From the Boltzmann equation for gas mixture to the two-fluid incompressible hydrodynamic system

The Boltzmann equation for the gas mixture (BEGM) describes the time evolution of the density distribution of the gas mixture of two different species [13], which reads:

where $f_{l}(t,x,v),\,l\in\{1,2\}$ denotes the density distribution functions of the gas molecules at time $t\geq 0$, with position $x\in\mathbb{R}^{3}$ and velocity $v\in\mathbb{R}^{3}$. The collision operator $Q_{ln}(f_{l},g_{n})$ is given by, for $l,n\in\{1,2\}$,

where $f_{l}=f_{l}(t,x,v),\,g_{n*}=g_{n}(t,x,v_{*}),\,f^{\prime}_{l}=f_{l}(t,x,v^{\prime}),$ $g^{\prime}_{n*}=g_{n}(t,x,v^{\prime}_{*})$ with $(v^{\prime},v^{\prime}_{*})$ and $(v,v_{*})$ representing the velocity pairs before and after the collisions, respectively. For simplicity, we assume that particles of different species have the same mass $m$, but different radii (see [13, Chapter II, Section 4]) such that the conservation of momentum and energy hold in the sense that

This allows us to express $(v^{\prime},v_{*}^{\prime})$ in terms of $(v,v_{*})$ and unit vector $\sigma\in\mathbb{S}^{2}$ using the following relations:

The collision kernel $B_{ln}(v-v_{*},\sigma)$ describes the intensity of collisions, which will be assumed to satisfy the following properties: for $l,n\in\{1,2\}$,

• •

  The collisions are symmetric between species:

  $$B_{ln}(v-v_{*},\sigma)=B_{nl}(v-v_{*},\sigma).$$

• •

  $B_{ln}$ can be separated into the kinetic part $\Phi$ and angular part $b$ in the case of the inverse power law:

  $$B_{ln}(v-v_{*},\sigma)=b_{ln}(\cos\theta)\,\Phi_{ln}(|v-v_{*}|),\quad\text{with}\ \cos\theta=\sigma\cdot\frac{v-v_{*}}{|v-v_{*}|},$$

  where kinetic collision part $\Phi(|v-v_{*}|)=|v-v_{*}|^{\gamma}$ includes hard potential $(\gamma>0)$, Maxwellian molecule $(\gamma=0)$ and soft potential $(\gamma<0)$. Note that, for the rigorous justification throughout this paper, we consider the well-known hard sphere model, i.e., $\gamma=1$, and $b_{ln}$ to satisfy the Grad’s cutoff assumption.

We refer the readers for more details about the collision kernel $B_{ln}$ in [55].

To figure out the scaling that will be applied in the hydrodynamic limit process, we present the complete derivation of the dimensionless form of the BEGM equation.

Suppose the reference temperature $T_{0}$ and reference average macroscopic density $R_{0}$ are identical with the mixture particles, we can first define the macroscopic velocity $U_{0}$ and microscopic velocity $c_{0}$ as follows: for given macroscopic time $t_{0}$, length $L_{0}$, and Boltzmann contact $k$,

• •

  The macroscopic velocity $U_{0}:=\frac{L_{0}}{t_{0}}$;

• •

  The microscopic velocity $c_{0}:=\sqrt{\frac{5kT_{0}}{3m}}$.

Then, we can introduce the dimensionless variables of time $\tilde{t}$, space $\tilde{x}$ and velocity $\tilde{v}$:

and the dimensionless density distribution function $\tilde{f}_{l}(\tilde{t},\tilde{x},\tilde{v})$ follows that

To complete the derivation of the dimensionless BEGM equation, the following parameters will be introduced

• •

  Mean free time $\tau_{l}$. Since the Boltzmann kernel has units of the reciprocal product of density, then we define a mean free time $\tau_{l}$ of one single type of particle

  $$\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}\mathcal{M}_{[R_{0},0,T_{0}]}(v)\mathcal{M}_{[R_{0},0,T_{0}]}(v_{*})B_{ll}(|v-v_{*}|,\omega)\,d\omega\,dv_{*}\,dv=\frac{R_{0}}{\tau_{l}},$$

  where $\mathcal{M}_{[R_{0},0,T_{0}]}(v)$ the Maxwellian distribution:

  $$\mathcal{M}_{[R_{0},0,T_{0}]}(v):=\frac{R_{0}}{(2\pi T_{0})^{\frac{3}{2}}}\textup{e}^{-\frac{|v|^{2}}{2T_{0}}}.$$

