1 Introduction

This paper studies the global in time existence of weak solutions to the Cauchy problem of the ES-BGK model:

$$\begin{aligned} \begin{aligned} \partial _t f+v\cdot \nabla _x f&=A_{\nu }(\mathcal {M}_{\nu }(f)-f),\\f(x,v,0)&=f_0(x,v), \end{aligned} \end{aligned}$$
(1.1)

in the critical case \((\nu =-1/2)\). The particle distribution function f(x, v, t) is the number density of the molecules on the position \(x\in \mathbb {R}^{3}\), with the velocity \(v\in \mathbb {R}^{3}\) at time \(t \ge 0\). The Knudsen parameter \(\nu \) is chosen in the range \(-1/2\le \nu <1\), and \(A_{\nu }=1/(1-\nu )\). The non-isotropic Gaussian \(\mathcal {M}_{\nu }(f)\) parametrized by \(\nu \) is defined by

$$\begin{aligned} \begin{aligned} \mathcal {M}_{\nu }(f) = \frac{\rho }{\sqrt{\det (2\pi \mathcal {T}_\nu )}}\exp \left( -\frac{1}{2}(v-U)^{\top }\mathcal {T}_\nu ^{-1}(v-U)\right) . \end{aligned} \end{aligned}$$
(1.2)

Here the local density \(\rho \), momentum U, temperature T and stress tensor \(\Theta \) are defined through the following relations:

$$\begin{aligned} \begin{aligned} \rho (x,t)&= \int _{\mathbb {R}^{3}} f(x,v,t) dv, \\\rho (x,t) U (x,t)&= \int _{\mathbb {R}^{3}} f(x,v,t)v dv,\\3 \rho (x,t) T (x,t)&= \int _{\mathbb {R}^{3}} f(x,v,t)|v-U(x,t)|^{2} dv,\\\rho (x,t) \Theta (x,t)&= \int _{\mathbb {R}^{3}} f(x,v,t)\big (v-U(x,t)\big )\otimes \big (v-U(x,t)\big ) dv. \end{aligned} \end{aligned}$$
(1.3)

Note that elements of \(\Theta \) are given by \((1\le i,j \le 3)\)

$$\begin{aligned} \rho (x,t)\Theta _{ij}(x,t)=\int _{\mathbb {R}^{3}} f(x,v,t)\big (v-U_i(x,t)\big )\big (v-U_j(x,t)\big ) dv. \end{aligned}$$

The temperature tensor \(\mathcal {T}_{\nu }\) is given as a linear combination of the temperature and the stress tensor:

$$\begin{aligned} \mathcal {T}_{\nu }&=(1-\nu )T Id+\nu \Theta \\&=\left( \begin{array}{ccc}(1-\nu ) T+\nu \Theta _{11}&{}\nu \Theta _{12}&{}\nu \Theta _{13}\\ \nu \Theta _{21}&{}(1-\nu )T+\nu \Theta _{22}&{}\nu \Theta _{23}\\ \nu \Theta _{31}&{}\nu \Theta _{32}&{}(1-\nu ) T+\nu \Theta _{33}\end{array}\right) , \end{aligned}$$

where Id is the \(3\times 3\) identity matrix. We note that on (x, t) where \(\rho =0\), \({\mathcal M}_{\nu }(f)\) is defined to be zero. The range of \(\nu \) is restricted to \(1/2\le \nu <1\) since it is the minimum condition that guarantees the non-negative definiteness of the temperature tensor at least at the formal level [2]. We also mention that the horizontal cross-section of the non-isotropic Gaussian \(\mathcal {M}_{\nu }(f)\) is an ellipsoid, whereas the horizontal-cross section of the usual Maxwellian is a sphere. This is why the model is called the ellipsoidal BGK model.

