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Hydrodynamic Limit of the Incompressible Navier–Stokes–Fourier–Maxwell System with Ohm’s Law from the Vlasov–Maxwell–Boltzmann System: Hilbert Expansion Approach

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Abstract

We prove a global-in-time limit from the two-species Vlasov–Maxwell–Boltzmann system with a cutoff hard potential collision kernel to the two-fluid incompressible Navier–Stokes–Fourier–Maxwell system with Ohm’s law. Besides the techniques developed for the classical solutions to the Vlasov–Maxwell–Boltzmann equations in the past years, such as the nonlinear energy method and micro–macro decomposition are employed, in this paper, key roles are played by the decay properties of both the electric field and the wave equation with linear damping of the divergence free magnetic field. This is a companion paper of Jiang and Luo (Ann PDE, 8(1), Paper No. 4, 126, 2022) in which Hilbert expansion was not employed but only the hard sphere case was considered.

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Acknowledgements

The author N. Jiang is supported by grants from the National NSFC under contract Nos. 11971360 and 12221001 and the Strategic Priority Research Program of Chinese Academy of Sciences, Grant No. XDA25010404. Y.-L. Luo is supported by grants from the National Natural Science Foundation of China under contract No. 12201220, the Guang Dong Basic and Applied Basic Research Foundation under contract No. 2021A1515110210, and the Science and Technology Program of Guangzhou, China under the contract No. 202201010497. T.-F. Zhang is supported by grants from the National Natural Science Foundation of China under contract Nos. 11701534, and 11871203. The authors would like to thank the anonymous referees for careful reading and helpful comments.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, People’s Republic of China

    Ning Jiang

  2. School of Mathematics, South China University of Technology, Guangzhou, 510641, People’s Republic of China

    Yi-Long Luo

  3. School of Mathematics and Physics, China University of Geosciences, Wuhan, 430074, People’s Republic of China

    Teng-Fei Zhang

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Appendix A: Estimates on \((\rho _1, u_1, \theta _1, E_1, B_1)\): Proof of Lemma 3.5

Appendix A: Estimates on \((\rho _1, u_1, \theta _1, E_1, B_1)\): Proof of Lemma 3.5

This section aims to derive the energy bounds of the linear Maxwell equations (2.43)–(2.44) with initial data (1.12), namely, it aims to prove Lemma 3.5.

Proof of Lemma 3.5

The proof will be divided into two steps.

Step 1. Estimates on \((\rho _1, u_1, \theta _1)\).

We first prove the bound (3.20) of \( u_1\). Recalling the relations (2.44), one sees that

$$\begin{aligned} \begin{aligned} u_1 (t,x) = \nabla _x \phi (t,x) , \ \Delta _x \phi = \partial _t \theta _0 , \int _{{\mathbb {T}^3}} \phi \mathop {}\!\textrm{d}x = 0 . \end{aligned} \end{aligned}$$

The standard elliptic theory implies \(\Vert \phi (t,\cdot )\Vert _{H_x^{M+2}} \lesssim \Vert \partial _t \theta _0(t,\cdot ) \Vert _{H_x^M}\), provided that \(\partial _t \theta _0(t,\cdot ) \in {H_x^M}\). Hence, it follows from the third equation of the NSMF system (2.29) that, for \(M \ge 1\),

$$\begin{aligned} \Vert u_1 \Vert _{H^{M+1}_x}= & {} \Vert \nabla _x \phi \Vert _{H^{M+1}_x}(t) \lesssim \Vert \partial _t \theta _0(t,\cdot ) \Vert _{H_x^M} \nonumber \\= & {} \Vert \kappa \Delta _x \theta _0 - u_0 \cdot \nabla _x \theta _0\Vert _{H^M_x} \nonumber \\\lesssim & {} \Vert \nabla _x \theta _0\Vert _{H^{M+1}_x} + \Vert u_0\Vert _{H^M_x} \Vert \nabla _x \theta _0\Vert _{H^M_x} . \end{aligned}$$
(5.137)

Then, from the definitions of \(\mathcal {E}_{0,M+1} (t)\) and \(\mathcal {D}_{0,M+1} (t) \) in (3.10) and (3.11) with \(s = M+1\), the inequality (3.20) holds.

