On Existence and Uniqueness to Homogeneous Boltzmann Flows of Monatomic Gas Mixtures

Abstract

We solve the Cauchy problem for the full non-linear homogeneous Boltzmann system of equations describing multi-component monatomic gas mixtures for binary interactions in three dimensions. More precisely, we show the existence and uniqueness of the vector value solution by means of an existence theorem for ODE systems in Banach spaces under the transition probability rates assumption corresponding to hard potentials rates in the interval (0, 1], with an angular section modeled by an integrable function of the angular transition rates modeling binary scattering effects. The initial data for the vector valued solutions needs to be a vector of non-negative measures with finite total number density, momentum and strictly positive energy, as well as to have a finite $L^1_{k_*}(\mathbb {R}^3)$-integrability property corresponding to a sum across each species of $k_*$-polynomial weighted norms depending on the corresponding mass fraction parameter for each species as much as on the intermolecular potential rates, referred as to the scalar polynomial moment of order $k_*$. The rigorous existence and uniqueness results rely on a new angular averaging lemma adjusted to vector values solution that yield a Povzner estimate with constants that decay with the order of the corresponding dimensionless scalar polynomial moment. In addition, such initial data yields global generation of such scalar polynomial moments at any order as well as their summability of moments to obtain estimates for corresponding scalar exponentially decaying high energy tails, referred as to scalar exponential moments associated to the system solution. Such scalar polynomial and exponential moments propagate as well.

Appendices

Appendix A. Existence and Uniqueness Theory for ODE in Banach Spaces

Theorem A.1

Let $E:=(E,\left\| \cdot \right\| )$ be a Banach space, $\mathcal {S}$ be a bounded, convex and closed subset of E, and $\mathcal {Q}:\mathcal {S}\rightarrow E$ be an operator satisfying the following properties:

• (a) The Hölder continuity condition

  $$\begin{aligned} \left\| \mathcal {Q}[u] - \mathcal {Q}[v] \right\| \leqq C \left\| u-v \right\| ^{\beta }, \ \beta \in (0,1), \ \forall u, v \in \mathcal {S}; \end{aligned}$$

• (b) The Sub-tangent condition

  $$\begin{aligned} \lim \limits _{h\rightarrow 0+} \frac{\text {dist}\left( u + h \mathcal {Q}[u], \mathcal {S} \right) }{h} =0, \ \forall u \in \mathcal {S}; \end{aligned}$$

• (c) The One-sided Lipschitz condition

  $$\begin{aligned} \left[ \mathcal {Q}[u] - \mathcal {Q}[v], u - v \right] \leqq C \left\| u-v \right\| , \ \forall u, v \in \mathcal {S}, \end{aligned}$$

  where $\left[ \varphi ,\phi \right] =\lim _{h\rightarrow 0^-} h^{-1}\left( \left\| \phi + h \varphi \right\| - \left\| \phi \right\| \right)$.

Then the equation

$$\begin{aligned} \begin{aligned} \partial _t u = \mathcal {Q}[u], \ \text {for} \ t\in (0,\infty ), \ \text {with initial data} \ u(0)= u_0 \ \text {in} \ \mathcal {S}, \end{aligned} \end{aligned}$$

has a unique solution in $C([0,\infty ),\mathcal {S})\cap C^1((0,\infty ),E)$.

The proof of this Theorem on ODE flows on Banach spaces can be found in the unpublished notes [10] or in [3].

Remark 9

In Sect. 5, we will concentrate on $E:=L_2^1$. Therefore, for the one-sided Lipschitz condition, we will use the following inequality:

$$\begin{aligned} \left[ \varphi ,\phi \right] \leqq \sum _{i=1}^I \int _{ \mathbb {R}^3} \varphi _i(v) \, \text {sign}(\phi _i (v))\left\langle v \right\rangle _i^2 \mathrm {d}v, \end{aligned}$$

for $\varphi =\left[ \varphi _i\right] _{1\leqq i\leqq I}$ and $\phi =\left[ \phi _i\right] _{1\leqq i\leqq I}$, as pointed out in [3].