A direct computation shows that the ellipsoidal Gaussian satisfies

$$\begin{aligned} \int _{\mathbb {R}^{3}}\left\{ \mathcal {M}_{\nu }(f)(x,v,t)-f(x,v,t)\right\} \left( \begin{array}{c}1\\ v\\ |v|^2\end{array}\right) dv=0, \end{aligned}$$

which leads to the conservation laws of mass, momentum and energy:

$$\begin{aligned} \frac{d}{dt}\int _{\mathbb {R}^3\times \mathbb {R}^3}f(x,v,t){\left( \begin{array}{c}1\\ v\\ |v|^2\end{array}\right) }dxdv=0. \end{aligned}$$

The celebrated H-theorem was verified by Andries et al [2]:

$$\begin{aligned} \frac{d}{dt}\int _{\mathbb {R}^3\times \mathbb {R}^3}f\ln f dvdx \le 0. \end{aligned}$$
(1.4)

The Boltzmann equation is the fundamental model for the description of gases at the mesoscopic level. In practice, the BGK model [4] is widely used in place of the Boltzmann equation due to its reliable performance in numerical simulations at much lower computational costs. But the compressible Navier-Stokes limit of the original BGK model shows that the Prandtl number—The ratio between the heat conductivity and the viscosity—is not computed correctly. Holway managed this problem by introducing a free parameter \(\nu \in [-1/2,1)\) and generalizing the local Maxwellian into a non-isotropic Gaussian [13]. When \(\nu =0\), (1.1) reduces to the original BGK model [4] and \(\nu =-1/2\) is the choice that yields the correct Prantl number. The ES-BGK model, however, was not employed popularly in the community since the H-theorem was not known. The H-theorem was verified later in [2], and the model got popularized [1, 10,11,12, 15, 18, 23]. To motivate the current work, we briefly review the results that are directly relevant to this work. Brull et al. derived ES-BGK model systematically using an entropy minimization argument [5]. The entropy production estimate for ES-BGK model was obtained in [22] for the non-critical case \(-1/2<\nu <1\) and in [14] for the critical case \(\nu =-1/2\). The weak solutions and the unique mild solution in the non-critical case, were established in [16], and [8, 19, 20] respectively. The existence of classical solutions near-equilibrium was studied in [21] for \(-1/2\le \nu <1\). The results on the stationary solution for the ES-BGK in a bounded interval can be found in [3] for the non-critical case and in [6] for the critical case.

All in all, the existence of the ES-BGK model in the non-critical case has been rather thoroughly studied, while many problems remain open for the critical case. One of the main reasons is that, in the non-critical case \((-1/2<\nu <1)\), the temperature tensor enjoys the following equivalence type estimate [6, 19, 21]:

$$\begin{aligned} \min \{1-\nu ,1+2\nu \}T Id\le \mathcal {T}_{\nu }\le \max \{1-\nu ,1+2\nu \}T Id. \end{aligned}$$

Therefore, many estimates of the temperature tensor can be reduced to similar estimates of the local temperature. In the critical case \(\nu =-1/2\), however, such estimate breaks down, and the temperature tensor has to be treated with more care. Especially, the existence of weak solutions for (1.1) in the critical case \((\nu =-1/2)\) has not been addressed, which is the main purpose of this work. In this regard, our main result is as follows:

Theorem 1.1

Let \(\nu =-1/2\). Suppose that \(f_0(x,v)\ge 0\) satisfies

$$\begin{aligned} \int _{\mathbb {R}^{6}}(1+|v|^2+|x|^2+|\ln f_0|)f_0 dxdv < \infty . \end{aligned}$$

