Global well-posedness of the quantum Boltzmann equation for bosons interacting via inverse power law potentials

We recall some existing mathematical results on the quantum Boltzmann equation in this subsection. The quantum Boltzmann equations include two models: one for bosons or Bose-Einstein(B-E) particles, the other for fermions or Fermi-Dirac(F-D) particles. The quantum Boltzmann equation for B-E particles is written by (1.7), (1.8) and (1.9) and usually referred as “BBE equation”. If the four “$+$” in (1.8) and the one “$+$” in (1.9) are replaced by minus sign “$-$”, we will get the quantum Boltzmann equation for F-D particles which is usually referred as “BFD equation”.

We begin with the mathematical results on BFD equations. In spatially homogeneous case $F=F(t,v)$, we refer the monograph Escobedo-Mischler-Valle [18] for global existence and weak convergence to equilibrium. We also refer Lu [31] and Lu-Wennberg [40] for weak and strong convergence to equilibrium of mild solutions. On the whole space $x\in\mathbb{R}^{3}$, Dolbeault [14] established global existence and uniqueness of mild solutions for globally integrable kernels. Lions [29] proved global existence of distributional solutions for locally integrable kernels. Alexandre [1] proved global existence of H-solutions for non-cutoff kernels that allow angular singularity given by inverse power law potentials. Based on the averaging compactness result in [34], on the torus $x\in\mathbb{T}^{3}$, Lu [35] proved global existence of weak solutions for kernels with very soft potentials. Recently in a perturbation framework, on the whole space $x\in\mathbb{R}^{3}$, Jiang-Xiong-Zhou [27] went further to consider the incompressible Navier-Stokes-Fourier limit from the BFD equation with hard sphere collisions.

Unlike Fermi-Dirac particles whose density has a natural bound $0\leq F\leq 1$ due to Pauli’s exclusion principle, density of Bose-Einstein particles may blow up in finite time, which corresponds to the intriguing phenomenon: Bose-Einstein condensation(BEC) in low temperature. As a consequence, the mathematical study of BBE equations is much more difficult than BFD equations. As a result, most of existing results on BBE equations are concerned with isotropic solutions $F(t,v)=F(t,|v|)$ in the homogeneous case with non-singular kernel, for instance hard sphere model or hard potentials with some cutoff. Lu [30] proved global existence of $L^{1}$ solutions for some kernels with strong cutoff assumption (not satisfied by the hard sphere model). Based on the results in [30], Lu [32] further established global existence of conservative distributional (measure-valued) isotropic solutions for some kernels including the hard sphere model. Also see Escobedo-Mischler-Velázquez [19] and Escobedo-Mischler-Velázquez [20] for some singular (near $|v|=0$) solutions.

In terms of long time behavior of the conservative measure-valued isotropic solutions, we refer Lu [33] for strong convergence to equilibrium and single point concentration. With some local condition on the initial datum, long time strong convergence to equilibrium was proved in [38] and [39]. Remarkably for any temperature and without any local condition on the initial datum, Cai-Lu [12] provided algebraic rate of strong convergence to equilibrium.

BEC is an interesting physical phenomenon that deserves deep mathematical understanding. It is fortunate to see many excellent mathematical results (for instance, Spohn [45], Lu [36, 37], Escobedo-Velázquez [21, 22]) on this topic.

Note that all the above results on BBE equations are concerned with isotropic solutions $F(t,v)=F(t,|v|)$. In the homogeneous case, local existence of anisotropic solutions was first proved in Briant-Einav [11], while global existence was first proved in Li-Lu [28] for very high temperature.

Note that all the above results on BBE equations are obtained in the homogeneous case. In the inhomogeneous case, there are at least two recent results: Bae-Jang-Yun [5] and Ouyang-Wu [44]. These two works have different focuses and will be revisited later.