We next derive the inequality (3.21). By (2.44), \(\rho _1\) and \(\theta _1\) satisfy

$$\begin{aligned} \begin{aligned} \rho _1 = \theta _1 , \ \Delta _x \rho _1 = \tfrac{1}{6} \Delta _x |u_0|^2 - \tfrac{1}{2} \textrm{div}_x ( u_0 \cdot \nabla _x u_0 - \tfrac{1}{2} j_0 \times B_0) , \ \int _{{\mathbb {T}^3}} \rho _1 \mathop {}\!\textrm{d}x = 0 . \end{aligned} \end{aligned}$$

Then, together with the last third equation of the NSMF system (2.29), the elliptic theory yields

$$\begin{aligned} \Vert \rho _1 \Vert _{H^{M+1}_x} = \Vert \theta _1 \Vert _{H^{M+1}_x} \lesssim&\Vert \tfrac{1}{6} \Delta _x |u_0|^2 - \tfrac{1}{2} \textrm{div}_x ( u_0 \cdot \nabla _x u_0 - \tfrac{1}{2} j_0 \times B_0 ) \Vert _{H_x^{M-1}} \nonumber \\ \lesssim&\big ( \Vert \nabla _x u_0 \Vert _{H_x^{M+1}} + \Vert \nabla _x n_0 \Vert _{H_x^{M+1}}\nonumber \\&\quad + \Vert E_0 \Vert _{H_x^{M+1}} \big ) \big ( \Vert u_0 \Vert _{H_x^{M+1}} + \Vert B_0 \Vert _{H_x^{M+1}} \big ) \nonumber \\&\quad + \big ( \Vert \nabla _x u_0 \Vert _{H_x^{M+1}} + \Vert n_0 \Vert _{H_x^{M+1}} \big ) \nonumber \\&\quad \big ( \Vert u_0 \Vert ^2_{H_x^{M+1}} + \Vert B_0 \Vert ^2_{H_x^{M+1}} \big ) \,. \end{aligned}$$
(5.138)

Thus, the inequality (3.21) follows from the definition of \(\mathcal {E}_{0,M+1} (t) \) and \(\mathcal {D}_{0,M+1} (t) \) in (3.10) and (3.11), respectively.

Step 2. Estimates on \((E_1, B_1)\).

It remains to derive the inequalities (3.22)–(3.23). The key point is to seek some essential decay structures. In (2.43), denote by \(j_1 = u_2^+ - u_2^-\).

From applying \(\textrm{div}_x\) to the first equation of (2.43) and using \(\textrm{div}_x E_1 = n_1\), it follows that

$$\begin{aligned} \partial _t n_1 - \tfrac{1}{2}\sigma \Delta _x n_1 + \sigma n_1 = - \textrm{div}_x j_1. \end{aligned}$$
(5.139)

Moreover, from taking the time derivative on the \( B_1\)-equation in (2.43),

$$\begin{aligned} \partial _{tt} B_1 + \nabla _x \times (\nabla _x \times B_1) - \sigma \nabla _x \times E_1 = \nabla _x \times j_1 \,. \end{aligned}$$

Noticing that \(\textrm{div}_x B_1 = 0\) yields \(\nabla _x \times (\nabla _x B_1) = - \Delta _x B_1\), one has

$$\begin{aligned} \partial _{tt} B_1 - \Delta _x B_1 + \sigma \partial _t B_1 = \nabla _x \times j_1 \,, \end{aligned}$$
(A.1)

which possesses a decay effect \(\sigma \partial _t B_1\).

Then, we turn to focus on the system (2.43), (A.3) and (A.4). For all \(|m| \le M\), applying \(\partial ^m\) to the first and second equations of (2.43), and then taking \(L^2_x\)-inner product with \(\partial ^m E_1\) and \(\partial ^m B_1\), respectively, one infers that

$$\begin{aligned}&\tfrac{1}{2} \tfrac{\mathop {}\!\textrm{d}}{\mathop {}\!\textrm{d}t} \big ( \Vert \partial ^m E_1 \Vert ^2_{L^2_x} + \Vert \partial ^m B_1 \Vert ^2_{L^2_x} \big ) + \sigma \Vert \partial ^m E_1 \Vert ^2_{L^2_x} + \tfrac{1}{2} \sigma \Vert \partial ^m n_1 \Vert ^2_{L^2_x} \\&\qquad = - \int _{{\mathbb {T}^3}} \partial ^m j_1 \cdot \partial ^m E_1 \mathop {}\!\textrm{d}x \,, \end{aligned}$$