Appendix B. Upper and Lower Bound of the Cross Section

In this section, we derive an upper and lower estimate for the non-angular part of the cross section, $\left| v-v_*\right| ^{\gamma _{ij}}$, $\gamma _{ij} \in (0,1]$, with $1\leqq i, j, \leqq I$. First, for the upper estimate, by triangle inequality, we have

$$\begin{aligned} \sqrt{\frac{m_i}{\sum _{i=1}^I m_i}} \sqrt{\frac{m_j}{\sum _{i=1}^I m_i}} \left| v-v_*\right|&\leqq \min \left\{ \sqrt{\frac{m_i}{\sum _{i=1}^I m_i}}, \sqrt{\frac{m_j}{\sum _{i=1}^I m_i}} \right\} \left| v-v_*\right| \nonumber \\&\leqq \min \left\{ \sqrt{\frac{m_i}{\sum _{i=1}^I m_i}}, \sqrt{\frac{m_j}{\sum _{i=1}^I m_i}} \right\} \left( \left| v\right| + \left| v_*\right| \right) \nonumber \\&\leqq \sqrt{\frac{m_i}{\sum _{i=1}^I m_i}} \left| v\right| + \sqrt{\frac{m_j}{\sum _{i=1}^I m_i}} \left| v_*\right| \nonumber \\&\leqq \sqrt{1 + \frac{m_i}{\sum _{i=1}^I m_i} \left| v\right| ^2} + \sqrt{1 + \frac{m_j}{\sum _{i=1}^I m_i} \left| v_*\right| ^2}. \end{aligned}$$

(B.1)

Therefore,

$$\begin{aligned} \left| v-v_*\right| ^{\gamma _{ij}} \leqq \left( \frac{\sum _{i=1}^I m_i}{\sqrt{m_i m_j}} \right) ^{\gamma _{ij}} \left( \left\langle v \right\rangle _i^{\gamma _{ij}} + \left\langle v_* \right\rangle _j^{\gamma _{ij}}\right) \end{aligned}$$

(B.2)

for $\gamma _{ij}\in (0,1]$, and any $i, j \in \left\{ 1,\dots ,I \right\}$.

From (B.1) it also follows that

$$\begin{aligned}&\sqrt{\frac{m_i}{\sum _{i=1}^I m_i}} \sqrt{\frac{m_j}{\sum _{i=1}^I m_i}} \left| v-v_*\right| \\&\quad \leqq \sqrt{\frac{m_i}{\sum _{i=1}^I m_i}} \left| v\right| + \sqrt{\frac{m_j}{\sum _{i=1}^I m_i}} \left| v_*\right| \\&\quad = \left( \frac{m_i}{\sum _{i=1}^I m_i} \left| v\right| ^2 + \frac{m_j}{\sum _{i=1}^I m_i} \left| v_*\right| ^2 + 2 \frac{\sqrt{m_i m_j}}{\sum _{i=1}^I m_i} \left| v\right| \left| v_*\right| \right) ^{1/2}\\&\quad \leqq \left\langle v \right\rangle _i \left\langle v_* \right\rangle _j. \end{aligned}$$

Therefore,

$$\begin{aligned} \left| v-v_*\right| ^{\gamma _{ij}} \leqq \left( \frac{\sum _{i=1}^I m_i}{\sqrt{m_i m_j}} \right) ^{\gamma _{ij}} \left\langle v \right\rangle _i^{\gamma _{ij}} \left\langle v_* \right\rangle _j^{\gamma _{ij}} \end{aligned}$$

(B.3)

for $\gamma _{ij} \in (0,1]$ and $1\leqq i, j \leqq I$.

