Extending Cercignani’s Conjecture Results from Boltzmann to Boltzmann–Fermi–Dirac Equation

Abstract

We establish a connection between the relative Classical entropy and the relative Fermi–Dirac entropy, allowing to transpose, in the context of the Boltzmann or Landau equation, any entropy–entropy production inequality from one case to the other; therefore providing entropy–entropy production inequalities for the Boltzmann–Fermi–Dirac operator, similar to the ones of the Classical Boltzmann operator. We also provide a generalized version of the Csiszár–Kullback–Pinsker inequality to weighted $L^p$ norms, $1 \le p \le 2$ and a wide class of entropies.

Appendices

Appendix A Similar Results in the Bose–Einstein Case

An upper-bound inequality similar to (34) can also be obtained in the Bose–Einstein case. This latter case formally corresponds to taking $- \varepsilon$ instead of $\varepsilon$ in our formulas. First define for any $x \in {\mathbb {R}}_+$ and $\varepsilon > 0$,

$$\begin{aligned} \varphi ^{BE}_{\varepsilon }(x) := \frac{x}{1 + \varepsilon x}, \qquad \Phi _{\varepsilon }^{BE}(x) := \int _0^x \log \varphi ^{BE}_{\varepsilon }(y) \, \textrm{d}y, \end{aligned}$$

and the Bose–Einstein entropy of $0 \le f \in L^1_2({\mathbb {R}}^3)$:

$$\begin{aligned} H^{BE}_{\varepsilon }(f) := \int _{{\mathbb {R}}^3} \Phi _{\varepsilon }^{BE}(f) \, \textrm{d}v. \end{aligned}$$

Lu proved in [21] that, under the condition

$$\begin{aligned} T \ge \frac{\zeta (\frac{5}{2})}{\zeta (\frac{3}{2})} \, T_c, \qquad \quad T_c := \frac{1}{2 \pi } \left( \frac{\rho \, \varepsilon }{\zeta (\frac{3}{2})} \right) ^{2/3}, \end{aligned}$$

(101)

where $\zeta$ is the Riemann Zêta function and $T_c$ is called the critical temperature, there exists a unique $\varepsilon$-Bose–Einstein statistics ${\mathcal {M}}_{\varepsilon }^{BE,f}$ associated to f - that is a distribution such that $\log \varphi ^{BE}_{\varepsilon } ({\mathcal {M}}_{\varepsilon }^{BE,f})$ is a linear combination of $v\mapsto 1$, $v\mapsto v$ and $v\mapsto |v|^2$ and sharing the same normalization in $v\mapsto 1$, $v\mapsto v$ and $v\mapsto |v|^2$ as f. We can obtain from Proposition 15 in Appendix B that, denoting $H^{BE}_{\varepsilon }[f|{\mathcal {M}}_{\varepsilon }^{BE,f}] := H^{BE}_{\varepsilon }(f) - H^{BE}_{\varepsilon }({\mathcal {M}}_{\varepsilon }^{BE,f})$, we have

$$\begin{aligned}&H^{BE}_{\varepsilon }[f|{\mathcal {M}}_{\varepsilon }^{BE,f}] \nonumber \\&= \int _0^1 (1-\tau ) \left( \int _{{\mathbb {R}}^3} \left( f(v)-{\mathcal {M}}_{\varepsilon }^{BE,f}(v) \right) ^2 \, {\Phi ^{BE}_{\varepsilon }}'' \left( (1-\tau ){\mathcal {M}}_{\varepsilon }^{BE,f}(v) + \tau f(v) \right) \, \textrm{d}v \right) \textrm{d}\tau \nonumber \\&= \int _{{\mathbb {R}}^3} \int _{{\mathcal {M}}_{\varepsilon }^{BE,f}(v)}^{f(v)} \frac{f(v) - x}{\varphi ^{BE}_{\varepsilon }(x)} \, {\varphi ^{BE}_{\varepsilon }}'(x) \, \textrm{d}x \, \textrm{d}v, \end{aligned}$$

(102)

where the last equality comes from ${\Phi ^{BE}_{\varepsilon }}'' = \frac{{\varphi ^{BE}_{\varepsilon }}'}{\varphi ^{BE}_{\varepsilon }}$ and the change of variables $x = {\mathcal {M}}_{\varepsilon }^{BE,f}(v) + \tau (f(v) - {\mathcal {M}}_{\varepsilon }^{BE,f})$.

In the following Proposition, we provide a link between the relative entropies to equilibrium of the Bose–Einstein and the classical cases. Although I believe that both inequalities could be obtained, we only present here the “upper-bound” inequality as its proof is rather short. Further work may allow to obtain the lower-bound inequality, with a constant that probably depends on an $L^{\infty }$ bound on f. This constitutes another reason why we did not investigate this other inequality, as, although $L^{\infty }$ bounds are natural to use in the Fermi–Dirac context, due to Pauli’s exclusion principle, they are not in the Bose–Einstein one, due to the phenomenon of condensation.

Proposition 12

Upper-bound in the Bose–Einstein case. For any $\varepsilon > 0$ and nonnegative $f \in L^1_2({\mathbb {R}}^3) \cap L \log L ({\mathbb {R}}^3) \setminus \{0\}$ which density and temperature satisfy (101), we have

$$\begin{aligned} H_0\left[ \left. \frac{f}{1 + \varepsilon f} \right| M^{\frac{f}{1 + \varepsilon f}} \right] \le H^{BE}_{\varepsilon } \left[ f|{\mathcal {M}}_{\varepsilon }^{BE,f} \right] . \end{aligned}$$

(103)

Proof

The starting point of the proof are Equation (102) and the inequality $|y-z| \ge |\varphi ^{BE}_{\varepsilon }(y) - \varphi ^{BE}_{\varepsilon }(z)|$ for all $(y,z) \in {\mathbb {R}}_+^2$, yielding

$$\begin{aligned}{} & {} H^{BE}_{\varepsilon } \left[ f|{\mathcal {M}}_{\varepsilon }^{BE,f} \right] \\{} & {} \quad = \int _{{\mathbb {R}}^3} \int _{{\mathcal {M}}_{\varepsilon }^{BE,f}}^f \frac{f - x}{\varphi ^{BE}_{\varepsilon }(x)} \, {\varphi ^{BE}_{\varepsilon }}'(x) \, \textrm{d}x \, \textrm{d}v \ge \int _{{\mathbb {R}}^3} \int _{{\mathcal {M}}_{\varepsilon }^{BE,f}}^f \frac{\varphi ^{BE}_{\varepsilon }(f) - \varphi ^{BE}_{\varepsilon }(x)}{\varphi ^{BE}_{\varepsilon }(x)} \, {\varphi ^{BE}_{\varepsilon }}'(x) \, \textrm{d}x \, \textrm{d}v. \end{aligned}$$

Applying the change of variables $y = \varphi ^{BE}_{\varepsilon }(x)$ and using formula (27), we obtain

$$\begin{aligned} H^{BE}_{\varepsilon } \left[ f|{\mathcal {M}}_{\varepsilon }^{BE,f} \right] \ge H_0\left[ \varphi ^{BE}_{\varepsilon }(f)|M^{\varphi ^{BE}_{\varepsilon }(f)} \right] + \int _{{\mathbb {R}}^3} \int _{\varphi ^{BE}_{\varepsilon }({\mathcal {M}}_{\varepsilon }^{BE,f}) }^{M^{\varphi ^{BE}_{\varepsilon }(f)} } \frac{\varphi ^{BE}_{\varepsilon }(f) - y}{y} \, \textrm{d}y \, \textrm{d}v. \end{aligned}$$

