Stationary quantum BGK model for bosons and fermions in a bounded interval

TL;DR

This paper proves the existence of solutions for a stationary quantum BGK model describing bosons and fermions in a bounded interval, ensuring the quantum states are well-defined without phase transitions.

Contribution

It establishes the existence of mild solutions for the quantum BGK model with boundary data, addressing quantum-specific equilibrium states and phase transition issues.

Findings

Existence of mild solutions for quantum BGK models

Well-defined quantum local equilibrium states

No phase transition issues in the solution space

Abstract

In this paper, we consider the existence problem for a stationary relaxational models of the quantum Boltzmann equation. More precisely, we establish the existence of mild solution to the fermionic or bosonic quantum BGK model in a slab with inflow boundary data. Unlike the classical case, it is necessary to verify that the quantum local equilibrium state is well-defined, and the transition from the non-condensed state to the condensated state (Bosons), or from the non-saturated state to the saturated state (Fermions) does not arise in our solution space.

Equations

$$p_1\frac{\partial f}{\partial x}$$

$$f(0,p)=f_L(p)\text{for}{p}_1>0,f(1,p)=f_R(p)\text{for}{p}_1<0.$$

$$\begin{array}{r}{N}_f(x)={\int }_{{\mathbb{R}}^3}f(x,p)dp,P_f(x)={\int }_{{\mathbb{R}}^3}f(x,p)pdp,E_f(x)={\int }_{{\mathbb{R}}^3}f(x,p)|p{|}^{2}dp.\end{array}$$

$$\begin{array}{r}\frac{N_f(x)}{{\left(E_f(x)-\frac{|P_f(x){|}^2}{N_f(x)}\right)}^{\frac{3}{5}}}=\frac{{\int }_{{\mathbb{R}}^3}\frac{1}{e^{|p{|}^2+c(x)}\pm 1}dp}{{\left({\int }_{{\mathbb{R}}^3}\frac{|p{|}^2}{e^{|p{|}^2+c(x)}\pm 1}dp\right)}^{\frac{3}{5}}},\end{array}$$

$$a(x)$$

$${\beta }_{\mathcal{K}}(c)=\frac{{\int }_{{\mathbb{R}}^3}\frac{1}{e^{|p{|}^2+c}\pm 1}dp}{{\left({\int }_{{\mathbb{R}}^3}\frac{|p{|}^2}{e^{|p{|}^2+c}\pm 1}dp\right)}^{\frac{3}{5}}}.$$

$$\displaystyle\begin{cases}\mathcal{B}_{1}(f)=\left(e^{a(x)\big{|}p-\frac{P(x)}{N(x)}\big{|}^{2}+c(x)}-1\right)^{-1},~{}&\textit{if}\quad\frac{N}{\left(E-\frac{|P|^{2}}{N}\right)^{3/5}}\leq\beta_{\mathcal{B}}(0),\cr\mathcal{B}_{2}(f)=\left(e^{a(x)\big{|}p-\frac{P(x)}{N(x)}\big{|}^{2}}-1\right)^{-1}+k(x)\delta_{p=\frac{P(x)}{N(x)}},\quad&\textit{otherwise},\end{cases}$$

$$k(x)=N(x)-{\beta }_{\mathcal{B}}(0){\left(E(x)-\frac{P(x{)}^2}{N(x)}\right)}^{\frac{3}{5}}.$$

$$\displaystyle\begin{cases}\mathcal{F}_{1}(f)=\left(e^{a(x)\big{|}p-\frac{P(x)}{N(x)}\big{|}^{2}+c(x)}+1\right)^{-1},~{}&\textit{if}\quad\frac{N}{\left(E-\frac{|P|^{2}}{N}\right)^{3/5}}<\beta(-\infty),\cr\mathcal{F}_{2}(f)=\chi_{\big{|}p-\frac{P(x)}{N(x)}\big{|}\leq\left(\frac{3N(x)}{4\pi}\right)^{\frac{1}{3}}},\quad&\textit{otherwise},\end{cases}$$

$$\underset{x}{\sup }||f|{|}_{L_2^1}$$

$$||f|{|}_{L_2^{\infty }}$$

$$\displaystyle\beta_{\mathcal{B}}^{-1}=\big{(}\beta_{\mathcal{B}}\big{|}_{[0,\infty)}\big{)}^{-1},\quad\beta_{\mathcal{F}}^{-1}=\big{(}\beta_{\mathcal{F}}\big{|}_{(-\ln 3,\infty)}\big{)}^{-1}.$$