In weak-coupling regime for bosons, the Boltzmann kernel $B_{\phi}$ depends on the potential function $\phi$ via (1.9). As mentioned before, existing results on BBE equations are mostly concerned with the hard sphere model. This amounts to taking the Dirac delta function as the potential function $\phi(x)=\delta(x)$ and thus $B_{\phi}=(v-v_{*},\sigma)=C|v-v_{*}|$ for some universal constant $C>0$. Observe that the kernel in the quantum case is the same as that in the classical case. Another physically relevant potential is the inverse power law $\phi(x)=|x|^{-p}$ which has been extensively studied in the research of classical Boltzmann equations but has rarely been considered in the quantum case. To our best knowledge, this article seems to be the first to study BBE equation with inverse power law.

Note that $p=1$ corresponds to the famous Coulomb potential which is the critical case for Boltzmann equation to be meaningful. In this article, we work in dimension 3 where $p=3$ is the critical value such that $|x|^{-p}$ is locally integrable near $|x|=0$. Fix $1<p<3$, then the Fourier transform of $\phi(x)=|x|^{-p}$ is $\hat{\phi}(\xi)=\hat{\phi}(|\xi|)=C|\xi|^{p-3}.$ Let $\theta$ be the angle between $v-v_{*}$ and $\sigma$, then $|v-v_{*}|\sin\frac{\theta}{2}=|v-v^{\prime}|,|v-v_{*}|\cos\frac{\theta}{2}=|v-v^{\prime}_{*}|$. As a result, the Boltzmann kernel given by (1.9) is

$\displaystyle B_{\phi}(v-v_{*},\sigma)=C|v-v_{*}|^{2p-5}(\sin^{p-3}\frac{\theta}{2}+\cos^{p-3}\frac{\theta}{2})^{2}.$

Here and in the rest of the article $C$ denotes a constant that depends only on fixed parameters and could change from line to line. Due the symmetry structure of (1.9), we can always assume $0\leq\theta\leq\pi/2$. Then $\sin\frac{\theta}{2}\leq\cos\frac{\theta}{2}$, since $p<3$, we get

$\displaystyle B_{\phi}(v-v_{*},\sigma)\leq C|v-v_{*}|^{2p-5}\sin^{2p-6}\frac{\theta}{2}.$

Since $p>1$, the following integral over $\mathbb{S}^{2}$ is bounded,

$\displaystyle\int B_{\phi}(v-v_{*},\sigma)\sin^{2}\frac{\theta}{2}\mathrm{d}\sigma\leq C|v-v_{*}|^{2p-5}.$

(1.10)

As in (1.10), in the rest of the article, when taking integration, the range of some frequently used variables will be omitted if there is no ambiguity. For instance, the usual ranges $\sigma\in\mathbb{S}^{2},x\in\mathbb{T}^{3},v,v_{*}\in\mathbb{R}^{3}$ will be consistently used unless otherwise specified. Whenever a new variable appears, we will specify its range once and then omit it thereafter.

By (1.10), for $1<p<3$ the kernel $B_{\phi}$ has finite momentum transfer which is a basic condition for the classical Boltzmann equation to be well-posed. The constant $C$ in (1.10) blows up as $p\rightarrow 1^{+}$ since $\int_{0}^{\sqrt{2}/{2}}t^{2p-3}\mathrm{d}t\sim\frac{1}{p-1}$ for $1<p<3$. If $p>2$, the angular function $\sin^{2p-6}\frac{\theta}{2}$ satisfies Grad’s angular cutoff assumption

$\displaystyle\int\sin^{2p-6}\frac{\theta}{2}\mathrm{d}\sigma\lesssim\frac{1}{p-2}.$

Note that $p=2$ is the critical value such that Grad’s cutoff assumption fails. To summarize, $2<p<3$ can be seen as angular cutoff while $1<p<2$ can be seen as angular non-cutoff. We consider the much harder case $1<p<2$ in this article. By taking $s=2-p,\gamma=2p-5$, it suffices to consider the more general kernel,