Remark that, as $\log M^{\varphi ^{BE}_{\varepsilon }(f)} - \log \varphi ^{BE}_{\varepsilon }({\mathcal {M}}_{\varepsilon }^{BE,f})$ is a linear combination of conserved quantities (namely, $v \mapsto 1$, $v \mapsto v$ and $v \mapsto |v|^2$), we have by definition of $M^{\varphi ^{BE}_{\varepsilon }(f)}$ that

$$\begin{aligned} \int _{{\mathbb {R}}^3} \varphi ^{BE}_{\varepsilon }(f) \, \log \left( \frac{M^{\varphi ^{BE}_{\varepsilon }(f)}}{\varphi ^{BE}_{\varepsilon }({\mathcal {M}}_{\varepsilon }^{BE,f})}\right) \textrm{d}v = \int _{{\mathbb {R}}^3} M^{\varphi ^{BE}_{\varepsilon }(f)} \, \log \left( \frac{M^{\varphi ^{BE}_{\varepsilon }(f)}}{\varphi ^{BE}_{\varepsilon }({\mathcal {M}}_{\varepsilon }^{BE,f})}\right) \textrm{d}v, \end{aligned}$$

so that

$$\begin{aligned} \int _{{\mathbb {R}}^3} \int _{\varphi ^{BE}_{\varepsilon }({\mathcal {M}}_{\varepsilon }^{BE,f}) }^{M^{\varphi ^{BE}_{\varepsilon }(f)} } \frac{\varphi ^{BE}_{\varepsilon }(f) - y}{y} \, \textrm{d}y \, \textrm{d}v = \int _{{\mathbb {R}}^3} \int _{\varphi ^{BE}_{\varepsilon }({\mathcal {M}}_{\varepsilon }^{BE,f}) }^{M^{\varphi ^{BE}_{\varepsilon }(f)} } \frac{M^{\varphi ^{BE}_{\varepsilon }(f)} - y}{y} \, \textrm{d}y \, \textrm{d}v \ge 0, \end{aligned}$$

ending the proof. $\square$

Proposition 13

Bose–Einstein CKP inequality. Let $\varpi : {\mathbb {R}}^3 \rightarrow {\mathbb {R}}_+$ be measurable, and $r \in [1,2]$. We recall the definition (88) of the function $\Lambda$,

$$\begin{aligned} \Lambda (\lambda ) := {\left\{ \begin{array}{ll} \qquad \quad 2 &{}\text { if } \lambda = 1,\\ \displaystyle \frac{(\lambda -1)^2}{\lambda \log \lambda - \lambda + 1} \qquad &{}\text { if } \lambda \in {\mathbb {R}}_+ \setminus \{1\}. \end{array}\right. } \end{aligned}$$

Then for any $\varepsilon > 0$ and $0\le f \in L^1_2({\mathbb {R}}^3) \cap L \log L ({\mathbb {R}}^3) \setminus \{0\}$ which density and temperature satisfy (101), assuming that the norms below are finite,

$$\begin{aligned} \left\| \left( f-{\mathcal {M}}\right) \, \varpi \right\| ^2_{L^r} \le \left\| {\mathcal {M}}(1 + \varepsilon {\mathcal {M}}) \, \varpi ^2 \right\| _{L^{\frac{r}{2-r}}} \Lambda \left( \frac{\left\| f(1 + \varepsilon f) \, \varpi ^2 \right\| _{L^{\frac{r}{2-r}}}}{\left\| {\mathcal {M}}(1 + \varepsilon {\mathcal {M}}) \, \varpi ^2 \right\| _{L^{\frac{r}{2-r}}}} \right) H^{BE}_{\varepsilon } \left[ f \left| {\mathcal {M}}\right. \right] , \end{aligned}$$

(104)

where we denoted ${\mathcal {M}}\equiv {\mathcal {M}}_{\varepsilon }^{BE,f}$ the $\varepsilon$-Bose–Einstein distribution associated to f, and $H^{BE}_{\varepsilon }$ is the Bose–Einstein entropy. When $r=2$, $L^{\frac{r}{2-r}}$ shall be understood as $L^{\infty }$.

Proof

The proof is similar to the one of Proposition 10. Let $\varepsilon > 0$.

We apply Corollary 17 with $\Phi (x) \equiv \Phi ^{BE}_{\varepsilon }(x) = \int _0^x \log \frac{y}{1 + \varepsilon y} \, \textrm{d}y$, $J = {\mathbb {R}}_+^*$ and

$$\begin{aligned} {\mathcal {F}}= \left\{ 0 \le g \in L^1_2({\mathbb {R}}^3) \left| \; \int _{{\mathbb {R}}^3} g(v) \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} \textrm{d}v = \begin{pmatrix} \rho \\ \rho \, u \\ 3 \, \rho T + \rho \, |u|^2 \end{pmatrix}\right. \right\} , \end{aligned}$$

with $\rho , T$ and $\varepsilon$ such that (101) is satisfied, ensuring the existence of ${\mathcal {M}}_{\varepsilon }^{BE,f}$. In the rest of this proof, we denote for the sake of clarity ${\mathcal {M}}\equiv {\mathcal {M}}_{\varepsilon }^{BE,f}$. Then (114) writes, since $H_{\Phi ^{BE}_{\varepsilon }} \equiv H^{BE}_{\varepsilon }$ the $\varepsilon$-Bose–Einstein entropy, and $\displaystyle \frac{1}{{\Phi ^{BE}_{\varepsilon }}''}(x) = x(1+\varepsilon x)$ is convex,

$$\begin{aligned}{} & {} \left\| (f-{\mathcal {M}}) \, \varpi \right\| _{L^r}^2 \\{} & {} \quad \le \left( \int _0^1 (1-\tau ) \, \left\| ((1-\tau ){\mathcal {M}}(1 + \varepsilon {\mathcal {M}}) + \tau f(1 + \varepsilon f)) \, \varpi ^2 \right\| _{L^{\frac{r}{2-r}}}^{-1} \, \textrm{d}\tau \right) ^{-1} \, H^{BE}_{\varepsilon }[f|{\mathcal {M}}]. \end{aligned}$$

We focus on the term with the integral in $\tau$. From Minkowski’s inequality, it is smaller than

$$\begin{aligned}&\left( \int _0^1 \frac{1-\tau }{ (1-\tau )\left\| {\mathcal {M}}(1 + \varepsilon {\mathcal {M}}) \, \varpi ^2 \right\| _{L^{\frac{r}{2-r}}} + \tau \left\| f(1 + \varepsilon f) \, \varpi ^2 \right\| _{L^{\frac{r}{2-r}}}} \, \textrm{d}\tau \right) ^{-1} \\ = \;&\left\| {\mathcal {M}}(1 + \varepsilon {\mathcal {M}}) \, \varpi ^2 \right\| _{L^{\frac{r}{2-r}}} \, \Lambda \left( \frac{\left\| f(1 + \varepsilon f) \, \varpi ^2 \right\| _{L^{\frac{r}{2-r}}}}{\left\| {\mathcal {M}}(1 + \varepsilon {\mathcal {M}}) \, \varpi ^2 \right\| _{L^{\frac{r}{2-r}}}}\right) , \end{aligned}$$

where $\Lambda$ is defined in (88) and appears thanks to a Taylor expansion of $\lambda \mapsto \lambda \log \lambda - \lambda$ around 1 like in the proof of Proposition 10, allowing to conclude. $\square$

From the previous proposition, we easily deduce the following standard inequalities.

Corollary 14

Standard Bose–Einstein CKP inequalities. For any $\varepsilon > 0$, $\alpha \ge 0$ and $0\le f \in L^1_2({\mathbb {R}}^3) \cap L \log L ({\mathbb {R}}^3) \setminus \{0\}$ satisfying (101),

$$\begin{aligned}&\Vert ({\mathcal {M}}-f)_+ \Vert ^2_{L^1_{\alpha }} \le 2 \left( \left\| {\mathcal {M}}\right\| _{L^{1}_{2 \alpha }} + \varepsilon \left\| {\mathcal {M}}\right\| ^2_{L^{2}_{\alpha }} \right) \, \times H^{BE}_{\varepsilon }[f|{\mathcal {M}}], \end{aligned}$$

(105)

$$\begin{aligned}&\Vert f-{\mathcal {M}}\Vert ^2_{L^1} \qquad \le 8 \, \left( \left\| {\mathcal {M}}\right\| _{L^{1}} + \varepsilon \Vert {\mathcal {M}}\Vert _{L^2}^2 \right) \quad \times H^{BE}_{\varepsilon }[f|{\mathcal {M}}], \end{aligned}$$

(106)

$$\begin{aligned}&\Vert f-{\mathcal {M}}\Vert ^2_{L^1_2} \qquad \le 8 \, \left( \left\| {\mathcal {M}}\right\| _{L^{1}_{4}} + \varepsilon \left\| {\mathcal {M}}\right\| ^2_{L^{2}_{2}} \right) \, \times H^{BE}_{\varepsilon }[f|{\mathcal {M}}], \end{aligned}$$

(107)

where we denoted for clarity ${\mathcal {M}}\equiv {\mathcal {M}}_{\varepsilon }^{BE,f}$ the $\varepsilon$-Bose–Einstein distribution associated to f.

