. We have that DF(f, u) is continuous

$$\begin{aligned} DF(f,u): C_K({\overline{\mathcal {P}}^+ }\mathcal {C}^+ )\times C_b( {\overline{\mathcal {P}}^+ }M ) \rightarrow C_b( {\overline{\mathcal {P}}^+ }M ), \end{aligned}$$

by Lemma B.4. Finally, note that the Frechét differential in the second variable of F at (0, 0)

$$\begin{aligned} DF_2(0,0):C_b( {\overline{\mathcal {P}}^+ }M ) \rightarrow C_b( {\overline{\mathcal {P}}^+ }M ), \end{aligned}$$

given by $DF_2(0,0)=DF(0,0)(0,\,\cdot \,)$, is just the identity map.

By the implicit function theorem in Banach spaces there exist open neighbourhoods $B_1\subset C_K({\overline{\mathcal {P}}^+ }\mathcal {C}^+ )$ and $B_2\subset C_b( {\overline{\mathcal {P}}^+ }M )$ of the respective origins and a continuously (Frechét) differentiable map $T: V\rightarrow U$ such that for $f\in B_1$, the function $u=T(f)\in B_2$ is the unique solution to $F(f,u)=0$. Further, since T is continuously differentiable, there exists $c_{A,K}>0$ such that

$$\begin{aligned} \Vert u \Vert _{C_b({\overline{\mathcal {P}}^+ }M ) }\le c_{A,K}\Vert f \Vert _{C_K({\overline{\mathcal {P}}^+ }M)}. \end{aligned}$$

This concludes the proof. $\square$

Next we show that the source-to-solution mapping of the Boltzmann equation can be used to compute the source-to-solution mappings of the first and second linearizations of the Boltzmann equation.

Lemma 3.4

Assume as in Theorem 1.3 and adopt its notation. Let $\Phi :B_1 \rightarrow B_2 \subset C_b({\overline{\mathcal {P}}^+ }M)$, $B_1\subset C_K({\overline{\mathcal {P}}^+ }\mathcal {C}^+ )$, be the source-to-solution map of the Boltzmann equation.

The map $\Phi$ is twice Frechét differentiable at the origin of $C_K({\overline{\mathcal {P}}^+ }\mathcal {C}^+ )$. If $f,\, h\in B_1$, then we have:

• (1) The first Frechét derivative $\Phi '$ of the source-to-solution map $\Phi$ at the origin satisfies

  $$\begin{aligned} \Phi '(0;f) = \Phi ^{L}(f), \end{aligned}$$

  where $\Phi ^L$ is the source-to-solution map of the Vlasov Eq. (3.6).

• (2) The second Frechét derivative $\Phi ''$ of the source-to-solution map $\Phi$ at the origin satisfies

  $$\begin{aligned} \Phi ''(0;f,h) = \Phi ^{2L}(f,h), \end{aligned}$$

  where $\Phi ^{2L}(f,h)\in C({\overline{\mathcal {P}}^+ }M)$ is the unique solution to the equation

  $$\begin{aligned} \mathcal {X}\Phi ^{2L}(f,h)&= \mathcal {Q}[\Phi ^L(f),\Phi ^L(h)] + \mathcal {Q}[\Phi ^L(h),\Phi ^L(f)],&\text {on}\quad {\overline{\mathcal {P}}^+ }M, \nonumber \\ \Phi ^{2L}(f,h)&= 0,&\text {on}\quad {\overline{\mathcal {P}}^+ }\mathcal {C}^-. \end{aligned}$$

  (B.125)

Proof of Lemma 3.4

Proof of (1). We adopt the notation of Theorem 1.3. Let $f\in C_K({\overline{\mathcal {P}}^+ }\mathcal {C}^+ )$ and let $f_0\in B_1$. Then by Theorem 1.3 there exists a neighbourhood $B_2$ of the origin in $C_b({\overline{\mathcal {P}}^+ }M)$ and $\epsilon _0>0$ such that for all $\epsilon _0>\epsilon >0$ the problem

$$\begin{aligned} \mathcal {X}u_{\epsilon }-\mathcal {Q}[u_{\epsilon },u_{\epsilon }]&= \epsilon f \quad \text {on } {\overline{\mathcal {P}}^+ }M \\ u_{\epsilon }&= 0 \quad \text {on } {\overline{\mathcal {P}}^+ }\mathcal {C}^-, \nonumber \end{aligned}$$

(B.126)

has a unique solution $u_{\epsilon }\in B_2$ satisfying $\Vert u_\epsilon \Vert _{C_b({\overline{\mathcal {P}}^+ }M)}\le c_{A,K}\epsilon \Vert f \Vert _{C_K({\overline{\mathcal {P}}^+ }M)}$. Let us define functions $r_\epsilon \in C_b({\overline{\mathcal {P}}^+ }M)$, for $\epsilon <\epsilon _0$, by

$$\begin{aligned} u_\epsilon =\epsilon v_0 +r_\epsilon , \end{aligned}$$

where $v_0\in C_b({\overline{\mathcal {P}}^+ }M)$ solves

$$\begin{aligned} \mathcal {X}v_{0}&= f \quad \text {on } {\overline{\mathcal {P}}^+ }M\nonumber \\ v_{0}&= 0 \quad \text {on } {\overline{\mathcal {P}}^+ }\mathcal {C}^-. \end{aligned}$$