Then, for any final time \(T^f\)there exists a non-negative weak solution \(f(x,v,t)\in L^1([0,T^f],\mathbb {R}^3\times \mathbb {R}^3)\) to (1.1):

$$\begin{aligned}&-\int _{\mathbb {R}^3\times \mathbb {R}^3}f_0\phi (0)dxdv-\int ^{T^f}_0\int _{\mathbb {R}^3\times \mathbb {R}^3}f(\partial _t\phi +v\cdot \nabla _x\phi )dxdvdt\\&=A_{\nu }\int ^{T^f}_0\int _{\mathbb {R}^3\times \mathbb {R}^3}\big (\mathcal {M}_{\nu }(f)-f\big )\phi dxdvdt \end{aligned}$$

for every \(\phi \in C^1_c(\mathbb {R}^3\times \mathbb {R}^3\times \mathbb {R}^+)\) with \(\phi (x,v,T^f)=0\). Moreover, f satisfies

$$\begin{aligned} \int ^{T^f}_0\int _{\mathbb {R}^{6}}(1+|v|^2+|x|^2+|\ln f|)f dxdvdt < \infty , \end{aligned}$$

the conservation laws:

$$\begin{aligned} \int _{\mathbb {R}^{6}}f(t)\left( \begin{array}{c}1\\ v\\ |v|^2\end{array}\right) dxdv =\int _{\mathbb {R}^{6}}f_0\left( \begin{array}{c}1\\ v\\ |v|^2\end{array}\right) dxdv, \end{aligned}$$

and the entropy dissipation \((t_2\ge t_1\ge 0)\):

$$\begin{aligned} \int _{\mathbb {R}^{6}}f(t_2)\ln f(t_2) dxdv \le \int _{\mathbb {R}^{6}}f(t_1)\ln f(t_1) dxdv. \end{aligned}$$

2 Proof of Theorem 1.1

2.1 Approximate problem

For \(n=1,2,\cdots \), we set up our approximate problem of (1.1) by

$$\begin{aligned} \begin{aligned} \partial _t f_{n}+v\cdot \nabla _x f_{n}&=A_{-1/2+1/n}\big (\mathcal {M}_{-1/2+1/n}(f_{n})-f_{n}\big ),\\f_{n}(x,v,0)&=f_{0,n}(x,v), \end{aligned} \end{aligned}$$
(2.1)

where \(f^{n}_0\) is the regularized initial data:

$$\begin{aligned} f_{0,n}(x,v)= f_0(x,v) + \frac{1}{n} m(x,v), \end{aligned}$$

with m(x, v) is defined by \((q>5)\)

$$\begin{aligned} m(x,v)= e^{-|v|^2}(1+|x|^2)^{-q/2}, \end{aligned}$$

and \(\mathcal {M}_{-1/2+1/n}(f_n)\) corresponds to the non-isotropic Gaussian defined in (1.2) with \(\nu =-1/2+1/n\):

$$\begin{aligned} \begin{aligned} \mathcal {M}_{-1/2+1/n}(f_n) = \frac{\rho _n}{\sqrt{\det (2\pi \mathcal {T}_{-1/2+1/n,n})}}\exp \left( -\frac{1}{2}(v-U_n)^{\top }\mathcal {T}_{-1/2+1/n,n}^{-1}(v-U_n)\right) , \end{aligned} \end{aligned}$$

where \(\rho _n\), \(U_n\), \(T_n\) and \(\Theta _{n}\) are the macroscopic fields constructed from the particle distribution function \(f_n\) through the relation (1.3), and \(\mathcal {T}_{-1/2+1/n,n}\) is the temperature tensor constructed from \(f_n\) in the case \(\nu =-1/2+1/n\):

$$\begin{aligned} \mathcal {T}_{-1/2+1/n,n}&=\Big (1-\Big (\frac{1}{2}-\frac{1}{n}\Big )\Big )T_n{Id}+\Big (\frac{1}{2}-\frac{1}{n}\Big )\Theta _n\\&=\Big (\frac{1}{2}+\frac{1}{n}\Big )T_n{Id}+\Big (\frac{1}{2}-\frac{1}{n}\Big )\Theta _n. \end{aligned}$$

We note that the approximate equation (2.1) corresponds to the ES-BGK model with non-critical Prandtl parameter \((-1/2<\nu <1)\), whose existence theory is considered in [16]:

Proposition 2.1

Let \(T^f\) be any final time. For each \(n=1,2,3,\cdots \), there exists a global weak solution \(f_n(x,v,t)\ge 0\) to (2.1):

$$\begin{aligned}&-\int _{\mathbb {R}^3\times \mathbb {R}^3}f_{0,n}\phi (0)dxdv-\int ^{T^f}_0\int _{\mathbb {R}^3\times \mathbb {R}^3}f_n(\partial _t\phi +v\cdot \nabla _x\phi )dxdvdt\\&=A_{-1/2+1/n}\int ^{T^f}_0\int _{\mathbb {R}^3\times \mathbb {R}^3}\big (\mathcal {M}_{-1/2+1/n}(f_n)-f_n\big )\phi dxdvdt \end{aligned}$$

for every \(\phi \in C^1_c(\mathbb {R}^3\times \mathbb {R}^3\times \mathbb {R}^+)\) with \(\phi (x,v,T^f)=0\). Moreover

  1. 1.

    \(f_n\) satisfies

    $$\begin{aligned} \int ^{T^f}_0\int _{\mathbb {R}^{6}}(1+|v|^2+|x|^2+|\ln f_n|)f_n dxdvdt < C, \end{aligned}$$

    for some \(C>0\) independent of n.

  2. 2.

    The conservation laws hold:

    $$\begin{aligned} \int _{\mathbb {R}^{6}}f_n(t)\left( \begin{array}{c}1\\ v\\ |v|^2\end{array}\right) dxdv =\int _{\mathbb {R}^{6}}f_{0,n}\left( \begin{array}{c}1\\ v\\ |v|^2\end{array}\right) dxdv. \end{aligned}$$
  3. 3.

    \(f_n\) satisfies the entropy dissipation:

    $$\begin{aligned} \int _{\mathbb {R}^{6}}f_n(t_2)\ln f_n(t_2) dxdv \le \int _{\mathbb {R}^{6}}f_n(t_1)\ln f_n(t_1) dxdv.\quad (t_2\ge t_1) \end{aligned}$$
  4. 4.

    For any compact set \(K_x\subseteq \mathbb {R}^3_x\), \(f_n\) satisfies the following moment estimate:

    $$\begin{aligned} \int _{0}^{T^f} \int _{K_{x}}\int _{\mathbb {R}^3}|v|^{3}f_{n}(x,v,t)dvdxdt \le C_{K_{x}}. \end{aligned}$$
  5. 5.

    \(\mathcal {T}_{-1/2+1/n,n}\) is strictly positive definite:

    $$\begin{aligned} \kappa ^{\top } \mathcal {T}_{-1/2+1/n,n}(x,t)\kappa \ge C_{T^f,f_{0,n},n}{(1+|x|^2)^{-q/2}}>0, \text{ for } \text{ any } \kappa \in \mathbb {S}^2. \end{aligned}$$

Remark 2.1

(1) The 3rd moment is established by Perthame in [17]. (2) The strictly positive definiteness in (5) holds due to the fact that the regularized initial data \(f_{0,n}\) has a strict lower bound. See Theorem 2.1. in [16].

The following estimate is also crucially used for the weak \(L^1\) compactness of \(\mathcal {M}_{-1/2+1/n}\).

2.2 Weak compactness of \(f_n\) and \(\mathcal {M}_{-1/2+1/n}(f_n)\)

We deduce from Proposition 2.1 and Dunford-Pettis theorem [7, 9] that there exists \(f\in L^1\) such that \(f_n\), \(f_n v\) converge to f, fv weakly \(L^1(\mathbb {R}^3 \times \mathbb {R}^3 \times [0,T^f] )\). This, combined with the velocity averaging lemma gives

$$\begin{aligned} \rho _n = \int _{\mathbb {R}^{3}}f_n dv&\rightharpoonup \int _{\mathbb {R}^{3}}f dv =\rho \quad \text {in} \quad L^1([0,T^f],\mathbb {R}^3_x), \\\rho _n U_n=\int _{\mathbb {R}^{3}}f_n v dv&\rightharpoonup \int _{\mathbb {R}^{3}}fv dv= \rho U \quad \text {in} \quad L^1([0,T^f],\mathbb {R}^3). \end{aligned}$$