We prove the above inequalities similarly as we did for Corollary 11. For (105) we apply Proposition 13 with $r = 1$ and $\varpi (v) = (1+|v|^2)^{\frac{\alpha }{2}} \, \textbf{1}_{f \le {\mathcal {M}}}$ and notice that $\Lambda \le 2$ on [0, 1]. We then obtain (106)–(107) by decomposing $|f-{\mathcal {M}}| = f - {\mathcal {M}}+ 2({\mathcal {M}}-f)_+$, using the fact that f and ${\mathcal {M}}$ share the same normalization in $v\mapsto 1$, $v\mapsto v$ and $v\mapsto |v|^2$, and (105) with respectively $\alpha = 0$ and $\alpha =2$.

Appendix B A General Discussion About Entropies and Equilibria

In this section we intend to provide general considerations on the entropy, which are much more general than the scope of this paper, but give a good understanding of the notions we used, and could also be helpful in the study of weak turbulence, where various kinds of unusual entropies can emerge (see [7]). Our setting is laid down quite generally. Consider a measured space $({\mathcal {E}},{\mathcal {A}},\mu )$, an open interval $J \subset {\mathbb {R}}$ which closure we denote by $\bar{J}$, and a function $\Phi \in {\mathcal {C}}^2(J) \cap {\mathcal {C}}^0(\bar{J})$ such that $\Phi '' > 0$ on J. Remark that $\Phi '$ is then a ${\mathcal {C}}^1$-diffeomorphism from J onto $\Phi '(J)$.

Entropy. We define the $\Phi$-entropy, for any (${\mathcal {A}}$, Bor$(\bar{J})$)-measurable $f : {\mathcal {E}}\rightarrow \bar{J}$ such that the following integral makes sense and is finite, by

$$\begin{aligned} H_{\Phi }(f) := \int _{{\mathcal {E}}} \Phi (f(\zeta )) \, \textrm{d}\mu (\zeta ), \end{aligned}$$

(108)

and we denote by $E_{\Phi }$ the set of such f. We let the relative $\Phi$-entropy of f and g to be

$$\begin{aligned} H_{\Phi }[f|g] = H_{\Phi }(f) - H_{\Phi }(g). \end{aligned}$$

(109)

We also define, for (${\mathcal {A}}$, Bor$(\bar{J})$)-measurable $f,g : {\mathcal {E}}\rightarrow \bar{J}$, the $\Phi$-relative-entropy (which in general differs from the relative $\Phi$-entropy) of f and g by

$$\begin{aligned} {\mathcal {H}}_{\Phi }[f|g] := \int _0^1 (1-\tau ) \left( \int _{g \in J} (f-g)^2 \, \Phi ''((1-\tau )g + \tau f) \, \textrm{d}\mu (\zeta ) \right) \textrm{d}\tau . \end{aligned}$$

(110)

Note that ${\mathcal {H}}_{\Phi }[f|g]$ is possibly infinite, but always well-defined, as

$$\begin{aligned} g \in J \implies \; \forall \, \tau \in (0,1), \; \; (1-\tau )g + \tau f \in J, \end{aligned}$$

and that ${\mathcal {H}}_{\Phi }[f|g]$ is always nonnegative. The following Proposition 15 gives a quite simple but general result linking entropies, conserved quantities and equilibrium distributions, under a sole existence assumption.

Proposition 15

Let I be a countable set, $(\phi _i)_{i \in I}$ a family of measurable real functions, $(\omega _i)_{i \in I}$ a family of real numbers, and

$$\begin{aligned} {\mathcal {F}}= \left\{ f \in E_{\Phi } \text { s.t. } \forall \, i \in I, \; \; \int _{{\mathcal {E}}} |f(\zeta )| \, |\phi _i(\zeta )| \, \textrm{d}\mu (\zeta ) < \infty \text { and } \int _{{\mathcal {E}}} f(\zeta ) \, \phi _i(\zeta ) \, \textrm{d}\mu (\zeta ) = \omega _i \right\} . \end{aligned}$$

Assume that $\Phi '(J) = {\mathbb {R}}$, and that there exists $(\alpha _i(\omega )) \in {\mathbb {R}}^I$ such that $M^{{\mathcal {F}}}_{\Phi } \in {\mathcal {F}}$, where

$$\begin{aligned} M^{{\mathcal {F}}}_{\Phi } := (\Phi ')^{-1} \left( \sum _{i \in I}\alpha _i(\omega ) \, \phi _i \right) . \end{aligned}$$

(111)

Then the following four propositions are equivalent. Let $g \in {\mathcal {F}}$.

$$\begin{aligned} (i) \;&g \in J \quad \mu \text {-a.e.} \; \text { and } \; \; \forall \, f \in {\mathcal {F}}, \; H_{\Phi }[f|g] = {\mathcal {H}}_{\Phi }[f|g], \\ (ii) \;&g \in J \quad \mu \text {-a.e.} \; \text { and } \; \; \forall \, f \in {\mathcal {F}}, \; \; {\mathcal {H}}_{\Phi }[f|g] < \infty \; \text { and } \; \; \int _{{\mathcal {E}}} (f-g) \, \Phi '(g) \, \textrm{d}\mu (\zeta ) = 0, \\ (iii) \;&H_{\Phi }(g) = \underset{h \in {\mathcal {F}}}{\min }\ \; H_{\Phi }(h), \\ (iv) \;&g = M^{{\mathcal {F}}}_{\Phi } \quad \mu \text {-a.e.} \end{aligned}$$

In particular, $M^{{\mathcal {F}}}_{\Phi }$ is the unique minimizer of $H_{\Phi }$ under the constraints of the set ${\mathcal {F}}$.

The above proposition actually proves the following (under the assumptions $\Phi '(J) = {\mathbb {R}}$ and of existence of $M^{{\mathcal {F}}}_{\Phi }$). An admissible distribution g is an equilibrium relative to the conserved quantities $\phi _i$, in the sense of the minimization of the $\Phi$-entropy [(iii)], if and only if $\Phi '(g)$ is a linear combination of the functions $\phi _i$ [(iv)], if and only if the relative $\Phi$-entropy between any admissible distribution f and g is given by (110) [(i)], if and only if the quantity $\Phi '(g)$ is conserved amongst all admissible distributions [(ii)] - indeed, (ii) is equivalent, assuming the following integrals make sense, to $\forall \, f_1,f_2 \in {\mathcal {F}}$, $\int _{{\mathcal {E}}} f_1 \, \Phi '(g) \, \textrm{d}\mu (\zeta ) = \int _{{\mathcal {E}}} f_2 \, \Phi '(g) \, \textrm{d}\mu (\zeta )$.

The reader may notice that the Classical case corresponds to the choice $\Phi (x) \equiv \Phi _0(x) = x \log x - x$, for which $(\Phi '_0)^{-1} = \exp$, hence $M^{{\mathcal {F}}}_{\Phi }$ is in this case a Maxwellian, since the conserved quantities (corresponding to the functions $\phi _i$ in the proposition) are $v \mapsto 1$, $v \mapsto v$ and $v \mapsto |v|^2$.