where the cancellation relation \(\int _{\mathbb {T}^3} (\nabla _x \times \partial ^m E_1 ) \cdot \partial ^m B_1 \mathop {}\!\textrm{d}x = \int _{\mathbb {T}^3} ( \nabla _x \times \partial ^m B_1 ) \cdot \partial ^m E_1 \mathop {}\!\textrm{d}x \) and the third equation of (2.43), i.e., \(\textrm{div}_x E_1 = n_1\) have been used. Then, summing up for \(|m| \le M\) implies that

$$\begin{aligned}&\tfrac{1}{2} \tfrac{\mathop {}\!\textrm{d}}{\mathop {}\!\textrm{d}t} \big ( \Vert E_1 \Vert ^2_{H^M_x} + \Vert B_1 \Vert ^2_{H^M_x} \big ) + \sigma \Vert E_1 \Vert ^2_{H^M_x} + \tfrac{1}{2} \sigma \Vert n_1 \Vert ^2_{H^M_x}\\&\qquad = - \sum _{|m| \le M} \int _{\mathbb {T}^3} \partial ^m j_1 \cdot \partial ^m E_1 \mathop {}\!\textrm{d}x \,. \end{aligned}$$
(A.2)

Performing the similar procedure as above to the equation (A.3), one has that

$$\begin{aligned} \tfrac{1}{2} \tfrac{\mathop {}\!\textrm{d}}{\mathop {}\!\textrm{d}t} \Vert n_1 \Vert ^2_{H^M_x} + \sigma \Vert n_1 \Vert ^2_{H^M_x} + \tfrac{1}{2} \sigma \Vert \nabla _x n_1 \Vert ^2_{H^M_x} = \sum _{|m| \le M} \int _{\mathbb {T}^3} \partial ^m j_1 \cdot \nabla _x \partial ^m n_1 \mathop {}\!\textrm{d}x \,. \end{aligned}$$
(A.3)

For the equation (A.4), from taking the operator \(\partial ^m\), multiplying by \(\partial _t \partial ^m B_1\), integrating by parts over \(x \in \mathbb {T}^3\) and summing up for \(|m| \le M\), it follows that

$$\begin{aligned}&\tfrac{1}{2} \tfrac{\mathop {}\!\textrm{d}}{\mathop {}\!\textrm{d}t} \big ( \Vert \partial _t B_1 \Vert ^2_{H^M_x} + \Vert \nabla _x B_1 \Vert ^2_{H^M_x} \big ) + \sigma \Vert \partial _t \partial ^m B_1 \Vert ^2_{H^M_x} \nonumber \\&\quad = \sum _{|m| \le M} \int _{\mathbb {T}^3} ( \nabla _x \times \partial ^m j_1 ) \cdot \partial _t \partial ^m B_1 \mathop {}\!\textrm{d}x \,. \end{aligned}$$
(A.4)

Furthermore, if one multiplies by \(\partial ^m B_1\) instead of \(\partial _t \partial ^m B_1\) in the arguments of (A.8), it holds that

$$\begin{aligned}&\tfrac{1}{2} \tfrac{\mathop {}\!\textrm{d}}{\mathop {}\!\textrm{d}t} \big ( \Vert \partial _t B_1 + B_1 \Vert ^2_{H^M_x} - \Vert \partial _t B_1 \Vert ^2_{H^M_x} - ( 1 - \sigma ) \Vert B_1 \Vert ^2_{H^M_x} \big ) - \Vert \partial _t B_1 \Vert ^2_{H^M_x}+ \Vert \nabla _x B_1 \Vert ^2_{H^M_x} \nonumber \\&\quad = \sum _{ |m| \le M} \int _{\mathbb {T}^3} \partial ^m j_1 \cdot ( \nabla _x \times \partial ^m B_1) \mathop {}\!\textrm{d}x \,, \end{aligned}$$
(A.5)

where the equality \(( \nabla _x \times \partial ^m j_1) \cdot \partial ^m B_1 = \textrm{div}_x ( \partial ^m j_1 \times \partial ^m B_1 ) + \partial ^m j_1 \cdot ( \nabla _x \times \partial ^m B_1)\) has been utilized.