$$f_{LR}(p)=f_L(p)1_{p_1>0}+f_R(p)1_{p_1<0},$$

$$\begin{array}{rl}& a_u=2{\int }_{{\mathbb{R}}^3}{f}_{LR}dp,a_l={\int }_{{\mathbb{R}}^3}{e}^{-\frac{a_u}{\tau |p_1|}}{f}_{LR}dp,a_s={\int }_{{\mathbb{R}}^3}\frac{1}{|p_1|}{f}_{LR}dp,\\ & c_u=2{\int }_{{\mathbb{R}}^3}{f}_{LR}|p{|}^{2}dp,c_l={\int }_{{\mathbb{R}}^3}{e}^{-\frac{a_u}{\tau |p_1|}}{f}_{LR}|p{|}^{2}dp,c_s={\int }_{{\mathbb{R}}^3}\frac{1}{|p_1|}{f}_{LR}|p{|}^{2}dp,\end{array}$$

$$k=\left({\int }_{p_1>0}{e}^{-\frac{a_u}{\tau |p_1|}}{f}_L(p)|p_1|dp\right)\left({\int }_{p_1<0}{e}^{-\frac{a_u}{\tau |p_1|}}{f}_R(p)|p_1|dp\right).$$

$$\begin{array}{rl}f(x,p)& =e^{-\frac{1}{\tau |p_1|}{\int }_0^x{N}_f(y)dy}{f}_L(p)\\ & +\frac{1}{\tau |p_1|}{\int }_0^x{e}^{-\frac{1}{\tau |p_1|}{\int }_y^x{N}_f(z)dz}{N}_f(y)\mathcal{K}(f)dy\text{if}{p}_1>0,\end{array}$$

$$\begin{array}{rl}f(x,p)& =e^{-\frac{1}{\tau |p_1|}{\int }_x^1{N}_f(y)dy}{f}_R(p)\\ & +\frac{1}{\tau |p_1|}{\int }_x^1{e}^{-\frac{1}{\tau |p_1|}{\int }_x^y{N}_f(z)dz}{N}_f(y)\mathcal{K}(f)dy\text{if}{p}_1<0.\end{array}$$

$$f_{LR}\ge 0,$$

$$f_{LR},\frac{1}{|p_1|}{f}_{LR}\in L_2^1,$$

$${\int }_{{\mathbb{R}}^2}{f}_L{p}_{i}d{p}_{2}d{p}_3={\int }_{{\mathbb{R}}^2}{f}_R{p}_{i}d{p}_{2}d{p}_3=0.(i=2,3)$$

$$\frac{{a_u}^{\frac{8}{5}}}{k^{\frac{3}{5}}}<{\beta }_{\mathcal{B}}(0)(\text{mbox}Boson),\frac{{a_u}^{\frac{8}{5}}}{k^{\frac{3}{5}}}<{\beta }_{\mathcal{F}}(-\ln 3)(\text{mbox}Fermion).$$

$$a_l\le N(x)\le a_u,c_l\le E(x)\le c_u,$$

$$E(x)N(x)-|P(x){|}^2\ge k.$$

$$f_L(p)=C_L{1}_{r_1\le p_1\le r_2}{e}^{-\frac{|p_2{|}^2}{2}-\frac{|p_3{|}^2}{2}},f_R(p)=C_R{1}_{-r_2\le p_1\le -r_1}{e}^{-\frac{|p_2{|}^2}{2}-\frac{|p_3{|}^2}{2}},$$

$$a_u$$

$$=2\left(C_L{\int }_{r_1}^{r_2}1d{p}_1{\left({\int }_{-\infty }^{\infty }{e}^{-\frac{|p_2{|}^2}{2}}d{p}_2\right)}^2+C_R{\int }_{-r_2}^{-r_1}1d{p}_1{\left({\int }_{-\infty }^{\infty }{e}^{-\frac{|p_2{|}^2}{2}}d{p}_2\right)}^2\right)$$

$$=4\pi (C_L+C_R)(r_2-r_1).$$

$${\int }_{p_1>0}{e}^{-\frac{a_u}{\tau |p_1|}}{f}_L(p)p_{1}dp$$

$$=C_L{e}^{-\frac{4\pi (C_L+C_R)(r_2-r_1)}{\tau r_1}}{\int }_{r_1}^{r_2}{p}_{1}d{p}_1{\left({\int }_{-\infty }^{\infty }{e}^{-\frac{|p_2{|}^2}{2}}d{p}_2\right)}^2$$

$$=\pi C_L{e}^{-\frac{4\pi (C_L+C_R)(r_2-r_1)}{\tau r_1}}(r_2^2-r_1^2),$$

$${\int }_{p_1<0}{e}^{-\frac{a_u}{\tau |p_1|}}{f}_R(p)|p_1|dp$$

$$k$$

$$={\pi }^2{C}_L{C}_R{e}^{-\frac{8\pi (C_L+C_R)(r_2-r_1)}{\tau r_1}}(r_2^2-r_1^2{)}^2.$$