Moreover, the Fermi–Dirac case corresponds to the choice $\Phi (x) \equiv \Phi _{\varepsilon }(x) \equiv \int _0^x \log \frac{y}{1 - \varepsilon y} \, \textrm{d}y$, for which $\displaystyle (\Phi '_{\varepsilon })^{-1}(x) = \frac{e^x}{1 + \varepsilon e^x}$, hence $M^{{\mathcal {F}}}_{\Phi }$ is in this case a Fermi–Dirac distribution, since again, the conserved quantities (corresponding to the functions $\phi _i$ in the proposition) are $v \mapsto 1$, $v \mapsto v$ and $v \mapsto |v|^2$. Proposition 1 then comes as a corollary to Proposition 15 with ${\mathcal {E}}= {\mathbb {R}}^3$ endowed with the Lebesgue measure, $J = (0,\varepsilon ^{-1})$, $\displaystyle \Phi (x) \equiv \Phi _{\varepsilon }(x) \equiv \int _0^x \log \varphi _{\varepsilon }(y) \, \textrm{d}y$, $M^{{\mathcal {F}}}_{\Phi } \equiv {{\mathcal {M}}_{\varepsilon }^f}$ and

$$\begin{aligned} {\mathcal {F}}= \left\{ 0 \le f \in L^1_2({\mathbb {R}}^3) \left| \; 1 - \varepsilon f \ge 0, \quad \int _{{\mathbb {R}}^3} f(v) \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} \textrm{d}v = \begin{pmatrix} \rho \\ \rho \, u \\ 3 \, \rho T + \rho \, |u|^2 \end{pmatrix}\right. \right\} , \end{aligned}$$

with $\rho , T$ and $\varepsilon$ such that $\displaystyle \gamma > \frac{2}{5}$ (ensuring the existence of ${{\mathcal {M}}_{\varepsilon }^f}$, as proven in [22]).

Remark 6

The Bose–Einstein case is also recovered with $\displaystyle \Phi (x) \equiv \Phi ^{BE}_{\varepsilon }(x) \equiv \int _0^x \log \frac{y}{1 + \varepsilon y} \, \textrm{d}y$, for which $\displaystyle (\Phi '_{-\varepsilon })^{-1}(x) = \frac{e^x}{1 - \varepsilon e^x}$, hence $M^{{\mathcal {F}}}_{\Phi }$ is in this case a Bose–Einstein distribution (when it exists).

Proof

We start by proving $(i) \iff (ii)$. Let $f,g \in {\mathcal {F}}$ such that $g \in J$ $\mu$-almost everywhere and ${\mathcal {H}}_{\Phi }[f|g]~<~\infty$. Using a Taylor expansion followed by Fubini’s Theorem, we get

$$\begin{aligned} H_{\Phi }[f|g]&= \int _{{\mathcal {E}}} (\Phi (f) - \Phi (g)) \, \textrm{d}\mu (\zeta ) = \int _{g \in J} (\Phi (f) - \Phi (g)) \, \textrm{d}\mu (\zeta ) \\&= \int _{g \in J} \left( (f-g) \Phi '(g) + (f-g)^2 \int _0^1 (1-\tau ) \Phi ''((1-\tau )g + \tau f) \, \textrm{d}\tau \right) \textrm{d}\mu (\zeta ) \\&= \int _{g \in J} (f-g) \Phi '(g) \, \textrm{d}\mu (\zeta ) + {\mathcal {H}}_{\Phi }[f|g], \end{aligned}$$

thus proving the announced equivalence. Remark that in the last equality we used the fact that ${\mathcal {H}}_{\Phi }[f|g] < \infty$ to ensure that $\int _{g \in J} (f-g) \Phi '(g) \, \textrm{d}\mu (\zeta )$ is well-defined. We now show $(iv) \implies (ii)$. Since $\textrm{Im}({\Phi '}^{-1}) = J$, we do have $M^{{\mathcal {F}}}_{\Phi } \in J$ $\mu$-almost everywhere, and

$$\begin{aligned} \int _{M^{{\mathcal {F}}}_{\Phi } \in J} (f-M^{{\mathcal {F}}}_{\Phi }) \, \Phi '(M^{{\mathcal {F}}}_{\Phi }) \, \textrm{d}\mu (\zeta )&= \int _{{\mathcal {E}}}(f-M^{{\mathcal {F}}}_{\Phi }) \sum _{i} \alpha _i(\omega _i) \, \phi _i \, \textrm{d}\mu (\zeta ) \\ {}&= \sum _i \alpha _i(\omega _i) \left( \int _{{\mathcal {E}}} f \, \phi _i \, \textrm{d}\mu (\zeta ) - \int _{{\mathcal {E}}} M^{{\mathcal {F}}}_{\Phi } \, \phi _i \, \textrm{d}\mu (\zeta ) \right) = 0, \end{aligned}$$

where the last equality comes from the fact that both f and $M^{{\mathcal {F}}}_{\Phi }$ belong to ${\mathcal {F}}$. We now focus on $(ii) \implies (iv)$. Assume the existence of $g \in {\mathcal {F}}$ such that $g \in J$ $\mu$-almost everywhere and

$$\begin{aligned} \forall \, f \in {\mathcal {F}}, \quad \int _{g \in J} (f-g) \, \Phi '(g) \, \textrm{d}\mu (\zeta ) = 0. \end{aligned}$$

Since we also have $M^{{\mathcal {F}}}_{\Phi } \in {\mathcal {F}}$, then $M^{{\mathcal {F}}}_{\Phi } \in J$ $\mu$-almost everywhere and we just proved that

$$\begin{aligned} \int _{M^{{\mathcal {F}}}_{\Phi } \in J} (g-M^{{\mathcal {F}}}_{\Phi }) \, \Phi ' (M^{{\mathcal {F}}}_{\Phi }) \, \textrm{d}\mu (\zeta ) = 0, \end{aligned}$$

allowing to deduce that

$$\begin{aligned} \int _{{\mathcal {E}}}(g-M^{{\mathcal {F}}}_{\Phi })(\Phi '(g) - \Phi '(M^{{\mathcal {F}}}_{\Phi })) \, \textrm{d}\mu (\zeta ) = 0. \end{aligned}$$

Since $\Phi '$ is increasing, this implies that $g = M^{{\mathcal {F}}}_{\Phi }$ $\mu$-almost everywhere. We now focus on $(iv)~\implies ~(iii)$. Since we already proved $(iv) \implies (ii) \implies (i)$, we have for any $f \in {\mathcal {F}}$,

$$\begin{aligned} H_{\Phi }[f|M^{{\mathcal {F}}}_{\Phi }] = {\mathcal {H}}_{\Phi }[f|M^{{\mathcal {F}}}_{\Phi }] \ge 0, \end{aligned}$$

thus

$$\begin{aligned} H_{\Phi }(f) \ge H_{\Phi }(M^{{\mathcal {F}}}_{\Phi }). \end{aligned}$$

Finally, we prove $(iii) \implies (iv)$. Assume $H_{\Phi }(g) = \underset{h \in {\mathcal {F}}}{\min }\ \; H_{\Phi }(h)$. We just proved that $H_{\Phi }(M^{{\mathcal {F}}}_{\Phi }) = \underset{h \in {\mathcal {F}}}{\min }\ \; H_{\Phi }(h)$, hence $H_{\Phi }(g) = H_{\Phi }(M^{{\mathcal {F}}}_{\Phi })$ and $H_{\Phi }[g|M^{{\mathcal {F}}}_{\Phi }] = 0$. Since we also proved $(iv) \implies (i)$, we know that

$$\begin{aligned} {\mathcal {H}}_{\Phi }[g|M^{{\mathcal {F}}}_{\Phi }] = H_{\Phi }[g|M^{{\mathcal {F}}}_{\Phi }], \end{aligned}$$

hence ${\mathcal {H}}_{\Phi }[g|M^{{\mathcal {F}}}_{\Phi }] = 0$, that is

$$\begin{aligned} \int _0^1 (1-\tau ) \left( \int _{M^{{\mathcal {F}}}_{\Phi } \in J} (g-M^{{\mathcal {F}}}_{\Phi })^2 \, \Phi ''((1-\tau )M + \tau g) \, \textrm{d}\mu (\zeta ) \right) \textrm{d}\tau = 0. \end{aligned}$$

This implies, since $M^{{\mathcal {F}}}_{\Phi } \in J$ and $(1-\tau )M^{{\mathcal {F}}}_{\Phi } + \tau g \in J$ $\mu$-almost everywhere for any $\tau \in (0,1)$, and $\Phi ''>0$ on J, that $g=M^{{\mathcal {F}}}_{\Phi }$ $\mu$ almost everywhere. $\square$

1.1 Appendix B.1 General Csiszár–Kullback–Pinsker Inequality

The famous Csiszár–Kullback–Pinsker inequality, linking the squared $L^1$ distance of two probabilities with their relative Classical entropy can in fact be generalized to the whole family of $\Phi$-entropies, and for weighted $L^r$, $1 \le r \le 2$ distances, by the following Proposition 16, and more specifically, in the context of convergence towards equilibrium, by Corollary 17. Again, such inequalities may be useful in the study of weak turbulence [7].