Choosing a positive constant \(\delta = \tfrac{1}{2} \min \{ 1,\sigma \} \in ( 0, \tfrac{1}{2} ] \), multiplying equality (A.9) by \(\delta \), and adding the resultant to the relations (A.5), (A.7) and (A.8), one has that

$$\begin{aligned}&\tfrac{1}{2} \tfrac{\mathop {}\!\textrm{d}}{\mathop {}\!\textrm{d}t} \big ( \Vert E_1 \Vert ^2_{H^M_x} + \Vert n_1 \Vert ^2_{H^M_x} + ( 1 - \delta + \delta \sigma ) \Vert B_1 \Vert ^2_{H^M_x} + \Vert \nabla _x B_1 \Vert ^2_{H^M_x} \nonumber \\&\qquad + ( 1 - \delta ) \Vert \partial _t B_1 \Vert ^2_{H^M_x} + \delta \Vert \partial _t B_1 + B_1 \Vert ^2_{H^M_x} \big ) \nonumber \\&\qquad + ( \sigma - \delta ) \Vert \partial _t B_1 \Vert ^2_{H^M_x} + \delta \Vert \nabla _x B_1 \Vert ^2_{H^M_x} + \sigma \Vert E_1 \Vert ^2_{H^M_x}\nonumber \\&\qquad + \tfrac{1}{2} \sigma \Vert \nabla _x n_1 \Vert ^2_{H^M_x} + \tfrac{3}{2} \sigma \Vert n_1 \Vert ^2_{H^M_x} \nonumber \\&\quad = \sum _{|m| \le M} \int _{\mathbb {T}^3} [ \partial ^m j_1 \cdot ( \nabla _x \partial ^m n_1 - \partial ^m n_1 + \delta \nabla _x \times \partial ^m B_1 )\nonumber \\&\qquad + ( \nabla _x \times \partial ^m j_1 ) \cdot \partial _t \partial ^m B_1 ] \mathop {}\!\textrm{d}x \nonumber \\&\quad \le \tfrac{\sigma }{2} \Vert n_1 \Vert ^2_{H^M_x} + \tfrac{\sigma }{4} \Vert \nabla _x n_1 \Vert ^2_{H^M_x} + \tfrac{\delta }{2} \Vert \nabla _x B_1 \Vert ^2_{H^M_x} + \tfrac{ \sigma - \delta }{2} \Vert \partial _t B_1 \Vert ^2_{H^M_x}\nonumber \\&\qquad + C \Vert j_1 \Vert ^2_{H^M_x} + C \Vert \nabla _x \times j_1 \Vert ^2_{H^M_x} \,, \end{aligned}$$
(A.6)

where we have used the Hölder inequality in the last line.

Recalling the expression of \(j_1 = u_2^+ - u_2^-\) in (2.43) and the fact \(\nabla _x \times (\nabla _x n_1) = 0\), it follows from the Sobolev embedding theory that

$$\begin{aligned} \Vert j_1 \Vert ^2_{H^M_x} + \Vert \nabla _x \times j_1 \Vert ^2_{H^M_x}&\le \Vert u_1 \Vert ^2_{H^{M+1}_x} + C \Vert B_0 \Vert ^2_{H^{M+1}_x} \Vert u_1 \Vert ^2_{H^{M+1}_x} \\&\qquad + C \sum \Vert \Gamma _0^- \Vert ^2_{H^{M+1}_x} \\&\quad + C ( \Vert u_0 \Vert ^2_{H^{M+1}_x} + \Vert \theta _0 \Vert ^2_{H^{M+1}_x} ) ( \Vert n_1 \Vert ^2_{H^M_x}\\&\quad + \Vert \nabla _x n_1 \Vert ^2_{H^M_x} + \Vert \nabla _x B_1 \Vert ^2_{H^M_x} ) \,, \end{aligned}$$

where Poincaré’s inequality \(\Vert B_1 \Vert _{L^2_x} \le C \Vert \nabla _x B_1 \Vert _{L^2_x}\) has been used. Recalling the definition of \(\Gamma _0^-\) in (2.41), one can deduce from the Sobolev embedding theory that