Full text

1. Introduction

The stationary quantum BGK model [27, 36, 37, 39, 40, 42, 48, 49, 51] in a bounded interval reads

[TABLE]

subject to boundary conditions:

[TABLE]

The momentum distribution function $f(x,p)$ depends on the position $x\in[0,1]$ and the momentum $p\in\mathbb{R}^{3}$. The Knudsen number $\tau>0$ measures how rarefied the gas system is, and is defined by the ratio between the characteristic length and mean free path. Throughout this paper, $\mathcal{K}$ denotes the local equilibrium of the system. For bosonic case, it represents the Bose-Einstein distribution without condensation, and in the fermionic case, it represents the non-saturated Fermi-Dirac distribution, which will be defined below. To present the exact form of $\mathcal{K}$, we first define the macroscopic mass, momentum and energy:

[TABLE]

We then introduce the equilibrium parameter $a$ and $c$ defined by ( $+$ sign is for fermion and $-$ sign is for boson, see [7, 39]):

[TABLE]

and

[TABLE]

Note that $c$ is determined implicitly. For the later convenience, we define

[TABLE]

The relations (1.4) and (1.5) arise from the requirement that $\mathcal{F}$, $\mathcal{B}$ must share the same mass, momentum and energy with $f$ (See [7]): Now we are ready to define the local quantum equilibriums. [7, 15, 32, 39, 52]

$\bullet$   Bose-Einstein distribution: The local equilibrium for bosons is defined as follows:

[TABLE]

where

[TABLE]

The dirac delta function corresponds to Bose-Einstein condensation. $\mathcal{B}_{1}$ corresponds to the non-condensation case, while $\mathcal{B}_{2}$ is referred as the condensation case.

$\bullet$   Fermi-Dirac distribution: The local equilibrium for fermions is defined as follows:

[TABLE]

where $\chi_{A}$ denotes the characteristic function on $A$, and the second case of $\mathcal{F}_{2}$ is called the saturated Fermi-Dirac distribution.

Throughout this paper, we will use $\mathcal{B}(f)$ to denote the Bose-Enstein distribution without condensation $\mathcal{B}_{1}(f)$, while $\mathcal{F}(f)$ is used to denote the non-saturated Fermi-Dirac distribution $\mathcal{F}_{1}(f)$. Also, $\mathcal{K}(f)$ denotes either $\mathcal{B}(f)$ or $\mathcal{F}(f)$.

1.1. Brief history

The slab problem corresponds to the situation where there is a gas flow between two parallel gas-emitting plates of infinite size. This arise often in science and engineering, and attracted the interest of many researchers. In the case of the Boltzmann equation, the first mathematical study can be traced back to [2], where the existence of a measure valued solution were investigated. In the framework of weak solutions, Arkeryd and Nouri considered the existence of $L^{1}$ solution for the inflow boundary conditions in [4, 6] and for the diffusive reflection conditions in [4]. These results were extended to gas mixture problem by Brull [12, 13]. Gomeshi studied the existence of unique mild solutions under the condition that the Knudsen number is sufficiently large in [24]. For the related 3d problem near equilibrium, see [19, 20].

In the case of BGK type model, Ukai studied stationary Boltzmann BGK model in slab for fixed large boundary data in [47] using a Schauder type fixed point theorem. Nouri [39] established the existence of weak solutions for the stationary quantum BGK model with a discretized condensation term in a slab. Bang and Yun obtained the existence and uniqueness of mild solutions for the ES-BGK model under the assumption that gas is sufficiently rarefied and inflow datas are not concentrated on $p=0$ in [8].

The mathematical reserach for the quantum relaxation model has just started, and the literature remains extremely limited. The first mathmatical study was carried out by Nouri as mentioned above. In [9, 10], Braukhoff obtained analytic solutions of quantum BGK type model arising in the study of ultracold fermionic clouds. The global existence and asymptotic behavior of fermionic quantum BGK model near a global Fermi-Dirac distribution were studied by the authors in [7]. Presently, authors are not aware of any further analytical results on the quantum BGK models. We refer to [22, 26, 28, 29, 37, 43, 48, 50] for numerical studies on the quantum BGK model.