Similarly, but this time combined with Proposition 2.1 (4), it can be shown that

$$\begin{aligned} \int _{\mathbb {R}^3} f_n v_iv_jdv \rightharpoonup \int _{\mathbb {R}^3}fv_iv_jdv \end{aligned}$$

in \(L^1([0,T^f],K_x \times \mathbb {R}^3)\), so that

$$\begin{aligned}&\rho _n\mathcal {T}_{-1/2+1/n,n}+\rho _n\left\{ \Big (\frac{1}{2}-\frac{1}{3n}\Big )|U_n|^2Id+\Big (-\frac{1}{2}+\frac{1}{n}\Big )\rho _n U_n\otimes U_n\right\} \\&\qquad =\int _{\mathbb {R}^{3}}f_n\left\{ \Big (\frac{1}{2}-\frac{1}{3n}\Big )|v|^2Id+\Big (-\frac{1}{2}+\frac{1}{n}\Big ) v\otimes v \right\} dv \\&\qquad \rightharpoonup \int _{\mathbb {R}^{3}}f\left\{ \frac{1}{2}|v|^2Id-\frac{1}{2}v\otimes v \right\} dv\\&\qquad =\rho \mathcal {T}_{-1/2}+\rho \left\{ \frac{1}{2}|U|^2Id-\frac{1}{2}\rho U\otimes U\right\} , \end{aligned}$$

in \(L^1([0,T^f],K_x \times \mathbb {R}^3)\). Therefore, we have almost everywhere convergence of macroscopic fields on a set where \(\rho \) does not vanish:

$$\begin{aligned} \begin{aligned} \rho _n&\rightarrow \rho \qquad \text{ a.e } \text{ on } \mathbb {R}^3\times [0,T^f],\\U_n&\rightarrow U \qquad \text{ a.e } \text{ on } \mathbb {E},\\\mathcal {T}_{-1/2+1/n,n}&\rightarrow \mathcal {T}_{-1/2} \qquad \text{ a.e } \text{ on } \mathbb {E}, \end{aligned} \end{aligned}$$
(2.2)

where \(\mathbb {E}\) is defined by

$$\begin{aligned} \mathbb {E}=\{(x,t) \in \mathbb {R}^3 \times (0, T^{f})\ |\ \rho (x,t) \ne 0 \}. \end{aligned}$$
(2.3)

On the other hand, the weak compactness of \(\mathcal {M}_{-1/2+1/n}(f_n)\) in \(L^1((0,T^f)\times \mathbb {R}^3\times \mathbb {R}^3)\) follows from the following inequality established in Lemma 2.3 of [16] with a \(C>0\) independent of n:

$$\begin{aligned} \int ^{T^f}_0\int _{\mathbb {R}^{6}}(1+|v|^2+|x|^2+|\ln \mathcal {M}_{-1/2+1/n}(f_n)|)\mathcal {M}_{-1/2+1/n}(f_n) dxdvdt < C. \end{aligned}$$

Therefore, we can find \(M\in L^1([0,T^f],\mathbb {R}^3\times \mathbb {R}^3)\) such that \(\mathcal {M}_{-1/2+1/n}\) converges weakly in \(L_1\) to M as \(n \rightarrow \infty \).