$\displaystyle B(v-v_{*},\sigma)\colonequals C|v-v_{*}|^{\gamma}(\sin^{-1-s}\frac{\theta}{2}+\cos^{-1-s}\frac{\theta}{2})^{2}\mathrm{1}_{0\leq\theta\leq\pi/2},\quad-3<\gamma<0<s<1,\gamma+2s\leq 0.$

(1.11)

The parameter pair $(\gamma,s)$ is commonly used in the study of classical Boltzmann equations with inverse power law. We find the resulting kernel (1.11) has close relation (see (2.18) for details) to the Boltzmann kernel $B^{ipl}(v-v_{*},\sigma)$ defined in (2.16). The condition $\gamma+2s\leq 0$ is referred as (very) soft potentials. From now on, the notation $B(v-v_{*},\sigma)$ or $B$ stands for the kernel in (1.11) unless otherwise specified.

To reiterate, in this article we will study the following Cauchy problem

$\displaystyle\partial_{t}F+v\cdot\nabla_{x}F=Q(F,F),~{}~{}t>0,x\in\mathbb{T}^{3},v\in\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3};\quad F|_{t=0}(x,v)=F_{0}(x,v),$

(1.12)

where the operator $Q$ is defined through (1.8) with $B_{\phi}$ replaced by $B$ in (1.11). Here $F_{0}$ is a given initial datum satisfying some high temperature condition which will be given in the next subsection.

Temperature plays an important role in the study of quantum Boltzmann equation. For example, for B-E particles, BEC will happen in low temperature. We now introduce some basic knowledge about temperature in the quantum context. Let us consider a homogeneous density $f=f(v)\geq 0$ with zero mean $\int vf(v)\mathrm{d}v=0$. For $k\geq 0$, we recall the moment function

$\displaystyle M_{k}(f)\colonequals\int|v|^{k}f(v)\mathrm{d}v.$

Let $M_{0}=M_{0}(f),M_{2}=M_{2}(f)$ for simplicity. Let $m$ be the mass of a particle, then $mM_{0}$ and $\frac{1}{2}mM_{2}$ are the total mass and kinetic energy per unit space volume. Referring [32], the kinetic temperature $\bar{T}$ and the critical temperature $\bar{T}_{c}$ of the particle system are defined by

$\displaystyle\bar{T}=\frac{1}{3k_{B}}\frac{mM_{2}}{M_{0}},\quad\bar{T}_{c}=\frac{m\zeta(5/2)}{2\pi k_{B}\zeta(3/2)}\big{(}\frac{M_{0}}{\zeta(3/2)}\big{)}^{\frac{2}{3}},$

(1.13)

where $k_{B}$ is the Boltzmann constant and $\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}$ is the Riemann zeta function.

We now recall some known facts about equilibrium distribution. The equilibrium of the classical Boltzmann equation is the Maxwellian distribution with density $\mu_{\rho,v_{0},T}$ function defined by

$\displaystyle\mu_{\rho,v_{0},T}(v)\colonequals\rho(2\pi T)^{-\frac{3}{2}}e^{-\frac{1}{2T}|v-v_{0}|^{2}},\quad\rho,T>0,v_{0}\in\mathbb{R}^{3}.$

Here $\rho$ is density, $v_{0}$ is mean velocity and $T$ is temperature. The famous Bose-Einstein distribution has density function

$\displaystyle\mathcal{M}_{\rho,v_{0},T}\colonequals\frac{\mu_{\rho,v_{0},T}}{1-\mu_{\rho,v_{0},T}},\quad\rho(2\pi T)^{-\frac{3}{2}}\leq 1.$

(1.14)

Now $\rho,v_{0}$ and $T$ do not represent density, mean velocity and temperature anymore, but only three parameters. The ratio $\bar{T}/\bar{T}_{c}$ quantifies high and low temperature. In high temperature $\bar{T}/\bar{T}_{c}>1$, the equilibrium of BBE equation is the Bose-Einstein distribution (1.14) with $\rho(2\pi T)^{-\frac{3}{2}}<1$. In low temperature $\bar{T}/\bar{T}_{c}<1$, the equilibrium of BBE equation is the Bose-Einstein distribution (1.14) with $\rho(2\pi T)^{-\frac{3}{2}}=1$ plus some Dirac delta function. That is, the equilibrium contains a Dirac measure. In the critical case $\bar{T}/\bar{T}_{c}=1$, the equilibrium is (1.14) with $\rho(2\pi T)^{-\frac{3}{2}}=1$. One can refer to [33] for the classification of equilibria.