Proposition 16

Let some $({\mathcal {A}}$, $\textrm{Bor}(\bar{J}))$-measurable $\varpi : {\mathcal {E}}\rightarrow \bar{J}$ and $r \in [1,2]$. Then for any $({\mathcal {A}}$, $\textrm{Bor}(\bar{J}))$-measurable $f,g : {\mathcal {E}}\rightarrow \bar{J}$ such that all terms below are finite, we have

$$\begin{aligned} \left\| (f-g) \, \varpi \right\| _{L^r(g \in J)}^2 \le \left( \int _0^1 (1-\tau ) \, \left\| \frac{ \varpi ^2}{\Phi ''((1-\tau )g + \tau f) } \right\| _{L^{\frac{r}{2-r}}(g \in J)}^{-1} \, \textrm{d}\tau \right) ^{-1} \, {\mathcal {H}}_{\Phi }[f|g], \end{aligned}$$

(112)

where ${\mathcal {H}}_{\Phi }$ is the $\Phi$-relative-entropy defined in (110).

Proof

Let us recall the definition of ${\mathcal {H}}_{\Phi }[f|g]$, that is

$$\begin{aligned} {\mathcal {H}}_{\Phi }[f|g] = \int _0^1 (1-\tau ) \left( \int _{g \in J} (f-g)^2 \, \Phi ''((1-\tau )g + \tau f) \, \textrm{d}\mu (\zeta ) \right) \textrm{d}\tau . \end{aligned}$$

We fix $\tau \in (0,1)$. Let $p = \frac{2}{r}\in [1,2]$ and $q = \frac{2}{2-r} \in [2,+\infty ]$, so that $\displaystyle \frac{1}{p} + \frac{1}{q} = 1$. By Hölder’s inequality, we have

$$\begin{aligned}&\int _{g \in J} |f-g|^{\frac{2}{p}} \, \varpi ^{\frac{2}{p}} \, \textrm{d}\mu (\zeta ) \\&\quad = \int _{g \in J} |f-g|^{\frac{2}{p}}\Phi ''((1-\tau )g + \tau f)^{\frac{1}{p}} \, \left[ \frac{\varpi ^2}{\Phi ''((1-\tau )g + \tau f)} \right] ^{\frac{1}{p}} \, \textrm{d}\mu (\zeta ) \\&\quad \le \left( \int _{g \in J} (f-g)^2 \, \Phi ''((1-\tau )g + \tau f) \, \textrm{d}\mu (\zeta ) \right) ^{\frac{1}{p}} \, \left\| \left[ \frac{\varpi ^2}{\Phi ''((1-\tau )g + \tau f)} \right] ^{\frac{1}{p}} \right\| _{L^q(g \in J)}. \end{aligned}$$

Raising the above inequality to the power p, we obtain

$$\begin{aligned}{} & {} \left( \int _{g \in J} |f-g|^{\frac{2}{p}} \, \varpi ^{\frac{2}{p}} \, \textrm{d}\mu (\zeta ) \right) ^p \nonumber \\{} & {} \quad \le \left( \int _{g \in J} (f-g)^2 \, \Phi ''((1-\tau )g + \tau f) \, \textrm{d}\mu (\zeta ) \right) \, \left\| \frac{\varpi ^2}{\Phi ''((1-\tau )g + \tau f)} \right\| _{L^{\frac{q}{p}}(g \in J)},\nonumber \\ \end{aligned}$$

(113)

where we used the fact that $\Vert \cdot ^{\frac{1}{p}} \Vert ^p_{L^q} = \Vert \cdot \Vert _{L^{\frac{q}{p}}}$. Since $\frac{2}{p} = r$ and $\frac{q}{p} = \frac{r}{2-r}$, (113) actually writes

$$\begin{aligned}{} & {} \left\| (f-g) \, \varpi \right\| _{L^r(g \in J)}^2 \\{} & {} \quad \le \left( \int _{g \in J} (f-g)^2 \, \Phi ''((1-\tau )g + \tau f) \, \textrm{d}\mu (\zeta ) \right) \, \left\| \frac{\varpi ^2}{\Phi ''((1-\tau )g + \tau f)} \right\| _{L^{\frac{r}{2-r}}(g \in J)}. \end{aligned}$$

If $\varpi$ is zero $\mu$-almost everywhere on $\{g \in J\}$, then the proposition is trivial. Else, since $\Phi '' > 0$ on $\{g \in J \}$ and $(1-\tau )g + \tau f \in J$ on $\{g \in J\}$ for any $\tau \in (0,1)$, we know that $\left\| \Phi ''((1-\tau )g + \tau f) \, \varpi ^2 \right\| _{L^{\frac{r}{2-r}}(g \in J)} > 0$. Since we assumed that the quantity

$$\begin{aligned} \left( \int _0^1 (1-\tau ) \, \left\| \frac{ \varpi ^2}{\Phi ''((1-\tau )g + \tau f) } \right\| _{L^{\frac{r}{2-r}}(g \in J)}^{-1} \, \textrm{d}\tau \right) ^{-1} \end{aligned}$$

is finite, we also know that for almost every $\tau \in (0,1)$ we have $\displaystyle \left\| \frac{\varpi ^2}{\Phi ''((1-\tau )g + \tau f)} \right\| _{L^{\frac{r}{2-r}}(g \in J)}$ $< \infty$. For these values of $\tau$, we then have

$$\begin{aligned}{} & {} \left\| (f-g) \, \varpi \right\| _{L^r(g \in J)}^2 \; \frac{1-\tau }{\left\| \frac{\varpi ^2}{\Phi ''((1-\tau )g + \tau f)} \right\| _{L^{\frac{r}{2-r}}(g \in J)}}\\{} & {} \quad \le (1 - \tau ) \left( \int _{g \in J} (f-g)^2 \, \Phi ''((1-\tau )g + \tau f) \, \textrm{d}\mu (\zeta ) \right) . \end{aligned}$$

Integrating in $\tau$ yields

$$\begin{aligned} \left\| (f-g) \, \varpi \right\| _{L^r(g \in J)}^2 \left( \int _0^1 (1-\tau ) \, \left\| \frac{ \varpi ^2}{\Phi ''((1-\tau )g + \tau f) } \right\| _{L^{\frac{r}{2-r}}(g \in J)}^{-1} \, \textrm{d}\tau \right) \le {\mathcal {H}}_{\Phi }[f|g]. \end{aligned}$$

Since, again by hypothesis, the integral in $\tau$ is nonzero (its inverse is finite), we obtain (112). $\square$

Remarking that $\{M_{\Phi }^{{\mathcal {F}}} \in J\} = {\mathcal {E}}$ and that for any $f\in {\mathcal {F}}$, ${\mathcal {H}}_{\Phi }[f|M_{\Phi }^{{\mathcal {F}}}] = H_{\Phi }[f|M_{\Phi }^{{\mathcal {F}}}]$ (see Proposition 15), we straightforwardly obtain the following corollary.