2.3 Conclusion of the proof

It remains to check that

$$\begin{aligned} M=\mathcal {M}_{-1/2}(\rho ,U,\mathcal {T}_{-1/2}). \end{aligned}$$

For this, we define

$$\begin{aligned} \mathbb {A}=\big \{(x,t)\in \mathbb {R}^3_x\times [0,T^f] |\ k^{\top }\mathcal {T}_{-1/2}k\ne 0 \text { for all non zero}\ k \in \mathbb {R}^3\big \} \end{aligned}$$

and consider (Recall that \(\mathbb {E}\) is defined in (2.3).)

$$\begin{aligned} \begin{aligned}&\int _{0}^{T^f}\int _{\mathbb {R}^3_x}\int _{\mathbb {R}^3_v}\mathcal {M}_{-1/2+1/n}(f_n)\phi dvdxdt\\&\qquad = \int _{\mathbb {A}\cap \mathbb {E}}\int _{\mathbb {R}^3_v}\mathcal {M}_{-1/2+1/n}(f_n)\phi dvdxdt + \int _{\mathbb {A}\cap \mathbb {E}^c}\int _{\mathbb {R}^3_v}\mathcal {M}_{-1/2+1/n}(f_n)\phi dvdxdt\\&\quad \qquad +\int _{\mathbb {A}^c\cap \mathbb {E}^c}\int _{\mathbb {R}^3_v}\mathcal {M}_{-1/2+1/n}(f_n)\phi dvdxdt + \int _{\mathbb {A}^c\cap \mathbb {E}}\int _{\mathbb {R}^3_v}\mathcal {M}_{-1/2+1/n}(f_n)\phi dvdxdt \\&\qquad := I_1 +I_2+I_3 +I_4. \end{aligned} \end{aligned}$$

Below, we consider each integrals separately to show that M coincides with \(\mathcal {M}_{-1/2}\) on each subset of \(\mathbb {R}^3 \times \mathbb {R}^3\).

\(\bullet \) \(I_1\): Since \(\rho \ne 0\), we find from (2.2) that \(\mathcal {M}_{-1/2+1/n}(\rho _n,U_n,{\mathcal {T}}_{-1/2+1/n,n})\) converges almost everywhere to \(\mathcal {M}_{-1/2}(\rho ,U,\mathcal {T}_{-1/2})\). Therefore, using Fatou’s Lemma, we get

$$\begin{aligned} \int _{\mathbb {A}\cap \mathbb {E}}\int _{\mathbb {R}^3_v}\mathcal {M}_{-1/2}(\rho ,U,{\mathcal {T}}_{-1/2})\phi dvdxdt&\le \lim _{n \rightarrow \infty } \int _{\mathbb {A}\cap \mathbb {E}}\int _{\mathbb {R}^3_v}\mathcal {M}_{-1/2+1/n}(f_n)\phi dvdxdt. \end{aligned}$$

But we have from the definition of M that

$$\begin{aligned} \int _{\mathbb {A}\cap \mathbb {E}}\int _{\mathbb {R}^3_v}\mathcal {M}_{-1/2+1/n}(f_n)\phi dvdxdt = \int _{\mathbb {A}\cap \mathbb {E}}\int _{\mathbb {R}^3_v}M\phi dvdxdt. \end{aligned}$$

This yields

$$\begin{aligned} \int _{\mathbb {A}\cap \mathbb {E}}\int _{\mathbb {R}^3_v}\mathcal {M}_{-1/2+1/n}(\rho ,U,{\mathcal {T}}_{-1/2+1/n})\phi dvdxdt \le \int _{\mathbb {A}\cap \mathbb {E}}\int _{\mathbb {R}^3_v}M\phi dvdxdt. \end{aligned}$$
(2.4)

To show the reverse inequality, we choose \(\phi =1\) and observe from the definition of M that

$$\begin{aligned} \int _{\mathbb {A}\cap \mathbb {E}}\int _{\mathbb {R}^3_v}Mdvdxdt&=\lim _{n \rightarrow \infty } \int _{\mathbb {A}\cap \mathbb {E}}\int _{\mathbb {R}^3_v}\mathcal {M}_{-1/2+1/n}(f_n) dvdxdt. \end{aligned}$$