(B.127)

We show that $r_\epsilon =\mathcal {O}(\epsilon ^2)$ in $C_b({\overline{\mathcal {P}}^+ }M)$. To show this, we first calculate

$$\begin{aligned} \mathcal {X}r_\epsilon =\mathcal {X}(u_\epsilon -\epsilon v_0)=\mathcal {Q}[u_{\epsilon },u_{\epsilon }]+\epsilon f -\epsilon \mathcal {X}v_0=\mathcal {Q}[u_{\epsilon },u_{\epsilon }]. \end{aligned}$$

We integrate this equation along the flow of $\mathcal {X}$ to obtain

$$\begin{aligned} r_\epsilon (x,p)=\int _{-\infty }^0\mathcal {Q}[u_{\epsilon },u_{\epsilon }]\big ( \gamma _{(x,p)} (s) , \dot{\gamma }_{(x,p)} (s) \big ) ds. \end{aligned}$$

(As before, the integral is actually over a finite interval since $u_\epsilon$ vanishes in $\pi ^{-1}(\mathcal {C}^-)$.) By Lemma B.4, we have that the right hand side is at most

$$\begin{aligned} C_1\Vert u_\epsilon \Vert ^2_{C_b({\overline{\mathcal {P}}^+ }M))}. \end{aligned}$$

Since $\Vert u_\epsilon \Vert _{C_b({\overline{\mathcal {P}}^+ }M)}\le C\epsilon \Vert f \Vert _{C_K( {\overline{\mathcal {P}}^+ }M )}$ by Theorem 1.3, we have that $r_\epsilon =\mathcal {O}(\epsilon ^2)$ in $C_b({\overline{\mathcal {P}}^+ }M)$ as claimed. Consequently, we have that

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}\frac{\Phi (\epsilon f)-\Phi (0)}{\epsilon } = \lim _{\epsilon \rightarrow 0}\frac{u_{\epsilon }-0}{\epsilon }=\lim _{\epsilon \rightarrow 0}\frac{\epsilon v_0 +r_\epsilon }{\epsilon }=v_0=\Phi ^L(f), \end{aligned}$$

where the limit is in $C_b( {\overline{\mathcal {P}}^+ }M )$. This proves Part (1).

Proof of (2). Let f, $u_\epsilon$ and $v_0$ be as before. We first prove that

$$\begin{aligned} u_\epsilon = \epsilon v_0 + \epsilon ^2w + \mathcal {O}(\epsilon ^3) \end{aligned}$$

in $C_b({\overline{\mathcal {P}}^+ }M)$, where w is the unique solution to

$$\begin{aligned} \mathcal {X}w&= \mathcal {Q}[v_0,v_0] \quad \text {on}\quad {\overline{\mathcal {P}}^+ }M\nonumber \\ w&= 0, \quad \text {on}\quad {\overline{\mathcal {P}}^+ }\mathcal {C}^-. \end{aligned}$$

(B.128)

A unique solution to (B.129) exists by using the formula (B.115) and noting the $\pi (\text {supp}A)$ is compact. To show this, we define $R_\epsilon \in C_b({\overline{\mathcal {P}}^+ }M)$ by

$$\begin{aligned} u_\epsilon =\epsilon v_0-\epsilon ^2w + R_\epsilon . \end{aligned}$$

(B.129)

To show that $R_\epsilon =\mathcal {O}(\epsilon ^3)$ in ${C_b({\overline{\mathcal {P}}^+ }M)}$ we apply $\mathcal {X}$ to the Eq. B.130 above. We have that

$$\begin{aligned}&\mathcal {X}R_\epsilon =\mathcal {X}(u_\epsilon -\epsilon v_0-\epsilon ^2w)=\mathcal {Q}[u_\epsilon ,u_\epsilon ]+\epsilon f-\epsilon \mathcal {X}v_0 \nonumber \\&-\epsilon ^2\mathcal {Q}[v_0,v_0] =\mathcal {Q}[u_\epsilon ,u_\epsilon ]-\epsilon ^2\mathcal {Q}[v_0,v_0]. \end{aligned}$$

(B.130)

Since the collision operator $\mathcal {Q}$ is linear in both of its arguments, we have that

$$\begin{aligned}&\mathcal {Q}[u_\epsilon ,u_\epsilon ]-\epsilon ^2\mathcal {Q}[v_0,v_0]= \mathcal {Q}[(u_\epsilon -\epsilon v_0),u_\epsilon ]-\mathcal {Q}[\epsilon v_0,(u_\epsilon -\epsilon v_0)]. \end{aligned}$$

(B.131)