Quantum Boltzmann equation, on the other hand, has seen more progress. We refer to [11, 16, 17, 18, 31, 32, 33, 35, 34, 38, 44] for homogeneous problem, and [1, 3, 5, 14] for inhomogeneous problems.

1.2. Notations

We define notations and norms that are frequently used throughout this paper.

• •

Throughout this paper, we fix $\mathcal{B}(f)=\mathcal{B}_{1}(f)$ and $\mathcal{F}(f)=\mathcal{F}_{1}(f)$. Also, $\mathcal{K}(f)$ denotes either $\mathcal{B}(f)$ or $\mathcal{F}(f)$.

• •

Every constants $C$ are defined generically. We also use $C_{a,b,\cdots}$ when it is necessary to explicitly show the dependence on $a,b,\cdots$. Especially, we denote $C_{l,u}$ when the constant depends only on the constants defined in (2.2) and $k$.

• •

When there’s no risk of confusion, we suppress the dependence of the macroscopic fields on $f$, and denote $N$, $P$, $E$ instead of $N_{f}$, $P_{f}$ and $E_{f}$.

• •

We define our weighted $L^{1}$ norm and weighted $L^{\infty}$ as follows:

[TABLE]

• •

We use the following notation (See Ch $3$):

[TABLE]

This paper is organized as follows. In Section 2 we present the main result and give an example of boundary data satisfying the assumption of main theorem. Section 3 is devoted to the fixed point setup of the problem. We define the solution space and prove that the equilibrium is well defined in this space. Some useful estimates are also introduced in this section. In Section 4, we establish that the solution operator maps the solution space into itself. We prove the main theorem in the final Section 5 by showing that the solution operator is a contraction mapping.

2. Main result

In this section we present our main results. For brevity we denote

[TABLE]

and define the following quantities:

[TABLE]

and

[TABLE]

Definition 2.1**.**

We say that $f\in L^{1}_{2}([0,1]\times\mathbb{R}^{3}_{p})$ is a mild solution of (1.1) if $f$ satisfies

[TABLE]

and

[TABLE]

Now we state our main results.

Theorem 2.2**.**

Assume $f_{L}$ and $f_{R}$ satisfy the following conditions:

1. (1)

Boundary data are non-negative:

[TABLE] 2. (2)

Boundary data satisfy the following integrability conditions:

[TABLE] 3. (3)

Contributions of the inflow from the boundary in $p_{2}$ and $p_{3}$ directions are negligible:

[TABLE]

We assume further that

[TABLE]

Then for sufficiently large $\tau$, there exists a unique non-negative mild solution $f$ of (1.1) satisfying

[TABLE]

and

[TABLE]

Remark 2.3*.*

(1) The meaning of assumption (2.4) will be considered in Chapter 3. (2) Note that in (2.4), the fermion case is restricted to $\beta(-\ln 3)$. This is because we don’t know yet whether $\beta(c)$ for fermion is a strictly monotone decreasing function in the whole range, even though the numerics indicate in that way. This is left as a future preject. (3) Extending this result to include the condensated state (Boson) and the saturated state (Fermion) will be interesting, and is left for the future.

Before we move on the the proof of the theorem, we present a simple example of boundary data which satisfies the assumption of Theorem 2.2 (1), (2), (3) and (2.4) for bosons. Example for fermionic particles can be constructed similarly. We define

[TABLE]

for some $C_{L},C_{R}>0$ and $r_{1},r_{2}>0$ to be determined soon. Since it can be readily checked that they satisfy the conditions $(1)$, $(2)$, $(3)$ of Theorem 2.2, we check the condition (2.4) only. We first compute $a_{u}$ as

[TABLE]

We then compute

[TABLE]

and, similarly,

[TABLE]

to get

[TABLE]

Hence we derive

[TABLE]

This shows that a proper choice of $C_{L}$, $C_{R}$ and $r_{1}$, $r_{2}$ gives the desired condition.

3. Fixed point set-up

We define the solution space by

[TABLE]

endowed with the metric $d(f,g)=\sup_{x\in[0,1]}||f-g||_{L^{1}_{2}}$.