Since \(\mathcal {T}_{-1/2+1/n}\) is strictly positive definite by Proposition 2.1 (5), we can take the change of variable:

$$\begin{aligned} X=\mathcal {T}^{-1/2}_{-1/2+1/n}(v-U) \end{aligned}$$

to compute

$$\begin{aligned} \lim _{n \rightarrow \infty } \int _{\mathbb {A}\cap \mathbb {E}}\int _{\mathbb {R}^3_v}\mathcal {M}_{-1/2+1/n}(f_n) dvdxdt = \lim _{n \rightarrow \infty } \int _{\mathbb {A}\cap \mathbb {E}}\int _{\mathbb {R}^3_v}\rho _{n} dvdxdt = \int _{\mathbb {A}\cap \mathbb {E}}\int _{\mathbb {R}^3_v}\rho dvdxdt. \end{aligned}$$

The last line comes from (2.2). Now, since \(\mathcal {T}_{-1/2}(x,t)\) is also strictly positive definite on \(\mathbb {A}\). We can take the change of variable:

$$\begin{aligned} {Y}=\mathcal {T}^{-1/2}_{-1/2}(v-U), \end{aligned}$$

to get

$$\begin{aligned} \int _{\mathbb {A}\cap \mathbb {E}} \rho dxdt =\int _{\mathbb {A}\cap \mathbb {E}}\int _{\mathbb {R}^3_v}\mathcal {M}_{-\frac{1}{2}}(\rho ,U,\mathcal {T}_{-1/2}) dvdxdt. \end{aligned}$$

In summary, we have on \(\mathbb {A}\cap \mathbb {E}\)

$$\begin{aligned} \int _{\mathbb {A}\cap \mathbb {E}}\int _{\mathbb {R}^3_v}M dvdxdt = \int _{\mathbb {A}\cap \mathbb {E}}\int _{\mathbb {R}^3_v}\mathcal {M}_{-\frac{1}{2}}(\rho ,U,\mathcal {T}_{-1/2}) dvdxdt. \end{aligned}$$
(2.5)

From (2.4) and (2.5), we conclude that

$$\begin{aligned} M=\mathcal {M}_{-\frac{1}{2}}(\rho ,U,\mathcal {T}_{-1/2}) \end{aligned}$$

almost everywhere on \(\mathbb {A}\cap \mathbb {E}\).

\(\bullet \) \(I_2\): Using the same argument of case \(I_1\), we find

$$\begin{aligned} \int _{\mathbb {A}\cap \mathbb {E}^c}\int _{\mathbb {R}^3_v}Mdvdxdt&=\lim _{n \rightarrow \infty } \int _{\mathbb {A}\cap \mathbb {E}^c}\int _{\mathbb {R}^3_v}\mathcal {M}_{-1/2+1/n}(f_n) dvdxdt =\lim _{n \rightarrow \infty } \int _{\mathbb {A}\cap \mathbb {E}^c} \rho _n dxdt \\&=\int _{\mathbb {A}\cap \mathbb {E}^c} \rho dxdt =0. \end{aligned}$$

Therefore, \(M=0=\mathcal {M}_{-1/2}(\rho ,U,\mathcal {T}_{-1/2})\).

\(\bullet \) \(I_3\) : Since \(\rho =0\), we have \(\mathcal {M}_{-1/2}(f)=0 \) by definition. Therefore, by the Fatou’s lemma and the fact that \(\mathcal {M}_{\nu }(f_n)\) converges in weak \(L^1\) to M, we have

$$\begin{aligned} 0&=\int _{\mathbb {A}^c\cap \mathbb {E}^c}\int _{\mathbb {R}^3_v}\mathcal {M}_{-1/2}(f)\phi dvdxdt\\&\le \lim _{n\rightarrow \infty } \int _{\mathbb {A}^c\cap \mathbb {E}^c}\int _{\mathbb {R}^3_v}\mathcal {M}_{-1/2+1/n}(f_n)\phi dvdxdt\\&= \int _{\mathbb {A}^c\cap \mathbb {E}^c}\int _{\mathbb {R}^3_v}M\phi dvdxdt. \end{aligned}$$