• •

  Strength of interactions between species $\delta(l,n)$. Note that the particles of different species will also collide with each other in the gas mixture, we need to introduce the following dimensionless parameter $\delta(l,n)$ to describe the strength of interactions between the particles of different species [4]: for $l,n\in\{1,2\}$,

  $$\delta(l,n)=\left\{\begin{array}[]{cc}1,&\qquad\textup{if}\,\,l=n,\\[4.0pt] \bar{\delta},&\qquad\textup{if}\,\,l\neq n,\\[4.0pt] \end{array}\right.$$

  (1.3)

  with the constant $\bar{\delta}>0$. Hence, we can define the mean free time for the gas mixture as

  $$\tau_{ln}=\frac{\sqrt{\tau_{l}\tau_{n}}}{\delta^{2}(l,n)}.$$

• •

  Mean free path $\lambda_{ln}$. The scale of the average time that particles in the equilibrium density $\mathcal{M}_{[R_{0},0,T_{0}]}$ spend traveling freely between two collisions, which is related to the length scale of the mean free path $\lambda_{ln}$

  $$\lambda_{ln}=c_{0}\tau_{ln}.$$

  From the definition of $\tau_{ln}$, we have $\lambda_{ln}=\lambda_{nl}$ for each $l,n\in\{1,2\}$ and denote $\lambda_{l}:=\lambda_{ll}$ for convenience.

Finally, the corresponding dimensionless collision operator is derived as follows:

where the dimensionless Boltzmann collision kernel $\tilde{B}_{ln}$ is

Dropping all tildes, we deduce the BEGM equation in the dimensionless form:

where the Strouhal number St and Knudsen number $\textup{Kn}_{l}$ [52] are give by

Based on [4], we will distinguish the following three cases:

• •

  Strong inter-species interactions: $\bar{\delta}=O(1)$;
  

• •

  Weak inter-species interactions: $\bar{\delta}=o(1)$ and $\frac{\bar{\delta}}{(\textup{Kn}_{1}\textup{Kn}_{2}\textup{St}^{2})^{\frac{1}{4}}}$ is unbounded;
  

• •

  Very weak inter-species interactions: $\bar{\delta}=O\big{(}(\textup{Kn}_{1}\textup{Kn}_{2}\textup{St}^{2})^{\frac{1}{4}}\big{)}$.
  

In this paper, we only consider the third case with the same collision kernel, i.e.,

for $l,n\in\{1,2\}$.

Furthermore, for $l\in\{1,2\}$, we choose $\textup{St}=\varepsilon$, $\textup{Kn}_{l}=\varepsilon^{c_{l}}$, and $\bar{\delta}=\varepsilon^{q}$ with $c_{l},\,q>0$ and consider

which implies that

Therefore, the scaled BEGM equation (1.4) can be rewritten as

or, that is to say,

Then, suppose $f^{\varepsilon}_{l}\to f_{l}$ as $\varepsilon\to 0$, the right-hand side of (1.6) vanishes, which formally implies that

for $l,n\in\{1,2\}$ and $l\neq n$; on the other hand, we have

Thus, combining with (1.7) and (1.8), we obtain

In this subsection, we undertake a comprehensive review of prior research pertaining to the well-posedness and hydrodynamic limit of the classical Boltzmann equation, alongside relevant results regarding its application to gas mixtures, which serve as a significant impetus for our work. Furthermore, our contribution and novelty of this paper will be introduced as well.

Previous results of “well-posedness” Standing as a cornerstone in kinetic theory, the Boltzmann equation has been attracting enduring attention for a long history, particularly the studies regarding its well-posedness. In the realm of weak solutions, in [16], DiPerna-Lions laid the foundational groundwork by establishing renormalized solutions for the Boltzmann equation, accommodating general initial data under Grad’s cutoff assumption. Furthermore, Alexandre-Villani extended this understanding to the case that encompasses long-range interaction kernels [1]. In terms of the initial boundary problem, Mischler proved the well-posedness of the Boltzmann equation with Maxwell reflection boundary condition for the cutoff case in [47]. Within the realm of classical solutions, Ukai in [54] achieved a breakthrough with the first attainment of global well-posedness in the close-to-equilibrium sense, specifically for collision kernels featuring cut-off hard potentials. By using the nonlinear energy method, the same type of result for the soft potential case for both a periodic domain and for whole space was proved by Guo [25, 27]. In the absence of the Grad’s cutoff assumption, the existence and regularity of global classical solution near the equilibrium for the whole space were obtained by Gressman-Strain [24] and Alexandre-Morimoto-Ukai-Xu-Yang [51]. Recent years have seen significant strides in understanding strong and mild solutions of Boltzmann equations within bounded domains under various boundary conditions [18, 19, 29, 43]. For further exploration, we refer readers to additional pertinent progress concerning the regularity and other types of Boltzmann models [34, 35, 49, 50].