Then, for the lower estimate, we use the ideas of Lemma 2.1 in [4] to prove the next Lemma. Note that here functions F do not need to be solutions of the Boltzmann problem. Moreover, this lower bound may not hold for F being a singular measure, since the estimate degenerates as c goes to zero.

Lemma B.1

Let $\gamma _{ij} \in [0,2]$, for any $i, j \in \left\{ 1,\dots , I \right\}$, and assume that $0\leqq \left\{ F(t) = \left[ f_1(t) \dots f_I(t) \right] ^T \right\} _{t\geqq 0} \subset L_2^1$ satisfies

$$\begin{aligned} c\leqq & {} \sum _{i=1}^I \int _{\mathbb {R}^3} m_i \, f_i (t, v) \,\mathrm {d}v \leqq C, \quad c \leqq \sum _{i=1}^I \int _{\mathbb {R}^3} f_i (t, v) m_i \left| v\right| ^2 \,\mathrm {d}v \leqq C,\\&\sum _{i=1}^I \int _{\mathbb {R}^3} f_i (t, v) m_i v \,\mathrm {d}v=0 \end{aligned}$$

for some positive constants c and C. Assume also the boundedness of the moment

$$\begin{aligned} \sum _{i=1}^I \int _{\mathbb {R}^3} f_i (t, v) m_i \left| v\right| ^{2+\varepsilon } \,\mathrm {d}v \leqq B, \quad \varepsilon >0. \end{aligned}$$

Then, there exists a constant $c_{lb}$ characterized in (B.11) such that

$$\begin{aligned} \sum _{i=1}^I \int _{\mathbb {R}^3} m_i f_i (t, w) \left| v-w\right| ^{\gamma _{ij}} \,\mathrm {d}w \geqq c_{lb} \left\langle v \right\rangle _j^{\overline{\gamma }} \end{aligned}$$

(B.4)

for any $j \in \{1,\dots ,I\}$, with $\overline{\gamma }=\max _{1\leqq i, j \leqq I}\gamma _{ij}$.

Proof

Case $\gamma _{ij}=0$ is trivial, so take $\gamma _{ij} \in (0,2]$ for any $i,j,=1,\dots ,I$. Let us denote the open ball centered at the origin and of radius $r>0$ with $B(0,r) \subset \mathbb {R}^3$. We consider separately cases when $v \in B(0,r)$ and $v \in B(0,r)^c$, with r to be chosen later on depending on constants c, C, and $\gamma _{ij}$.

For $v \in B(0,r)^c$ we first consider the whole domain $\mathbb {R}^3$, and write, by the Young inequality, for any $v \in \mathbb {R}^3$ and $\gamma _{ij} \in (0,2]$,

$$\begin{aligned} \sum _{i=1}^I m_i \int _{\mathbb {R}^3} f_i(t,w) \left| v-w\right| ^{\gamma _{ij}} \,\mathrm {d} w \geqq \sum _{i=1}^I m_i \int _{\mathbb {R}^3} f_i(t,w) \left( \tilde{c} \left| v\right| ^{\gamma _{ij}} - \left| w\right| ^{\gamma _{ij}} \right) \,\mathrm {d} w, \end{aligned}$$

where $\tilde{c} =\min _{1\leqq i, j \leqq I} \left( \min \{1, 2^{1-\gamma _{ij}}\} \right)$. Since

$$\begin{aligned}&\sum _{i=1}^I m_i \int _{\mathbb {R}^3} f_i(t,w) \left| w\right| ^{\gamma _{ij}} \,\mathrm {d} w \\&\quad \leqq \sum _{i=1}^I m_i \int _{B(0,1)} f_i(t,w) \,\mathrm {d} w + \sum _{i=1}^I m_i \int _{B(0,1)^c} f_i(t,w) \left| w\right| ^2 \,\mathrm {d} w \leqq 2 C, \end{aligned}$$

we obtain that, for any $v\in \mathbb {R}^3$,

$$\begin{aligned}&\sum _{i=1}^I m_i \int _{\mathbb {R}^3} f_i(t,w) \left| v-w\right| ^{\gamma _{ij}} \,\mathrm {d} w \nonumber \\&\quad \geqq \tilde{c} \sum _{i=1}^I \left| v\right| ^{\gamma _{ij}} m_i \int _{\mathbb {R}^3} f_i(t,w) \,\mathrm {d} w - 2C. \end{aligned}$$