• •

($\mathcal{A}$) $f$ is non-negative:

[TABLE]

• •

($\mathcal{B}$) Mass and energy satisfy

[TABLE]

• •

($\mathcal{C}$) $f$ satisfies

[TABLE]

3.1. determination of $a$,$b$ and $c$

We first verify that for any distribution function $f$ that lies in $\Lambda$, the nonlinear relations (1.4) and $\eqref{a,c2}$ admit a unique set of solution $a$ and $c$, so that the local equilibrium $\mathcal{K}(f)$ is well defined. It is clear that $a$ is uniquely determined by (1.5) once the unique existence of $c$ is determined from (1.4). Note that, in view of the definition of (1.6), the nonlinear relation (1.4) is rewritten by

[TABLE]

Therefore, it is sufficient to show that $\beta_{\mathcal{F}}$ is a monotone function, and r.h.s of (3.1) lies in the range of $\beta_{\mathcal{F}}$. For this we recall the following lemma:

Lemma 3.1**.**

[7, 32]* The function $\beta_{\mathcal{B}}$ and $\beta_{\mathcal{F}}$ defined in (1.6) satisfy the following properties.*

1. (1)

$\beta_{\mathcal{B}}$* is strictly decreasing on $[0,\infty)$ and its range is $(0,\beta(0)]$.* 2. (2)

$\beta_{\mathcal{F}}$* is strictly decreasing on $(-\ln 3,\infty)$ and its range is $(0,\beta(-\ln 3))$.*

Proof.

Proof for (1) can be founded in [32], and the proof for (2) can be founded in [7]. ∎

Lemma 3.2**.**

Assume $f\in\Lambda$. Then $a$ and $c$ are uniquely determined from (1.4) and (1.5), and $\mathcal{K}(f)$ is well-defined. Moreover, $\mathcal{K}(f)$ is not condensated (Bosonic case) nor saturated (Fermionic case). That is, no transition from $\mathcal{B}_{1}(f)$ to $\mathcal{B}_{2}(f)$, or $\mathcal{F}_{1}(f)$ to $\mathcal{F}_{2}(f)$ occurs.

Proof.

$\bullet$ (Boson): We note from (2.4)1, (2.5) and (2.6) that

[TABLE]

Therefore, in view of Lemma 3.1, the interval \big{(}0,N^{\frac{8}{5}}/(EN-|P|^{2})^{\frac{3}{5}}\big{]} lies in the range of $\beta_{\mathcal{B}}$, and we can fix a unique $c$ satisfying (3.1) by the monotonicity of $\beta_{\mathcal{B}}$ obtained in Lemma 3.1, which in turn leads to the determination of $a$ by (1.5). Note also from (1.7) that this guarantees that the condensation does not arise if $f\in\Lambda$. In conclusion, $\mathcal{B}(f)$ is well-defined for $f\in\Lambda$.

$\bullet$ (Fermion): Similarly, combining second condition of (2.4)2 with (2.5) and (2.6) yields

[TABLE]

for fermion case. Therefore, by the exactly same argument, we can conclude that $a$ and $c$ are uniquely determined for $f\in\Lambda$, and the transition from the non-saturated state $\mathcal{F}_{1}(f)$ to the saturated state $\mathcal{F}_{2}(f)$ does not happen. ∎

In view of this consideration, we can uniquely determine $c$ satisfying (1.4). For brevity, we slightly abuse the notation to denote as

[TABLE]

and $\beta_{\mathcal{K}}^{-1}$ will denote

[TABLE]

We first consider the range of $a$ and $c$ when they are constructed from an element of $\Lambda$.

Lemma 3.3**.**

Let $f\in\Lambda$, and the boundary data $f_{LR}$ satisfy (2.4). Define $a_{*}$, $a^{*}$, $c_{*}$, $c^{*}$ by

[TABLE]

and

[TABLE]

Then, the equilibrium parameter $a$ and $c$ satisfy

[TABLE]

and

[TABLE]

In the case of fermion, we note that $-\ln 3\leq c_{*}$.

Proof.

(1) Estimates for $c$: From (1.4) and (1.6), we have

[TABLE]

Since $f\in\Lambda$ we have $a_{l}\leq N\leq a_{u}$ , $E\leq c_{u}$ and $EN-|P|^{2}\geq k$, so that

[TABLE]

Now, since Lemma 3.1 implies that $\beta_{\mathcal{K}}^{-1}$ is strictly decreasing, and the closed interval \big{[}{a_{l}^{{8}/{5}}}/{\left(a_{u}c_{u}\right)^{{3}/{5}}},{a_{u}^{{8}/{5}}}/{k^{{3}/{5}}}\big{]} lies in the range of $\beta_{\mathcal{K}}(c)$, we have

[TABLE]

to get the desired estimates for $c$.