Corollary 17

Let some $({\mathcal {A}}$, $\textrm{Bor}(\bar{J}))$-measurable $\varpi : {\mathcal {E}}\rightarrow \bar{J}$ and $r \in [1,2]$. With the same notations as in Proposition 15, assuming $M_{\Phi }^{{\mathcal {F}}}$ exists, we have for any $f \in {\mathcal {F}}$ such that the integral term below is finite,

$$\begin{aligned} \left\| (f-M_{\Phi }^{{\mathcal {F}}}) \, \varpi \right\| _{L^r}^2 \le \left( \int _0^1 (1-\tau ) \, \left\| \frac{ \varpi ^2}{\Phi ''((1-\tau )M_{\Phi }^{{\mathcal {F}}} + \tau f) } \right\| _{L^{\frac{r}{2-r}}}^{-1} \, \textrm{d}\tau \right) ^{-1} \, H_{\Phi }[f|M_{\Phi }^{{\mathcal {F}}}], \end{aligned}$$

(114)

where $H_{\Phi }$ is defined in (108)–(109) and $M_{\Phi }^{{\mathcal {F}}}$ is the equilibrium associated to $\Phi$ and the set ${\mathcal {F}}$, defined in (111).

Appendix C Technical Results

In this section, we consider ${{\mathcal {M}}_{\varepsilon }^f}$, the Fermi–Dirac distribution associated to some $\varepsilon >0$ and $0 \le f \in L^1_2({\mathbb {R}}^3)$ such that $1 - \varepsilon f \ge 0$. The existence of ${{\mathcal {M}}_{\varepsilon }^f}$ is provided by assuming $\displaystyle \gamma > \frac{2}{5}$ (see [22]), where we recall the notation

$$\begin{aligned} \gamma := \frac{T}{T_F(\rho , \varepsilon )}, \end{aligned}$$

(115)

where $\displaystyle T_F(\rho , \varepsilon ) = \frac{1}{2} \left( \frac{3 \rho \varepsilon }{4 \pi } \right) ^{2/3}$ is the Fermi temperature associated to $\rho$ and $\varepsilon$; and $\rho , T$ are respectively the density and temperature associated to the distribution f, defined in (12). We also recall the notation, for $x \in [0,\varepsilon ^{-1})$,

$$\begin{aligned} \displaystyle \varphi _{\varepsilon }(x) = \frac{x}{1- \varepsilon x}. \end{aligned}$$

1.1 Appendix C.1 $L^\infty$ bound for the Fermi–Dirac statistics

In this subsection, we provide an $L^\infty$ bound on the Fermi–Dirac statistics. The following result is very similar to [2, Lemma A.1].

Proposition 18

Let $\varepsilon >0$ and $f \in L^1_2({\mathbb {R}}^3)$ be a nonnegative distribution such that $1 - \varepsilon f \ge 0$ and $\displaystyle \gamma \ge \gamma ^{\dag }$, where $\gamma$ is given by (115) and

$$\begin{aligned} \gamma ^{\dag } := \left( \frac{4}{\pi }\right) ^{\frac{1}{3}} \left( \frac{5}{3} \right) ^{\frac{5}{3}}. \end{aligned}$$

(116)

Then the quantity $\varepsilon \Vert \varphi _{\varepsilon }({{\mathcal {M}}_{\varepsilon }^f}) \Vert _{\infty }$ satisfies

$$\begin{aligned} \varepsilon \Vert \varphi _{\varepsilon }({{\mathcal {M}}_{\varepsilon }^f}) \Vert _{\infty } \le \frac{2}{3} \left( \frac{\gamma }{\gamma ^{\dag }} \right) ^{-3/2}. \end{aligned}$$

(117)

Proof

Our proof is based on [22, proof of Proposition 3]. We introduce, for $s \ge 0$,

$$\begin{aligned} I_s(t) := \int _0^{\infty } \frac{r^s}{1 + t e^{r^2}} \, \textrm{d}r, \qquad P(t) := I_4(t) [I_2(t)]^{-5/3}, \quad t>0. \end{aligned}$$

It is proven in [22, proof of Proposition 3] that P is continuous and increasing on ${\mathbb {R}}_+$, and that

$$\begin{aligned} P \left( \frac{1}{\varepsilon \Vert \varphi _{\varepsilon }({{\mathcal {M}}_{\varepsilon }^f}) \Vert _{\infty }} \right) = 3 \rho ^{-2/3} \, T \, (4 \pi \varepsilon ^{-1})^{2/3} \equiv \frac{3^{5/3}}{2} \; \gamma . \end{aligned}$$

Let $\alpha > 0$ and

$$\begin{aligned} t = \alpha ^{-1} \times \frac{3 \sqrt{\pi }}{4} \, \gamma ^{3/2}. \end{aligned}$$

As, for any $r \ge 0$, it holds that

$$\begin{aligned} \frac{e^{-r^2}}{1+t} \le \frac{1}{1+t \, e^{r^2}} \le \frac{e^{-r^2}}{t}, \end{aligned}$$

we have

$$\begin{aligned}{} & {} P(t) \le \left( \frac{1}{t} \int _0^{\infty } r^4 \, e^{-r^2} \, \textrm{d}r \right) \left( \frac{1}{1+t} \int _0^{\infty } r^2 \, e^{-r^2} \, \textrm{d}r \right) ^{-5/3} \\{} & {} \quad = \frac{(1+t)^{5/3}}{t} \, \left( \frac{1}{2} \Gamma \left( \frac{5}{2} \right) \right) \left( \frac{1}{2} \Gamma \left( \frac{3}{2} \right) \right) ^{-5/3}, \end{aligned}$$

that is

$$\begin{aligned} P(t) \le \frac{(1+t)^{5/3}}{t} \times \frac{3 \times 2^{1/3}}{ \pi ^{1/3}}. \end{aligned}$$

We define

$$\begin{aligned} \gamma ^{\alpha } := \left( \frac{4}{3 \sqrt{\pi }} \times \frac{\alpha }{\alpha ^{2/5}-1} \right) ^{2/3}. \end{aligned}$$

Then, whenever $\gamma \ge \gamma ^{\alpha }$, we have $\displaystyle t \ge \frac{1}{\alpha ^{2/5} - 1}$, so that $1 + t \le \alpha ^{2/5} \, t$, implying

$$\begin{aligned}&P(t) \le t^{2/3} \times \frac{3 \times 2^{1/3} \, \alpha ^{2/3}}{\pi ^{1/3}} = \alpha ^{-2/3} \times \left( \frac{3 \sqrt{\pi }}{4} \right) ^{2/3} \gamma \times \frac{3 \times 2^{1/3} \, \alpha ^{2/3}}{\pi ^{1/3}} \\&\quad = \frac{3^{5/3}}{2} \, \gamma = P \left( \frac{1}{\varepsilon \Vert \varphi _{\varepsilon }({{\mathcal {M}}_{\varepsilon }^f}) \Vert _{\infty }} \right) . \end{aligned}$$

Since P is increasing, we deduce that, whenever $\gamma \ge \gamma ^{\alpha }$, we have

$$\begin{aligned} \alpha ^{-1} \times \frac{3 \sqrt{\pi }}{4} \, \gamma ^{3/2} = t \le \frac{1}{\varepsilon \Vert \varphi _{\varepsilon }({{\mathcal {M}}_{\varepsilon }^f}) \Vert _{\infty }}, \end{aligned}$$

that is

$$\begin{aligned} \varepsilon \Vert \varphi _{\varepsilon }({{\mathcal {M}}_{\varepsilon }^f}) \Vert _{\infty } \le \alpha \times \frac{4}{3 \sqrt{\pi }} \gamma ^{-3/2}. \end{aligned}$$

(118)

By computing the derivative of $\displaystyle \alpha \mapsto \frac{\alpha }{\alpha ^{2/5}-1}$, we can minimize $\alpha \mapsto \gamma ^{\alpha }$ and find that the minimum value is

$$\begin{aligned} \gamma ^{\dag } = \left( \frac{4}{\pi }\right) ^{\frac{1}{3}} \left( \frac{5}{3} \right) ^{\frac{5}{3}}, \end{aligned}$$

reached for $\alpha ^{\dag } = \left( \frac{5}{3} \right) ^{5/2}$, and, combined with (118), this proves (117). $\square$