(B.5)

Since, for any $i,j=1,\dots ,I$ and $v\in \mathbb {R}^3$, we have the following lower bound:

$$\begin{aligned} \left| v \right| ^{\gamma _{ij}} = \left| v \right| ^{\gamma _{ij}} \left( \mathbb {1}_{\left| v\right| <1}(v) + \mathbb {1}_{\left| v\right| \geqq 1}(v) \right) \geqq \left| v \right| ^{\overline{\gamma }} +1, \end{aligned}$$

where

$$\begin{aligned} \overline{\gamma }=\max _{1\leqq i, j \leqq I}\gamma _{ij}. \end{aligned}$$

Therefore, (B.5) becomes

$$\begin{aligned} \sum _{i=1}^I m_i \int _{\mathbb {R}^3} f_i(t,w) \left| v-w\right| ^{\gamma _{ij}} \,\mathrm {d} w\geqq & {} \tilde{c} \, c \, \left| v\right| ^{\overline{\gamma }} + \tilde{c} \, c - 2C \\\geqq & {} \tilde{c} \, c \left( \sqrt{\frac{m_j}{\sum _{i=1}^I m_i}} \left| v\right| \right) ^{\overline{\gamma }} + \tilde{c} \, c - 2C \end{aligned}$$

for every $j\in \{1,\dots ,I\}$. Since here $v \in B(0,r)^c$, we choose r in a such a way as to ensure that

$$\begin{aligned} \tilde{c} \, c \left( \sqrt{\frac{m_j}{\sum _{i=1}^I m_i}} \left| v\right| \right) ^{\overline{\gamma }} + \tilde{c} \, c - 2C \geqq \frac{\tilde{c} \, c}{2} \left( \sqrt{\frac{m_j}{\sum _{i=1}^I m_i}} \left| v\right| \right) ^{\overline{\gamma }}, \end{aligned}$$

which amounts to choosing

$$\begin{aligned} r:=r_* =\left( \frac{2 \, C}{\tilde{c} \, c} \right) ^\frac{1}{\overline{\gamma }}, \end{aligned}$$

(B.6)

since $C\geqq c$ by assumption, and $\tilde{c} \leqq 1$. Therefore, for $v \in B(0,r^*)^c$, we have

$$\begin{aligned} \sum _{i=1}^I m_i \int _{\mathbb {R}^3} f_i(t,w) \left| v-w\right| ^{\gamma _{ij}} \,\mathrm {d} w \geqq \frac{\tilde{c} \, c}{2} \left( \sqrt{\frac{m_j}{\sum _{i=1}^I m_i}} \left| v\right| \right) ^{\overline{\gamma }} \end{aligned}$$

(B.7)

for any $j\in \{1,\dots ,I\}$.

On the other hand, let us study the case $v \in B(0,r^*)$. First note that for any $R>0$,

$$\begin{aligned}&\sum _{i=1}^I m_i \int _{\left| v-w\right| \leqq R} f_i(t,w) \left| v-w\right| ^2 \,\mathrm {d} w \nonumber \\&\quad =\sum _{i=1}^I m_i \int _{\mathbb {R}^3} f_i(t,w) \left| v-w\right| ^2 \,\mathrm {d} w - \sum _{i=1}^I m_i \int _{\left| v-w\right| \geqq R} f_i(t,w) \left| v-w\right| ^2 \,\mathrm {d} w \nonumber \\&\quad \geqq c \left| v\right| ^2+ c - \sum _{i=1}^I m_i \int _{\left| v-w\right| \geqq R} f_i(t,w) \left| v-w\right| ^2 \,\mathrm {d} w \nonumber \\&\quad \geqq c (1+ \left| v\right| ^2) - \frac{1}{R^{\varepsilon }} \sum _{i=1}^I m_i \int _{\left| v-w\right| \geqq R} f_i(t,w) \left| v-w\right| ^{2+\varepsilon } \,\mathrm {d} w. \end{aligned}$$