(2) Estimates for $a$: We recall (1.5). Then from $a_{l}\leq N\leq a_{u}$ and estimates of $c$ established above, we find

[TABLE]

For boson case, $c_{*}\geq 0$ implies the positivity of $a_{*}$. For fermion case, positivity of $a_{*}$ is trivial. This completes the proof. ∎

3.2. Solution operator

By Lemma 3.2, the following solution operator $\Phi$ is well-defined on $\Lambda$:

Definition 3.4**.**

We defind our solution operator $\Phi$ as

[TABLE]

where

[TABLE]

and

[TABLE]

In the remaining sections, we show that $\Phi$ has a unique fixed point in $\Lambda$ if $\tau$ is sufficiently large. We first prove several estimates on the quantum local equilibrium.

Lemma 3.5**.**

Let $f\in\Lambda$, then there exists a constant $C_{l,u}$ depending only on the quantities in (2.2) and $k$ such that

[TABLE]

Proof.

We only consider $\mathcal{B}(f)|p|^{2}$. By an explicit computation, we have

[TABLE]

In last line, we used $a^{2}\leq 2|a-b|^{2}+2b^{2}$. Then, we observe

[TABLE]

which follows from $|P|\leq a_{u}+c_{u}$, and use the boundedness of $\frac{x^{2}+1}{e^{ax^{2}+c}-1}$ to get

[TABLE]

Now, since $|a-b|^{2}\geq a^{2}/2-b^{2}$, we have

[TABLE]

We then use (3.5) again to get the desired result:

[TABLE]

∎

The following decay estimates are crucially used throughout the paper. The proof can be found in [8]. We provide detailed proof for reader’s convenience.

Lemma 3.6**.**

We have

[TABLE]

Proof.

We divide the integral domain of $p_{1}$ into three parts:

[TABLE]

Integrating in $y$ first, we get

[TABLE]

We start similarly for $II$:

[TABLE]

Then we expand $e^{-\frac{a_{l}x}{\tau|p_{1}|}}$ in Taylor expansion to obtain

[TABLE]

Then, since

[TABLE]

We can bound $II$ by

[TABLE]

where we used $(\tau^{n}-1)/\tau^{n}\leq 1$ in second line. Finally, by using $e^{-\frac{a_{l}(x-y)}{\tau|p_{1}|}}<1$, we estimate $III$ as

[TABLE]

Combining the above estimates gives the desired results for sufficiently large $\tau$:

[TABLE]

∎

4. $\Phi$ maps $\Lambda$ into $\Lambda$

The main result of this section is stated in the following proposition.

Proposition 4.1**.**

Let $f_{LR}$ satisfies the assumptions in Theorem 2.2. Then, there exists $\tau_{0}$ such that if $\tau>\tau_{0}$, then the solution operator $\Phi$ maps $\Lambda$ into $\Lambda$.

Proof.

The proof is given in the following Lemma 4.1, 4.2, 4.3 and 4.5. ∎

Lemma 4.1**.**

Let $f\in\Lambda$. Assume $f_{LR}$ satisfies all the assumptions of the Theorem 2.2. Then $\Phi(f)$ satisfies the following estimates:

[TABLE]

Proof.

Thanks to Lemma 3.3, the local equilibrium is strictly positive:

[TABLE]

Therefore, we have from (3.3) and (3.4) that

[TABLE]

which gives desired result. ∎

Lemma 4.2**.**

Let $f\in\Lambda$. Assme $f_{LR}$ satisfies all the assumptions of the Theorem 2.2, then $\Phi(f)$ also satisfies the following inequality.

[TABLE]

Proof.

We only consider the second one. We see from (4.1) that

[TABLE]

Using $N_{f}\leq a_{u}$, we see that

[TABLE]

We then integrate with respect to $|p|^{2}dp$ to get the desired results:

[TABLE]

∎

Lemma 4.3**.**

Let $f\in\Lambda$. Assume $f_{LR}$ satisfies all the assumptions of the Theorem 2.2. Then $\Phi(f)$ satisfies the following estimates:

[TABLE]

for sufficiently large $\tau$.

Proof.

We only consider the second inequality. We integrate (3.3) with respect to $|p|^{2}dp$ to get

[TABLE]

Since $N_{f}\geq a_{l}$ and $x\geq 0$, we can estimate first term as

[TABLE]

We then recall Lemma 3.5 and use $a_{l}\leq N_{f}\leq a_{u}$ to bound the second term as

[TABLE]

Therefore, we have from Lemma 3.6 that

[TABLE]

Similarly, we can derive

[TABLE]

so that

[TABLE]

which gives the desired result for sufficiently large $\tau$. ∎

Lemma 4.4**.**

Let $f\in\Lambda$. Assume $f_{LR}$ satisfies all the assumptions of the Theorem 2.2. Then, for sufficiently large $\tau$, we have

[TABLE]

for $i=2,3$.