Previous results of “hydrodynamic limit” The hydrodynamic limit is the limiting process that connects the kinetic equation (scaled Boltzmann equation) with the fluid equation (Euler or Navier-Stokes equation). This concept can be traced back to Maxwell and Boltzmann, who initially founded the kinetic theory. The project of studying the hydrodynamic limit was then specifically formulated and addressed by Hilbert [33]. It aims to derive the fluid models as particles undergo an increasing number of collisions, causing the Knudsen number approaches to vanish.

Based on the existence of renormalized solutions [1, 16], one type of framework for studying the hydrodynamic limit pertains to weak solutions, particularly proving that the renormalized solution of the Boltzmann equation converges to the weak solution of the Euler or Navier-Stokes equations. In [6], Bardos-Golse-Levermore started with the formal derivation of the fluid equations, including compressible Euler equations, and incompressible Euler and Navier-Stokes equations. They also initialed the so-called BGL program to justify Leray’s solutions of the incompressible Navier-Stokes equations from renormalized solutions [5], which was somewhat completed by Golse-Saint Raymond under the cutoff assumption [22, 23]. A similar result was obtained by Arsenio [3] for the non-cutoff case as well. More results following this methodology can be found in [38, 40, 44, 46].

Another aspect of studying the hydrodynamic limit is from the perspective of classical solutions. The classical compressible Euler and Navier-Stokes equations can be formally derived from the scaled Boltzmann equation through the Hilbert and Chapman-Enskog expansions. The rigorous justification behind the asymptotic convergence was initially established by Caflish for the compressible Euler equations [12] and by De Masi-Esposito-Lebowitz for the incompressible Navier-Stokes equations [14]. By taking advantage of the energy method, Guo-Jang-Jiang also attained significant progress in understanding the acoustic limit [31, 32, 36]. Additionally, research on strong solutions near equilibrium is another avenue of exploration in the hydrodynamic limit. Nishida [48] established local-in-time convergence to the compressible Euler equations, while Bardos-Ukai in [7], as well as Gallagher-Tristani in their recent work [21], derived solutions for the incompressible Navier-Stokes equations. For comprehensive previous results, we refer to [28, 10, 41, 42, 30, 37] and the references cited therein.

Previous results for “Boltzmann equation for gas mixture” In recent years, the study of the Boltzmann equation for the gas mixture has achieved tremendous progress due to its more physical significance in describing the real world. When the different species of gas particles possess the same mass, the linearized Boltzmann operator can be completely decoupled such that the usual tools in studying the Boltzmann equation of a single species can still be applied, for instance, in [26], Guo first studied the well-posedness of the Vlasov-Maxwell-Boltzmann system of two types of gases around the Maxwellian, the decay rate of which was obtained by Wang in [56]. Aoki-Bardos-Takata studied the existence of the Knudsen layer of the Boltzmann equation for a gas mixture with the zero bulk velocity in [2], which was further extended by Bardos-Yang in [8] for the case of general drifting velocity. On the other hand, when the mass of the gas particles of different species are not identical, we refer to the work by Sotirov-Yu in [53], where they studied the Boltzmann equation for gas mixture in one space dimension via the Green’s function, as well as the work by Briant-Daus in [11] showing the existence of the solution to Boltzmann equation for the gas mixture around the bi-Mawellian.

In contrast with the Boltzmann equation of one single species, there are few results concerning the hydrodynamic limit of the Boltzmann equation of gas mixture. It is worth mentioning the recent work by Wu-Yang [57], where, for the scaling case $\textup{St}=O(1)$ and $\textup{Kn}_{l}=o(\varepsilon)$ in (1.4), the two-fluid compressible Euler equations was rigorously justified from the Boltzmann equation of gas mixture via the Hilbert expansion method. For more hydrodynamic limits from the Boltzmann equation for gas mixture, especially the formal derivation, we refer the readers to [17, 9] and the references therein.