(B.8)

Next, we have

$$\begin{aligned}&\sum _{i=1}^I m_i \int _{\left| v-w\right| \geqq R} f_i(t,w) \left| v-w\right| ^{2+\varepsilon } \,\mathrm {d} w \leqq 2^{1+\varepsilon } \max \{C, B\} \left( 1+ \left| v\right| ^{2+\varepsilon }\right) \\&\quad \leqq 2^{1+\varepsilon } \max \{C,B\} \left( 1+ \left| v\right| ^{2}\right) ^{\frac{2+\varepsilon }{2}} \leqq 2^{1+\varepsilon } \max \{C, B\} \left( 1+ r_*^{2}\right) ^{\frac{2+\varepsilon }{2}}. \end{aligned}$$

Choosing $R:=R(r_*, c, C, B)>0$ sufficiently large such that

$$\begin{aligned}&\frac{1}{R^\varepsilon }2^{1+\varepsilon } \max \{C, B\} \left( 1+ r_*^{2}\right) ^{\frac{2+\varepsilon }{2}} \leqq \frac{c}{2}, \quad \text {or} \ R \nonumber \\&\quad \geqq \left( 2^{2+\varepsilon } \left( \frac{\max \{C,B\}}{c} \right) \left( 1+ r_*^{2}\right) ^{\frac{2+\varepsilon }{2}}\right) ^{\frac{1}{\varepsilon }}, \end{aligned}$$

(B.9)

from (B.8) we have

$$\begin{aligned} \sum _{i=1}^I m_i \int _{\left| v-w\right| \leqq R} f_i(t,w) \left| v-w\right| ^2 \,\mathrm {d} w \geqq \frac{c}{2} \quad \forall v \in B(0,r_*). \end{aligned}$$

Moreover, for this choice of R, for any $\gamma _{ij}\in (0,2]$ we have

$$\begin{aligned}&\sum _{i=1}^I m_i \int _{\mathbb {R}^3} f_i(t,w) \left| v-w\right| ^{\gamma _{ij}} \,\mathrm {d} w \geqq \sum _{i=1}^I m_i \int _{\left| v-w\right| \leqq R} f_i(t,w) \left| v-w\right| ^{\gamma _{ij}} \,\mathrm {d} w \nonumber \\&\quad \geqq \sum _{i=1}^I R^{\gamma _{ij} -2 } \, m_i \int _{\left| v-w\right| \leqq R} f_i(t,w) \left| v-w\right| ^2 \,\mathrm {d} w. \end{aligned}$$

Since $R\geqq 1$, we can bound $R^{\gamma _{ij} -2 } \geqq R^{(\min _{1\leqq i, j \leqq I}\gamma _{ij}) -2}$, which yields the estimate

$$\begin{aligned} \sum _{i=1}^I m_i \int _{\mathbb {R}^3} f_i(t,w) \left| v-w\right| ^{\gamma _{ij}} \,\mathrm {d} w \geqq \frac{c}{2 R^{2-\min _{1\leqq i, j \leqq I}\gamma _{ij}}}, \quad \forall v \in B(0,r_*).\nonumber \\ \end{aligned}$$

(B.10)