Proof.

We only consider the case $i=2$. For this, we integrate (3.3) with respect to $p_{2}dp_{2}dp_{3}$:

[TABLE]

We note that the first term in r.h.s vanishes due to the assumption (3) of Theorem 2.2:

[TABLE]

For the second term, we use $a_{l}\leq N_{f}\leq a_{u}$ and employ Lemma 3.5 to derive

[TABLE]

Now we integrate with respect to $dp_{1}$ on $p_{1}>0$ to obtain

[TABLE]

Therefore, we have from Lemma 3.6 that

[TABLE]

Similarly, we have

[TABLE]

which gives the desired result. ∎

Lemma 4.5**.**

Let $f\in\Lambda$. Assume $f_{LR}$ satisfies all the assumptions of the Theorem 2.2. Then, for sufficiently large $\tau$, we have

[TABLE]

Proof.

We have from Cauchy-Schwarz inequality that

[TABLE]

where

[TABLE]

and

[TABLE]

From Lemma 4.3, we can bound $M_{1}$ as

[TABLE]

and $M_{2}$, $M_{3}$ decay as Lemma 4.4

[TABLE]

Therefore,

[TABLE]

On the other hand, we use $a^{2}-b^{2}=(a+b)(a-b)$ to get

[TABLE]

and observe that from the definition of $\Phi$, and property $(\mathcal{B})$ of $\Lambda$: $N_{f}\leq a_{u}$ that

[TABLE]

and

[TABLE]

We insert these lower bounds into (4.6) and recall the definition of $k$ in (2.3) to obtain

[TABLE]

From (4.4), (4.5) and (4.7), we have

[TABLE]

which, for sufficiently large $\tau$, gives the desired result. ∎

5. Continuity of quantum equilibrium $\mathcal{K}$

In this section, we establish the continuity property of the quantum equilibrium $\mathcal{K}$, which is crucially used to show the contractiveness of $\Phi$ in Section 5.

5.1. Transitional quantum local equilibrium $\mathcal{K}(\theta)$

In this subsection, we define a transitional quantum local equilibrium. We start with the convexity of our solution space.

Lemma 5.1**.**

Let $f,g\in\Lambda$, Then the linear combination $(1-\theta)f+\theta g$ lies in $\Lambda$ for $\theta\in[0,1]$.

Proof.

Since the conditions $(\mathcal{A})$ and $(\mathcal{B})$ of $\Lambda$ are trivially satisfied, we only consider $(\mathcal{C})$. For this, we define a functional $G$ by

[TABLE]

and a matrix $M$ by

[TABLE]

for $f\in\Lambda$. We note that

[TABLE]

Then, by Brum-Minkowski inequality, we have for $f,g\in\Lambda$

[TABLE]

Therefore, $\theta f+(1-\theta)g\in\Lambda$. ∎

We now define the transitional macroscopic fields constructed from the linear combination $\theta f+(1-\theta)g$ as

[TABLE]

for $\theta\in[0,1]$. Now, since we have shown in Lemma 5.1 that $\theta f+(1-\theta)g\in\Lambda$, the existence of the unique quantum equilibrium $\mathcal{K}(\theta)$:

[TABLE]

which shares the same mass, momentum and energy with $\theta f+(1-\theta)g$:

[TABLE]

is guaranteed by Lemma 3.2. We also recall from Lemma 3.3 that $a_{\theta}$ and $c_{\theta}$ are determined by

[TABLE]

and satisfy

[TABLE]

for some positive constants $a_{*}$, $a^{*}$, $c^{*}$ and $c_{*}$.

5.2. Derivatives of $\mathcal{F}(\theta)$

We now derive derivative estimates of $a_{\theta}$ and $c_{\theta}$, which will be needed later in the proof of the continuity estimate of $\mathcal{K}(\theta)$. We first need the following estimate of $\beta_{\mathcal{K}}$.

Lemma 5.2**.**

Let $f,g\in\Lambda$, then $\beta_{\mathcal{K}}$ defined in (1.6) satisfies

[TABLE]

where $C_{l,u}$ depends on constants of (2.2) and $k$.

Proof.

By definition given in (1.6), $\beta_{\mathcal{K}}$ is an infinitely differentiable function. On the other hand, Lemma 3.1 implies that $\beta^{\prime}_{\mathcal{K}}(c)<0$. Therefore, we see from Lemma 3.3 that $\beta^{\prime}_{\mathcal{K}}(c)$ is a strictly negative continuous function defined on a closed interval $[c_{*},c^{*}]$. Hence, there exists positive $C$ such that $|\beta_{\mathcal{K}}^{\prime}(c)|\geq C$, which gives the desired result. ∎

Lemma 5.3**.**

We have

[TABLE]

Proof.