Mathematical challenges and our contributions Motivated by the previous results above, this paper aims to study the hydrodynamic limit of the Boltzmann equation for gas mixture, mainly focusing on the following two aspects:
(i) In terms of formal derivation, this paper presents several novel hydrodynamic models, which, to the best of our knowledge, are derived for the first time from the Boltzmann equation for gas mixtures (BEGM equation) (1.5), employing various scalings and expansion forms. Notably, these include the Euler-Fourier equations coupled with the Navier-Stokes equations, among others (see Theorem 2.1 for additional models). To achieve this, the delicate dimensionless analysis in Section 1.2 has been applied, particularly focusing on describing the degree of collision between different species of particles. Inspired by [4], we introduce a dimensionless parameter $\delta(l,n)$ in (1.3) to accurately quantify the strength of interactions between two species.
(ii) In terms of the rigorous justification, we prove the two-fluid incompressible Navier-Stokes-Fourier system (2.21) as the hydrodynamic model stemming from the BEGM equation (2.15) under specific scaling and expansion forms (refer to Theorem 2.3). Here, we apply the methodology following the framework of the BGL program mentioned above as well as leverage the known hydrodynamic limit result of the single-species Boltzmann equation [42]. More precisely, by selecting the particular expansion form (2.1) of the solution to the BEGM equation (2.15), we can derive the reminder system (2.16) through prescribed scaling, the solution to which is shown to converge to the function constructed by the solutions to the incompressible Navier-Stokes-Fourier system (2.21), as $\varepsilon\to 0$. To this end, the key difficulty lies in establishing the uniform energy estimates of the solution to the reminder system (2.16); to address this, we apply the well-known Macro-Micro decomposition [45] and take full advantage of the dissipative structure of the linearized Boltzmann operator to close the refined energy estimates, thereby facilitating the proof of the limiting process through compact arguments.

(i) $A\lesssim B$ denotes $A\leq CB$ for some generic constants $C>0$ and $A\thicksim B$ denotes that there exist two generic constants $C_{1},C_{2}>0$ such that $C_{1}A\leq B\leq C_{2}A$.

(ii) $\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})$ is the multi-index in $\mathbb{N}^{3}$ with $|\alpha|=\alpha_{1}+\alpha_{2}+\alpha_{3}$. The $\alpha^{th}$ partial derivative denoted by

$\alpha\leq\tilde{\alpha}$ means each component of $\alpha\in\mathbb{N}^{3}$ is not greater than that of $\tilde{\alpha}$. $\alpha<\tilde{\alpha}$ means $\alpha\leq\tilde{\alpha}$ and $|\alpha|<|\tilde{\alpha}|$.

(iii) $L^{p}$ denotes the usual Lebesgue space, namely,

endowed with the norms:

where the weight function $\omega$ is either $1$ or the collisional frequency $\nu(v)$ given by

$L^{p}_{x}L^{q}_{v}(\omega)$ denotes Lebesgue space for $p,q\in[1,+\infty]$ (if $p=q$, $L^{p}_{x,v}(w)=L^{p}_{x}L^{p}_{v}(w)$) endowed with the norms:

$H^{s}_{x}L^{2}_{v}$ and $H^{s}_{x}L^{2}_{v}(\nu)$ denote the Sobolev spaces endowed with the norms:

(iv) $\langle\cdot,\cdot\rangle_{x,v}$, $\langle\cdot,\cdot\rangle_{v}$ and $\langle\cdot,\cdot\rangle_{x}$ denote the inner product in $L^{2}_{x,v}$, $L^{2}_{v}$ and $L^{2}_{x}$.

Our main results will be stated first of all.

Inspired by the Galilean transformation, we seek a special form of the solution to (1.5) around the global Maxwellian $M:=\mathcal{M}_{[1,0,1]}(v)$,

and by substituting (2.1) into (1.5), we obtain the remainder systems satisfied by $(g^{\varepsilon}_{1},g^{\varepsilon}_{2})$,

where the linearized Boltzmann operator $\hat{L}$ and the bilinear symmetric operator $\hat{\Gamma}$ are given by

for $l,n\in\{1,2\}$. Note that [20], for $i=1,2,3$,

therefore, denote the vectors $A=(A_{i})$ and tensors $B=(B_{ij})$ with $i,j=1,2,3$ as follows:

then $A_{i},B_{ij}\in\text{Ker}^{\perp}\hat{L}$. Also, $\hat{A}=(\hat{A}_{i})$ and $\hat{B}=(\hat{B}_{ij})$ are the unique $\hat{L}^{-1}$ of $A,B$ in $\text{Ker}^{\perp}\hat{L}$, i.e.,

Our first main theorem is about the hydrodynamic systems that can be derived from the BEGM equation (2.2) with various scalings.

Then, in terms of a particular case derived in Theorem 2.1 above, namely (2.9), we will provide the rigorous justification of the well-posedness and the hydrodynamic limit process.

More precisely, selecting $c_{1}=c_{2}=1$ in the scaled BEGM equation (1.5) leads to

if we choose $r=1$ in the expansion form (2.1), the reminder system satisfied by $(g^{\varepsilon}_{1},g^{\varepsilon}_{2})$ becomes