Finally, summarizing (B.7) and (B.10),

$$\begin{aligned}&\sum _{i=1}^I m_i \int _{\mathbb {R}^3} f_i(t,w) \left| v-w\right| ^{\gamma _{ij}} \,\mathrm {d} w \\&\quad \geqq \frac{c}{2 R^{2-\min _{1\leqq i, j \leqq I}\gamma _{ij}}} \mathbb {1}_{B(0,r_*)}(v) \\&\qquad +\, \frac{\tilde{c} \, c}{2} \left( \sqrt{\frac{m_j}{\sum _{i=1}^I m_i}} \left| v\right| \right) ^{\overline{\gamma }}\mathbb {1}_{B(0,r_*)^c}(v)\\&\quad \geqq \frac{\tilde{c} \, c}{2 R^{2-\min _{1\leqq i, j \leqq I}\gamma _{ij}}} \left( \mathbb {1}_{B(0,r_*)}(v)+ \left( \sqrt{\frac{m_j}{\sum _{i=1}^I m_i}} \left| v\right| \right) ^{\overline{\gamma }}\mathbb {1}_{B(0,r_*)^c}(v) \right) . \end{aligned}$$

Then there exists a constant $c_{lb}$ such that

$$\begin{aligned} \frac{\tilde{c} \, c}{2 R^{2-\min _{1\leqq i, j \leqq I}\gamma _{ij}}} \left( \mathbb {1}_{B(0,r_*)}(v)+ \left( \sqrt{\frac{m_j}{\sum _{i=1}^I m_i}} \left| v\right| \right) ^{\overline{\gamma }}\mathbb {1}_{B(0,r_*)^c}(v) \right) \geqq c_{lb} \left\langle v \right\rangle _j^{\overline{\gamma }} \end{aligned}$$

for any $j\in \left\{ 1,\dots ,I\right\}$. In fact, one may even construct $c_{lb}$ in order to ensure the last inequality. For example, $c_{lb}$ can take the value

$$\begin{aligned} c_{lb}= & {} \frac{c}{2} \tilde{c} \left( 2^{2+\varepsilon } \left( \frac{\max \{C,B\}}{c} \right) \left( 1+ \left( \frac{ 2 \, C}{\tilde{c} \, c} \right) ^\frac{2}{\overline{\gamma }}\right) ^{\frac{2+\varepsilon }{2}}\right) ^{\frac{-2+\min _{1\leqq i, j \leqq I}\gamma _{ij}}{\varepsilon }} \nonumber \\&\times \left( 1 + \frac{\max _{1\leqq j \leqq I} m_j}{\sum _{i=1}^I m_i} \left( \frac{ 2 \, C}{\tilde{c} \, c} \right) ^2 \right) ^{-\overline{\gamma }/2}, \end{aligned}$$

(B.11)

by taking into account (B.6) and (B.9). $\quad \square$

Appendix C. Some Technical Results

Lemma C.1

(Polynomial inequality I, Lemma 2 from [8].) Assume $p>1$, and let $n_p=\lfloor \frac{p+1}{2}\rfloor$. Then, for all $x, y>0$, the following inequality holds:

$$\begin{aligned} \left( x+y\right) ^p - x^p - y^p \leqq \sum _{n=1}^{n_p} \left( \begin{matrix} p \\ n \end{matrix} \right) \left( x^n y^{p-n} + x^{p-n} y^n \right) . \end{aligned}$$

Lemma C.2

(Polynomial inequality II.) Let $b+1\leqq a\leqq \frac{p+1}{2}$. Then, for any $x, y\geqq 0$,

$$\begin{aligned} x^a y^{p-a} + x^{p-a} y^a \leqq x^b y^{p-b} + x^{p-b} y^b. \end{aligned}$$