On the other hand, fixing \(\phi \) to 1 and proceeding as in the previous case, we get

$$\begin{aligned} \int _{\mathbb {A}^c\cap \mathbb {E}^c}\int _{\mathbb {R}^3_v}Mdvdxdt&=\lim _{n \rightarrow \infty } \int _{\mathbb {A}^c\cap \mathbb {E}^c}\int _{\mathbb {R}^3_v}\mathcal {M}_{-1/2+1/n}(f_n) dvdxdt =\lim _{n \rightarrow \infty } \int _{\mathbb {A}^c\cap \mathbb {E}^c} \rho _n dxdt \\&=\int _{\mathbb {A}^c\cap \mathbb {E}^c} \rho dxdt =0. \end{aligned}$$

\(\bullet \) \(I_4\): \((x,t)\in \mathbb {A}^c\) means that there exists a non-zero vector \(k(x,t)\in \mathbb {R}^3\) such that

$$\begin{aligned} k^{\top }(x,t) \mathcal {T}_{-1/2}(x,t)k(x,t)=0. \end{aligned}$$

We can find through an explicit computation using

$$\begin{aligned} Y^{\top }(X\otimes X) Y =\{X\cdot Y\}^2\quad (X,Y \in \mathbb {R}^3). \end{aligned}$$

Since \((x,t)\in \mathbb {A}\), the statement \(k^{\top } \mathcal {T}_{-1/2}k=0\), is equivalent to \( k^{\top }\left\{ \rho \mathcal {T}_{-1/2}\right\} k=0\). But

$$\begin{aligned} \begin{aligned} k^{\top } \left\{ \rho _{-1/2}\right\} \mathcal {T}_{-1/2}k&= k^{\top }\rho \left( \frac{3}{2}T_{-1/2}-\frac{1}{2}\Theta _{-1/2}\right) k\\&= k^{\top } \left\{ \frac{1}{2}\int _{\mathbb {R}^3_v}f|v-U|^2dv\right\} k - k^{\top } \left\{ \frac{1}{2}\int _{\mathbb {R}^3_v}f(v-U)\otimes (v-U) dv\right\} k\\&= \frac{1}{2}\int _{\mathbb {R}^3_v}f|v-U|^2|k|^2dv - \frac{1}{2}\int _{\mathbb {R}^3_v}f\{(v-U)\cdot k\}^2dv\\&= \frac{1}{2}\int _{\mathbb {R}^3_v}f\left\{ |v-U|^2|k|^2-\{(v-U)\cdot k\}^2\right\} dv. \end{aligned} \end{aligned}$$

Recalling

$$\begin{aligned} |v-U|^2|k|^2-\{(v-U)\cdot k\}^2\ge 0. \end{aligned}$$

One finds that

$$\begin{aligned} f(t,x,v)\left\{ |v-U|^2|k|^2-\{(v-U)\cdot k\}^2\right\} =0 \end{aligned}$$

on \((t,x,v)\in \mathbb {A}^c\cap \mathbb {E}\times \mathbb {R}^3_v\). If f is identically zero on the set, we are done. If not, there exists a measurable set B of strictly positive measure such that

$$\begin{aligned} f(t,x,v)>0 \text{ on } B. \end{aligned}$$

Therefore,

$$\begin{aligned} \left\{ |v-U|^2|k|^2-\{(v-U)\cdot k\}^2\right\} =0 \end{aligned}$$

on B, which is possible only when \(v-U(x,t)\) and k(x, t) are parallel on B. Combining the conclusion This is contradiction since k does not depend on v. From this, we conclude that

$$\begin{aligned} f(t,x,v)=0 \quad \text{ for } (x,t)\in \mathbb {E} ~ \text{ and } ~\forall v\in \mathbb {R}^3. \end{aligned}$$

Therefore, we have desired result from the same argument as in the case of \(I_2\).

Combining the arguments above, we conclude that \(M=\mathcal {M}_{-1/2} \text{ on } \mathbb {R}^3\times \mathbb {R}^3\), and the proof of main theorem is completed.