Recall that $c_{\theta}$ is function of $N_{\theta}$, $P_{\theta}$ and $E_{\theta}$:

[TABLE]

(1) By an explicit computation, we get

[TABLE]

We then use $N_{\theta}\leq a_{u}$, $E_{\theta}\leq c_{u}$ and $N_{\theta}E_{\theta}-P_{\theta}^{2}\geq k$ together with Lemma 5.2 to obtain

[TABLE]

(2) Similarly, we compute

[TABLE]

Since $|P_{\theta}|\leq a_{u}+c_{u}$, we have

[TABLE]

(3) In an almost identical manner, we compute

[TABLE]

∎

Lemma 5.4**.**

We have

[TABLE]

Proof.

(1) We recall (5.1) and compute

[TABLE]

It then follows directly from from $a_{l}\leq N_{\theta}\leq a_{u}$, Lemma 3.3, and Lemma 5.3 that

[TABLE]

(2) In a similar manner, we have

[TABLE]

(3) Replacing $\frac{\partial}{\partial P_{\theta i}}$ by $\frac{\partial}{\partial E_{\theta}}$ in (2), we get the same result for $\frac{\partial a_{\theta}}{\partial E_{\theta}}$. ∎

5.3. Continuity of $\mathcal{K}$

We now prove the main result of the this section:

Proposition 5.1**.**

Let $f,g\in\Lambda$. Then the quantum equilibrium $\mathcal{K}$ satisfies following property:

[TABLE]

Proof.

We apply taylor’s theorem around $\theta=0$ to have

[TABLE]

so that

[TABLE]

To estimate the first integral, we compute

[TABLE]

From Lemma 3.5 we observe $\mathcal{K}(f)\leq C_{l,u}$ to obtain

[TABLE]

With these computation and Lemma 3.3, Lemma 5.3 and Lemma 5.4, we get

[TABLE]

Since $|P|\leq a_{u}+c_{u}$ and $N_{\theta}\geq a_{\ell}$, we find

[TABLE]

which, thanks to Lemma 3.5, gives

[TABLE]

Similarly, we have $(i=1,2,3)$

[TABLE]

and

[TABLE]

Substituting these estimates into (5.2) yields the desired result:

[TABLE]

∎

6. $\Phi$ is contractive in $\Lambda$

It remains to show that $\Phi$ is a contraction mapping in $\Lambda$ for sufficiently large $\tau$.

Proposition 6.1**.**

Let $f,g\in\Lambda$ and $f_{LR}$ satisfies all the assumptions of the Theorem 2.2, then, for sufficietly large $\tau$, $\Phi$ satisfies

[TABLE]

for some constant $0<\alpha<1$.

Proof.

We only estimate $\Phi^{+}$. Let

[TABLE]

where

[TABLE]

and

[TABLE]

$\bullet$ Estimates for $I(f)-I(g)$: Consider

[TABLE]

which, by mean value theorem, can be rewritten as

[TABLE]

for some $0<\mu<1$. Since we have $N_{f},N_{g}\geq a_{l}$, we see that

[TABLE]

where we used

[TABLE]

Now we integrate each term with respect to $(1+|p|^{2})dp$ on $p_{1}>0$:

[TABLE]

to get the desired result.

$\bullet$ Estimates for $II(f)-II(g)$: We split it as

[TABLE]

(i) Estimate of $II_{1}$: In a similar manner as in (6.1), we get

[TABLE]

In last line, we used $xe^{-x}\leq Ce^{-\frac{x}{2}}$. From this, we see that

[TABLE]

We then apply Lemma 3.5

[TABLE]

and Lemma 3.6 to obtaini

[TABLE]

(i) Estimate of $II_{2}$: The estimate for $II_{2}$ is treated similarly:

[TABLE]

(iii) Estimate of $II_{3}$: we integrate with respect to $(1+|p|^{2})dp$ on $p_{1}>0$ to obtain

[TABLE]

We then apply the continuity property of $\mathcal{K}$ in Proposition 5.1:

[TABLE]

Combining all these estimates, we get the desired estimate for $\Phi_{+}$:

[TABLE]

The corresponding estimate for $\Phi_{-}$ can be derived in an identical manner:

[TABLE]

This gives the desired contractive estimate for $\Phi$ when $\tau$ is sufficiently large. ∎

Acknowledgement: S.-B. Yun was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B03935955)