We integrate the Eq. (B.131) for $R_\epsilon$ along the flow of $\mathcal {X}$ to obtain

$$\begin{aligned} R_\epsilon (x,p) \le C_1\Vert u_\epsilon -\epsilon v_0 \Vert _{C_b({\overline{\mathcal {P}}^+ }M)}\Vert u_\epsilon \Vert _{C_b({\overline{\mathcal {P}}^+ }M)}+C_1\Vert \epsilon v_0 \Vert _{C_b({\overline{\mathcal {P}}^+ }M)}\Vert u_\epsilon -\epsilon v_0 \Vert _{C_b({\overline{\mathcal {P}}^+ }M)}. \end{aligned}$$

Here we used (B.132) and Lemma B.4. By using the estimate $\Vert u_\epsilon -\epsilon v_0 \Vert _{C({\overline{\mathcal {P}}^+ }M)}\le C\epsilon ^2$ from Part (1) of this lemma, and by using that $\Vert u_\epsilon \Vert _{C_b({\overline{\mathcal {P}}^+ }M)}\le C\epsilon \Vert f \Vert _{C_K( {\overline{\mathcal {P}}^+ }M )}$ and that $\Vert v_0 \Vert _{C_b({\overline{\mathcal {P}}^+ }M)}\le C_2\Vert f \Vert _{C_K( {\overline{\mathcal {P}}^+ }M )}$ we obtain

$$\begin{aligned} \Vert R_\epsilon (x,p) \Vert _{C_b({\overline{\mathcal {P}}^+ }M)}\le C_3\epsilon ^3. \end{aligned}$$

We have shown that

$$\begin{aligned} u_\epsilon =\epsilon v_0-\epsilon ^2w + \mathcal {O}(\epsilon ^3). \end{aligned}$$

It follows that $\Phi$ is twice Frechét differentiable at the origin in $C_b({\overline{\mathcal {P}}^+ }M)$.

Let $f,\, h\in C_K({\overline{\mathcal {P}}^+ }\mathcal {C}^+)$. To prove Part (2) of the claim, we use “polarization identity of differentiation”, which says that any function F, which is twice differentiable at 0, satisfies

$$\begin{aligned} \frac{\partial ^2 }{\partial \epsilon _1\epsilon _2}\Big |_{\epsilon _1=\epsilon _2=0}F(\epsilon _1f_1+\epsilon _2f_2)=\frac{1}{4}\frac{\partial ^2}{\partial \epsilon ^2}\Big |_{\epsilon =0}[F(\epsilon (f_1+f_2))-F(\epsilon (f_1-f_2))]. \end{aligned}$$

For $f\in B_1\subset C_K({\overline{\mathcal {P}}^+ }\mathcal {C}^+ )$, we denote by $u_f$ the solution to the Boltzmann Eq. (B.127) with source f. We also denote by $v_f$ the solution to the Vlasov Eq. (B.128) with source f, and we denote similarly for $w_f$, where $w_f$ solves (B.129) where $v_0$ is replaced by $v_f$.

We need to show that

$$\begin{aligned} \frac{1}{\epsilon _1\epsilon _2}\left[ \Phi (\epsilon _1f_1+\epsilon _2f_2)-\Phi (\epsilon _2f_2)-\Phi (\epsilon _1f_1)+\Phi (0)\right] \longrightarrow w, \end{aligned}$$

(B.132)

as $\epsilon _1\rightarrow 0$ and $\epsilon _2\rightarrow 0$, where w solves

$$\begin{aligned} \mathcal {X}w&= \mathcal {Q}[ v_{f_1}, v_{f_2} ]+ \mathcal {Q}[ v_{f_2}, v_{f_1}], \quad \text {on} {\overline{\mathcal {P}}^+ }M\nonumber \\ w&= 0,\quad \text {on}\quad {\overline{\mathcal {P}}^+ }\mathcal {C}^-. \end{aligned}$$

(B.133)

By using the polarization identity (B.133) and the expansion of $u_{\epsilon (f_1\pm f_2)}$ for $\epsilon$ small, which we have already proven, we obtain

$$\begin{aligned}&\lim _{\epsilon _1,\, \epsilon _2\rightarrow 0}\frac{\Phi (\epsilon _1f_1+\epsilon _2f_2)-\Phi (\epsilon _2f_2)-\Phi (\epsilon _1f_1)+\Phi (0)}{\epsilon _1\epsilon _2} = \frac{1}{4}\frac{\partial ^2}{\partial \epsilon ^2}\Big |_{\epsilon =0}[u_{\epsilon (f_1+f_2)}-u_{\epsilon (f_1-f_2)}] \\&\quad =\frac{1}{2}w_{f_1+f_2}-\frac{1}{2}w_{f_1-f_2}. \end{aligned}$$

By denoting $w_{f_1-f_2}-w_{f_1+f_2}=2w$ and by using the linearity of $\mathcal {Q}$ in both of its argument, we finally have have that

$$\begin{aligned} \mathcal {X}w=\frac{1}{2}\mathcal {Q}[v_{f_1+f_2},v_{f_1+f_2}]-\frac{1}{2}\mathcal {Q}[v_{f_1-f_2},v_{f_1-f_2}]= \mathcal {Q}[v_{f_1},v_{f_2}]+\mathcal {Q}[v_{f_2},v_{f_1}]. \end{aligned}$$

Renaming $f_1$ and $f_2$ as f and h respectively proves the claim.  $\square$