1.2 Appendix C.2 Regularity in $\varepsilon$ of the Coefficients of the Fermi–Dirac Statistics

In this last subsection, for any $\varepsilon \ge 0$ and $0 \le g \in L^1_2({\mathbb {R}}^3) \cap L \log L ({\mathbb {R}}^3)$, we denote by ${\mathcal {M}}_{\varepsilon } \equiv {\mathcal {M}}_{\varepsilon }^{\varphi _{\varepsilon }^{-1}(g)}$ the $\varepsilon$-Fermi distribution associated to $\varphi _{\varepsilon }^{-1}(g)$. In particular, in the limit case $\varepsilon = 0$, ${\mathcal {M}}_0$ is the Maxwellian distribution associated to g. We also denote $a_{\varepsilon }, b_{\varepsilon }, \bar{u}_{\varepsilon }$ and $\rho _{\varepsilon }, u_{\varepsilon },T_{\varepsilon }$ the quantities such that, letting

$$\begin{aligned} {\mathcal {M}}_{\varepsilon } \equiv {\mathcal {M}}_{\varepsilon }^{\frac{g}{1 + \varepsilon g}}, \qquad M_{\varepsilon } = \frac{{\mathcal {M}}_{\varepsilon }}{1 - \varepsilon {\mathcal {M}}_{\varepsilon }}, \end{aligned}$$

(119)

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

$$\begin{aligned} M_{\varepsilon }(v) = \exp \left( a_{\varepsilon } + b_{\varepsilon } |v-\bar{u}_{\varepsilon }|^2 \right) , \end{aligned}$$

(120)

and

$$\begin{aligned} \int _{{\mathbb {R}}^3} \frac{g}{1 + \varepsilon g} \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} \textrm{d}v = \begin{pmatrix} \rho _{\varepsilon } \\ \rho _{\varepsilon } u_{\varepsilon } \\ 3 \rho _{\varepsilon } T_{\varepsilon } + \rho _{\varepsilon } |u_{\varepsilon }|^2 \end{pmatrix}. \end{aligned}$$

(121)

Lemma 19

Let $0 \le g \in L^1_2({\mathbb {R}}^3) \cap L \log L({\mathbb {R}}^3)$. Using the notation (121), the application $\varepsilon \mapsto (\rho _{\varepsilon },u_{\varepsilon },T_{\varepsilon })$ is continuous on ${\mathbb {R}}_+$ and ${\mathcal {C}}^1$ on ${\mathbb {R}}_+^*$.

Proof

The continuity of $\varepsilon \mapsto (\rho _{\varepsilon },u_{\varepsilon },T_{\varepsilon })$ on ${\mathbb {R}}_+$ comes by dominated convergence, as it holds for any $\varepsilon \ge 0$ and $v \in {\mathbb {R}}^3$ that

$$\begin{aligned} \frac{g}{1 + \varepsilon g} \, (1 + |v|^2) \le g \, (1 + |v|^2), \end{aligned}$$

and by hypothesis $0 \le g \in L^1_2({\mathbb {R}}^3)$. Similarly, for any $\varepsilon > 0$ and $v \in {\mathbb {R}}^3$ we have

$$\begin{aligned} \left| \partial _{\varepsilon } \frac{g}{1 + \varepsilon g}\right| \, (1 + |v|^2) = \left( \frac{g}{1 + \varepsilon g}\right) ^2 \, (1 + |v|^2) \le \varepsilon ^{-1} \, g \, (1 + |v|^2). \end{aligned}$$

Therefore, for any $\varepsilon > 0$, we have

$$\begin{aligned} \sup _{\varepsilon _* \in (\varepsilon /2,3\varepsilon /2)}\left| \partial _{\varepsilon _*} \frac{g}{1 + \varepsilon _* g}\right| \, (1 + |v|^2) \le 2 \varepsilon ^{-1} \, g \, (1 + |v|^2). \end{aligned}$$

The differentiability of $\varepsilon \mapsto (\rho _{\varepsilon },u_{\varepsilon },T_{\varepsilon })$ on ${\mathbb {R}}_+^*$ then comes by dominated convergence, as $g \in L^1_2({\mathbb {R}}^3)$. $\square$

The following lemmas provide the continuity of $\varepsilon \mapsto (a_{\varepsilon }, b_{\varepsilon }, \bar{u}_{\varepsilon })$ at the point $\varepsilon = 0$ and its differentiability on ${\mathbb {R}}_+^*$.

Lemma 20

Let $0 \le g \in L^1_2({\mathbb {R}}^3) \cap L \log L({\mathbb {R}}^3)$. Using the notations (119)–(120), the application $\varepsilon \mapsto (a_{\varepsilon }, b_{\varepsilon }, \bar{u}_{\varepsilon })$ is continuous at the point $\varepsilon = 0$.

Proof

Using the notations (119) and (121), we have

$$\begin{aligned} \left| \int _{{\mathbb {R}}^3} M_{\varepsilon } \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} \textrm{d}v - \int _{{\mathbb {R}}^3} {\mathcal {M}}_{\varepsilon } \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} \textrm{d}v \right|&= \left| \int _{{\mathbb {R}}^3} \frac{\varepsilon M_{\varepsilon }^2}{1 + \varepsilon M_{\varepsilon }} \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} \textrm{d}v \right| \le \varepsilon \Vert M_{\varepsilon }\Vert _{\infty } \int _{{\mathbb {R}}^3} {\mathcal {M}}_{\varepsilon } \begin{pmatrix} 1 \\ |v| \\ |v|^2 \end{pmatrix} \textrm{d}v \nonumber \\&\le \varepsilon \Vert M_{\varepsilon }\Vert _{\infty } (\rho _{\varepsilon } + 3 \rho _{\varepsilon } T_{\varepsilon } + \rho _{\varepsilon } |u_{\varepsilon }|^2). \end{aligned}$$

(122)

Recall the notation, in this case,

$$\begin{aligned} \gamma _{\varepsilon } = \frac{T_{\varepsilon }}{T_F(\rho _{\varepsilon }, \varepsilon )}, \end{aligned}$$

where $\displaystyle T_F(\rho _{\varepsilon }, \varepsilon ) = \frac{1}{2} \left( \frac{3 \rho _{\varepsilon } \, \varepsilon }{4 \pi } \right) ^{2/3}$. By continuity at the point $\varepsilon = 0$ of the application $\varepsilon \mapsto (\rho _{\varepsilon },T_{\varepsilon })$, given by Lemma 19, we have

$$\begin{aligned} \gamma _{\varepsilon } \xrightarrow {\varepsilon \rightarrow 0} + \infty . \end{aligned}$$

Thereby, there exists $\varepsilon ^* > 0$ such that $\gamma _{\varepsilon } \ge \gamma ^{\dag }$ for any $\varepsilon \in (0,\varepsilon ^*)$, where $\gamma ^{\dag }$ is a universal constant defined in Proposition 18. Then, from (117) in Proposition 18, for any $\varepsilon \in (0,\varepsilon ^*)$,

$$\begin{aligned} \varepsilon \Vert M_{\varepsilon }\Vert _{\infty } \le \frac{2}{3} \left( \frac{\gamma _{\varepsilon }}{\gamma _*}\right) ^{-\frac{3}{2}}, \end{aligned}$$

which vanishes as $\varepsilon \rightarrow 0$, since $\gamma _{\varepsilon }$ tends to $+ \infty$ in this limit. Combining this result, the continuity of $\varepsilon \mapsto (\rho _{\varepsilon },u_{\varepsilon },T_{\varepsilon })$ at $\varepsilon = 0$, given by Lemma 19, and Equation (122), we obtain

$$\begin{aligned} \left| \int _{{\mathbb {R}}^3} M_{\varepsilon } \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} \textrm{d}v - \int _{{\mathbb {R}}^3} {\mathcal {M}}_{\varepsilon } \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} \textrm{d}v \right| \xrightarrow {\varepsilon \rightarrow 0} 0. \end{aligned}$$

The continuity of $\varepsilon \mapsto (\rho _{\varepsilon },u_{\varepsilon },T_{\varepsilon })$ at $\varepsilon = 0$ being equivalent to the statement

$$\begin{aligned} \int _{{\mathbb {R}}^3} {\mathcal {M}}_{\varepsilon } \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} \textrm{d}v \xrightarrow {\varepsilon \rightarrow 0} \int _{{\mathbb {R}}^3} {\mathcal {M}}_0 \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} \textrm{d}v, \end{aligned}$$

we finally conclude that

$$\begin{aligned} \int _{{\mathbb {R}}^3} M_{\varepsilon } \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} \textrm{d}v \xrightarrow {\varepsilon \rightarrow 0} \int _{{\mathbb {R}}^3} {\mathcal {M}}_{0} \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} \textrm{d}v. \end{aligned}$$

Both $M_{\varepsilon }$ and ${\mathcal {M}}_0$ are gaussian distributions, which coefficients are continuously defined by the above moments, allowing to conclude to the continuity of these coefficients at the point $\varepsilon =0$. $\square$

Lemma 21

Let $0 \le g \in L^1_2({\mathbb {R}}^3) \cap L \log L({\mathbb {R}}^3)$. Using the notations (119)–(120), the application $\varepsilon \mapsto (a_{\varepsilon }, b_{\varepsilon }, \bar{u}_{\varepsilon })$ is ${\mathcal {C}}^1$ on ${\mathbb {R}}_+^*$.