$$\begin{aligned} \sum \Vert \Gamma _0^-\Vert ^2_{H^{M+1}_x}&\lesssim \Vert \nabla _x u_0 \Vert ^2_{H^{M+2}_x} + \Vert \nabla _x \theta _0 \Vert ^2_{H^{M+2}_x} + \Vert E_0 \Vert ^2_{H^{M+2}_x} \\&\qquad + \big ( \Vert \nabla _x n_0 \Vert ^2_{H^{M+2}_x} + \Vert \nabla _x \theta _0 \Vert ^2_{H^{M+2}_x} + \Vert E_0 \Vert ^2_{H^{M+2}_x} \big ) \\&\qquad \times \big ( \Vert n_0 \Vert ^2_{H^{M+2}_x} + \Vert \theta _0 \Vert ^2_{H^{M+2}_x} + \Vert u_0 \Vert ^2_{H^{M+2}_x} + \Vert B_0 \Vert ^2_{H^{M+2}_x} \big ) \\&\qquad + ( \Vert n_0 \Vert ^2_{H^{M+2}_x} + \Vert \nabla _x u_0 \Vert ^2_{H^{M+2}_x})\\&\qquad \cdot ( \Vert \theta _0 \Vert ^2_{H^{M+2}_x} + \Vert u_0 \Vert ^2_{H^{M+2}_x} + \Vert B_0 \Vert ^2_{H^{M+2}_x} ) \\&\lesssim \mathcal {D}_{0,M+2} (t) + \mathcal {E}_{0,M+2} (t) \big ( 1 + \mathcal {E}_{0,M+2} (t) \big ) \mathcal {D}_{0,M+2} (t) \,, \end{aligned}$$

where Ohm’s law \(j_0 = n_0 u_0 + \sigma (-\tfrac{1}{2} \nabla _x n_0 + E_0 + u_0 \times B_0) \) in (2.29) has been used. Therefore,

$$\begin{aligned}&\Vert j_1 \Vert ^2_{H^M_x} + \Vert \nabla _x \times j_1 \Vert ^2_{H^M_x}\nonumber \\&\qquad \lesssim \big ( 1 + \mathcal {E}_{0,M+2} (t) \big ) \Vert u_1 \Vert ^2_{H^{M+1}_x} + \big ( 1 + \mathcal {E}_{0,M+2} (t) \big )^2 \mathcal {D}_{0,M+2} (t) \nonumber \\&\quad \qquad + \mathcal {E}_{0,M+2} (t) \big ( \Vert n_1 \Vert ^2_{H^M_x} + \Vert \nabla _x n_1 \Vert ^2_{H^M_x} + \Vert \nabla _x B_1 \Vert ^2_{H^{M}_x} \big ) \,. \end{aligned}$$
(A.7)

Plugging the inequality (A.11) into (A.10) implies that

$$\begin{aligned}&\tfrac{1}{2} \tfrac{\mathop {}\!\textrm{d}}{\mathop {}\!\textrm{d}t} \big ( \Vert E_1 \Vert ^2_{H^M_x} + \Vert n_1 \Vert ^2_{H^M_x} + (1 - \delta + \delta \sigma ) \Vert B_1 \Vert ^2_{H^M_x} + \Vert \nabla _x B_1 \Vert ^2_{H^M_x} \nonumber \\&\qquad + ( 1 - \delta ) \Vert \partial _t B_1 \Vert ^2_{H^M_x} + \delta \Vert \partial _t B_1 + B_1 \Vert ^2_{H^M_x} \big ) \nonumber \\&\qquad + ( \sigma - \delta ) \Vert \partial _t B_1 \Vert ^2_{H^M_x} + \delta \Vert \nabla _x B_1 \Vert ^2_{H^M_x} + \sigma \Vert E_1 \Vert ^2_{H^M_x} \nonumber \\&\qquad + \tfrac{1}{2} \sigma \Vert \nabla _x n_1 \Vert ^2_{H^M_x} + \tfrac{3}{2} \sigma \Vert n_1 \Vert ^2_{H^M_x} \nonumber \\&\quad \le C \big ( 1 + \mathcal {E}_{0,M+2} (t) \big ) \Vert u_1 \Vert ^2_{H^{M+1}_x} + C \big ( 1 + \mathcal {E}_{0,M+2} (t) \big )^2 \mathcal {D}_{0,M+2} (t) \nonumber \\&\qquad + C \mathcal {E}_{0,M+2} (t) \big ( \Vert n_1 \Vert ^2_{H^M_x} + \Vert \nabla _x n_1 \Vert ^2_{H^M_x} + \Vert \nabla _x B_1 \Vert ^2_{H^M_x} \big ) \,. \end{aligned}$$
(A.8)