Proof

This Lemma is a modified version of Lemma A.1 from [22]. Indeed, the proof is the same; one just needs to observe that $a-b\geqq 0$ and $p-a-b\geqq 0$, and therefore that

$$\begin{aligned} \left( y^{a-b}-x^{a-b}\right) x^by^b\left( y^{p-a-b}-x^{p-a-b}\right) \geqq 0 \end{aligned}$$

for any $x,y\geqq 0$. $\quad \square$

Lemma C.3

(Interpolation inequality.) Let $k=\alpha k_1 + (1-\alpha ) k_2$, $\alpha \in (0,1)$, $0<k_1\leqq k \leqq k_2$. Then, for any $g \in L_{k,i}^1$

$$\begin{aligned} \left\| g \right\| _{L_{k,i}^1} \leqq \left\| g \right\| _{L_{k_1,i}^1}^{\alpha } \left\| g \right\| _{L_{k_2,i}^1}^{1-\alpha }. \end{aligned}$$

(C.1)

We can extend this interpolation inequality for vector functions $\mathbb {G}=\left[ g_i\right] _{1\leqq i \leqq I}$. Namely, under the same assumptions,

$$\begin{aligned} \left\| \mathbb {G} \right\| _{L_k^1} \leqq I \left\| \mathbb {G} \right\| _{L_{k_1}^1}^\alpha \left\| \mathbb {G} \right\| _{L_{k_2}^1}^{1-\alpha }. \end{aligned}$$

(C.2)

Lemma C.4

(Jensen’s inequality.) Let f(x) be positive and integrable in $\mathbb {R}^d$ and G a convex function. Then

$$\begin{aligned} G\left( \frac{1}{\int f(x) \,\mathrm {d}x} \int f(x) g(x) \,\mathrm {d}x \right) \leqq \frac{1}{\int f(x) \,\mathrm {d}x} \int f(x) G(g(x)) \,\mathrm {d}x \end{aligned}$$

for any positive function g.

We apply this lemma specifying that $g(x)=\left\langle x \right\rangle _i^k$ and $G(x)=x^{1+\frac{\lambda }{k}}$, $\lambda \in (0,1]$ and $k\geqq 1$. This implies

$$\begin{aligned} \int _{ \mathbb {R}^3} f_i(v) \left\langle v \right\rangle _i^{k+\lambda } \,\mathrm {d}v \geqq \left( \int _{ \mathbb {R}^3} f_i(v) \,\mathrm {d}v \right) ^{-\frac{\lambda }{k}} \left( \int _{ \mathbb {R}^3} f_i(v) \left\langle v \right\rangle _i^k \,\mathrm {d}v \right) ^{1+\frac{\lambda }{k}} . \end{aligned}$$

If, additionally, we have an upper bound on the zero order scalar polynomial moment, that is, if it holds that

$$\begin{aligned} \int _{ \mathbb {R}^3} f_i(v) \,\mathrm {d}v = \mathfrak {m}_{0,i}[\mathbb {F}] \leqq \mathfrak {m}_0[\mathbb {F}] \leqq C_{\mathfrak {m}_{0}}, \end{aligned}$$

then

$$\begin{aligned} \int _{ \mathbb {R}^3} f_i(v) \left\langle v \right\rangle _i^{k+\lambda } \,\mathrm {d}v \geqq C_{\mathfrak {m}_{0}}^{-\frac{\lambda }{k}} \left( \int _{ \mathbb {R}^3} f_i(v) \left\langle v \right\rangle _i^k \,\mathrm {d}v \right) ^{1+\frac{\lambda }{k}} . \end{aligned}$$

Summing over $i=1,\dots ,I$, after some manipulation we get a control from below for the moment $\mathfrak {m}_{k+\lambda }[\mathbb {F}]$. Indeed, we get that

$$\begin{aligned} \mathfrak {m}_{k+\lambda }[\mathbb {F}] \geqq \left( I C_{\mathfrak {m}_{0}} \right) ^{-\frac{\lambda }{k}} \mathfrak {m}_k[\mathbb {F}]^{1+\frac{\lambda }{k}}. \end{aligned}$$

(C.3)