Proof

Since the distribution $M_{\varepsilon }(\cdot + \bar{u}_{\varepsilon })$ is radially symmetric, so is ${\mathcal {M}}_{\varepsilon }(\cdot + \bar{u}_{\varepsilon })$, hence $\bar{u}_{\varepsilon } = u_{\varepsilon }$, which, by Lemma 19, is ${\mathcal {C}}^1$ on ${\mathbb {R}}_+^*$. We then define

$$\begin{aligned} g_{\varepsilon } : = g(\cdot + u_{\varepsilon }), \qquad {\mathcal {N}}_{\varepsilon } := {\mathcal {M}}_{\varepsilon }(\cdot + u_{\varepsilon }) \equiv {\mathcal {M}}_{\varepsilon }^{\varphi _{\varepsilon }^{-1}(g_{\varepsilon })} \quad \text {and} \quad N_{\varepsilon } := M_{\varepsilon }(\cdot + u_{\varepsilon }), \end{aligned}$$

so that, for any $\varepsilon > 0$, it holds that

$$\begin{aligned} N_{\varepsilon } = \exp \left( a_{\varepsilon } + b_{\varepsilon } |v|^2 \right) . \end{aligned}$$

As in (121), we let $\rho _{\varepsilon }, T_{\varepsilon } > 0$ be such that

$$\begin{aligned} \int _{{\mathbb {R}}^3} \frac{g_{\varepsilon }}{1 + \varepsilon g_{\varepsilon }} \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} \textrm{d}v = \begin{pmatrix} \rho _{\varepsilon } \\ 0 \\ 3 \rho _{\varepsilon } T_{\varepsilon } \end{pmatrix}. \end{aligned}$$

Let us now show that $\varepsilon \mapsto (a_{\varepsilon }, b_{\varepsilon })$ is ${\mathcal {C}}^1$ on ${\mathbb {R}}_+^*$. It is proven in [22, proof of Proposition 3] that, letting

$$\begin{aligned} I_s(\tau ) := \int _{0}^{\infty } \frac{r^s}{1 + \tau \, e^{r^2}} \, \textrm{d}r, \qquad P(\tau ) := I_4(\tau ) \, I_2(\tau )^{-5/3}, \qquad \tau \in {\mathbb {R}}_+^*, \end{aligned}$$

the function P is an increasing ${\mathcal {C}}^1$ function from ${\mathbb {R}}_+^*$ to $\left( \frac{3^{5/3}}{5}, +\infty \right)$, with $P' > 0$ on ${\mathbb {R}}_+^*$. Therefore it is invertible, and $P^{-1}$ is also ${\mathcal {C}}^1$. All the more, a dominated convergence argument ensures that $I_2$ is ${\mathcal {C}}^1$ on ${\mathbb {R}}_+^*$. It is moreover shown in [22, proof of Proposition 3] that

$$\begin{aligned} \left( \frac{\varepsilon }{4 \pi } \right) ^{2/3} P \left( \frac{1}{\varepsilon e^{a_{\varepsilon }}} \right) = \frac{3 \rho _{\varepsilon } T_{\varepsilon }}{\rho _{\varepsilon }^{5/3}}, \qquad b_{\varepsilon } = \left( \frac{4 \pi }{\varepsilon \rho _{\varepsilon }} \, I_2\left( \frac{1}{\varepsilon e^{a_{\varepsilon }}} \right) \right) ^{\frac{2}{3}}. \end{aligned}$$

By Lemma 19, the application $\varepsilon \mapsto (\rho _{\varepsilon }, T_{\varepsilon })$ is ${\mathcal {C}}^1$ on ${\mathbb {R}}_+^*$, hence so is the application $\varepsilon \mapsto (a_{\varepsilon },b_{\varepsilon })$, as a composition of ${\mathcal {C}}^1$ applications. $\square$

Lemma 22

Let $0 \le g \in L^1_2({\mathbb {R}}^3) \cap L \log L({\mathbb {R}}^3)$. Using the notation (119), for any $\overline{\varepsilon } > 0$, there exist $C>0$ and $\eta > 0$ such that for any $\varepsilon \in [0,\overline{\varepsilon }]$ and $v \in {\mathbb {R}}^3$, we have

$$\begin{aligned} M_{\varepsilon }(v) \le C \, e^{- \eta |v|^2}, \end{aligned}$$

(123)

and

$$\begin{aligned} |\log M_{\varepsilon }(v)| \le C (1 + |v|^2). \end{aligned}$$

(124)

Proof

We denote

$$\begin{aligned} \underline{a} = \sup _{\varepsilon \in [0,\overline{\varepsilon }]} |a_{\varepsilon }|, \qquad \underline{b} = -\sup _{\varepsilon \in [0,\overline{\varepsilon }]} |b_{\varepsilon }|, \qquad \overline{b} = -\inf _{\varepsilon \in [0,\overline{\varepsilon }]} |b_{\varepsilon }| \quad \text {and} \quad \underline{u} = \sup _{\varepsilon \in [0,\overline{\varepsilon }]} |u_{\varepsilon }|. \end{aligned}$$

Combining the results of Lemmas 20 and 21, the application $\varepsilon \mapsto (a_{\varepsilon },b_{\varepsilon },u_{\varepsilon })$ is continuous on ${\mathbb {R}}_+$, from which we deduce that $\underline{a}$, $\underline{b}$ and $\underline{u}$ are finite.

Moreover, as $M_{\varepsilon } \in L^1({\mathbb {R}}^3)$ for all $\varepsilon \in [0,\overline{\varepsilon }]$, the application $\varepsilon \mapsto b_{\varepsilon }$ is (strictly) negative on $[0,\overline{\varepsilon }]$, so that, as $\varepsilon \mapsto b_{\varepsilon }$ is continuous on $[0,\overline{\varepsilon }]$, we have $\overline{b} < 0$.

Therefore, for any $0 \le \varepsilon \le \overline{\varepsilon }$ and $v \in {\mathbb {R}}^3$, we have

$$\begin{aligned} |\log M_{\varepsilon }(v)| = |a_{\varepsilon } + b_{\varepsilon }|v-u_{\varepsilon }|^2| \le \underline{a} + 2 |\underline{b}| \, \underline{u}^2 + 2 |\underline{b}| \, |v|^2, \end{aligned}$$

and, since $|v-u_{\varepsilon }|^2 \ge \frac{1}{2} |v|^2 - |u_{\varepsilon }|^2 \ge \frac{1}{2} |v|^2 - \underline{u}^2$ and $\overline{b} < 0$,

$$\begin{aligned} M_{\varepsilon } (v) = e^{a_{\varepsilon } + b_{\varepsilon }|v-u_{\varepsilon }|^2} \le e^{\underline{a} + \overline{b}|v-u_{\varepsilon }|^2} \le e^{\underline{a} + \left| \overline{b} \right| \underline{u}^2 + \frac{1}{2}\overline{b}|v|^2}. \end{aligned}$$

Letting $\eta = -\frac{1}{2} \overline{b}$ and $C = \max \left( \underline{a} + 2 |\underline{b}| \, \underline{u}^2, \, e^{\underline{a} + \left| \overline{b} \right| \underline{u}^2} \right)$ yields the result. $\square$