By the monotonicity of \(\mathcal {E}_{0,s}(t)\) and \(\mathcal {D}_{0,s}(t)\) with respect to the index \(s \ge 0\), the inequality (3.20) reduces to

$$\begin{aligned} \Vert u_1 \Vert ^2_{H^{M+1}_x} (t) \le C \big ( 1 + \mathcal {E}_{0, M+2} (t) \big ) \mathcal {D}_{0,M+2} (t) \,, \end{aligned}$$
(A.9)

which, combined with the fact that \(\mathcal {E}_{0,M+2}(t) \le C \mathcal {E}_{0,M+2}^{\textrm{in}} \le C \lambda _0 (M+2)\) due to Lemma 3.4, enables us to get

$$\begin{aligned}&\tfrac{1}{2} \tfrac{\mathop {}\!\textrm{d}}{\mathop {}\!\textrm{d}t} \big ( \Vert E_1 \Vert ^2_{H^M_x} + \Vert n_1 \Vert ^2_{H^M_x} + ( 1 - \delta + \delta \sigma ) \Vert B_1 \Vert ^2_{H^M_x} + \Vert \nabla _x B_1 \Vert ^2_{H^M_x} \\&\qquad + ( 1 - \delta ) \Vert \partial _t B_1 \Vert ^2_{H^M_x} + \delta \Vert \partial _t B_1 + B_1 \Vert ^2_{H^M_x} \big ) \\&\qquad + ( \sigma - \delta ) \Vert \partial _t B_1 \Vert ^2_{H^M_x} + \delta \Vert \nabla _x B_1 \Vert ^2_{H^M_x} + \sigma \Vert E_1 \Vert ^2_{H^M_x} + \tfrac{1}{2} \sigma \Vert \nabla _x n_1 \Vert ^2_{H^M_x} + \tfrac{3}{2} \sigma \Vert n_1 \Vert ^2_{H^M_x} \\&\quad \le C ( 1 + \mathcal {E}_{0,M+2}^{\textrm{in}} )^2 \mathcal {D}_{0,M+2} (t) + C \mathcal {E}_{0,M+2}^{\textrm{in}} \big ( \Vert n_1 \Vert ^2_{H^M_x} + \Vert \nabla _x n_1 \Vert ^2_{H^M_x} + \Vert \nabla _x B_1 \Vert ^2_{H^M_x} \big ) \,. \end{aligned}$$

As a consequence, by the definitions of \(\mathcal {E}_{1,M}(t)\) and \(\mathcal {D}_{1,M}(t)\) in (3.17) and (3.18), respectively, it follows that

$$\begin{aligned} \tfrac{1}{2} \tfrac{\mathop {}\!\textrm{d}}{\mathop {}\!\textrm{d}t} \mathcal {E}_{1,M} (t) + 2 \mathcal {D}_{1,M} (t) \le C ( 1 + \mathcal {E}_{0,M+2}^{\textrm{in}} )^2 \mathcal {D}_{0,M+2} (t) + C \mathcal {E}_{0,M+2}^{\textrm{in}} \mathcal {D}_{1,M} (t) . \end{aligned}$$
(A.10)

Applying the energy inequality (3.14) with \(s=M+2\) in Lemma 3.4 yields

$$\begin{aligned} \tfrac{\mathop {}\!\textrm{d}}{\mathop {}\!\textrm{d}t}[ \mathcal {E}_{1, M} (t) + {\widetilde{C}}_M \mathcal {E}_{0, M+2} (t) ] + \left[ 2 \mathcal {D}_{1, M} (t) + \mathcal {D}_{0, M+2} (t) \right] \le C \mathcal {E}_{0,M+2}^{\textrm{in}} \mathcal {D}_{1,M} (t), \end{aligned}$$
(A.11)

with \({\widetilde{C}}_M= 1+ C ( 1 + \lambda _0 (M+2))^2 \ge 1\). Therefore, by choosing a small positive constant \(\lambda _1(M+2) \in ( 0, \lambda _0(M+2)] \) such that if \(\mathcal {E}_{0,M+2}^{\textrm{in}} \le \lambda _1(M+2)\), then \(C \mathcal {E}_{0,M+2}^{\textrm{in}} \le 1\), the above inequality will conclude (3.22). The bound (3.23) also follows immediately. The proof of Lemma 3.5 is completed. \(\square \)

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Jiang, N., Luo, YL. & Zhang, TF. Hydrodynamic Limit of the Incompressible Navier–Stokes–Fourier–Maxwell System with Ohm’s Law from the Vlasov–Maxwell–Boltzmann System: Hilbert Expansion Approach. Arch Rational Mech Anal 247, 55 (2023). https://doi.org/10.1007/s00205-023-01888-3

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