In this article, we work with high temperature and thus the equilibrium of (1.1) is $\mathcal{M}_{\rho,v_{0},T}$ defined in (1.14). For perturbation around equilibrium, we define

$\displaystyle\mathcal{N}_{\rho,v_{0},T}\colonequals\mathcal{M}_{\rho,v_{0},T}^{\frac{1}{2}}(1+\mathcal{M}_{\rho,v_{0},T})^{\frac{1}{2}}=\frac{\mu_{\rho,v_{0},T}^{\frac{1}{2}}}{1-\mu_{\rho,v_{0},T}}.$

(1.15)

We remark that the function $\mathcal{N}_{\rho,v_{0},T}$ serves as the multiplier in the expansion $F=\mathcal{M}_{\rho,v_{0},T}+\mathcal{N}_{\rho,v_{0},T}f$.

Recall that the solution of (1.1) conserves mass, momentum and energy. That is, for any $t\geq 0$,

$\displaystyle\int(1,v,|v|^{2})F(t,x,v)\mathrm{d}x\mathrm{d}v=\int(1,v,|v|^{2})F_{0}(x,v)\mathrm{d}x\mathrm{d}v.$

(1.16)

Once $F_{0}$ is appropriately given, the constants $\rho,T>0,v_{0}\in\mathbb{R}^{3}$ in (1.14) are uniquely determined through

$\displaystyle\int(1,v,|v|^{2})F_{0}(x,v)\mathrm{d}x\mathrm{d}v=\int(1,v,|v|^{2})\mathcal{M}_{\rho,v_{0},T}(v)\mathrm{d}x\mathrm{d}v.$

(1.17)

Without any loss of generality, we assume that $F_{0}$ has zero mean and thus gives $v_{0}=0$ from now on. Also without any loss of generality, we assume that $F_{0}$ gives $T=1$ in this article. Indeed, we can make the transform $f(v)\to f(T^{1/2}v)$ to reduce the general $T\neq 1$ case to the special case $T=1$. As a result, we only keep $\rho$ as a parameter. That is, we only consider those initial data with $T=1,v_{0}=0$ according to (1.17). Taking $\mu(v)\colonequals(2\pi)^{-\frac{3}{2}}e^{-\frac{1}{2}|v|^{2}},$ the equilibrium $\mathcal{M}_{\rho,v_{0},T}$ and the multiplier function $\mathcal{N}_{\rho,v_{0},T}$ reduce to

$\displaystyle\mathcal{M}_{\rho}\colonequals\frac{\rho\mu}{1-\rho\mu},\quad\mathcal{N}_{\rho}\colonequals\frac{\rho^{\frac{1}{2}}\mu^{\frac{1}{2}}}{1-\rho\mu}.$

(1.18)

When $\rho$ is small, it is easy to see $M_{2}(\mathcal{M}_{\rho})\sim\rho,M_{0}(\mathcal{M}_{\rho})\sim\rho$ and so by recalling (1.13),

$\displaystyle\frac{\bar{T}}{\bar{T}_{c}}\sim\rho^{-\frac{2}{3}}.$

In our main result Theorem 1.1, we will assume $0<\rho\leq\rho_{*}$ for some small constant $0<\rho_{*}\ll 1$ which means

$\displaystyle\frac{\bar{T}}{\bar{T}_{c}}\gtrsim\rho_{*}^{-\frac{2}{3}}\gg 1.$

(1.19)

That is, we need high temperature assumption. Note that high temperature assumption is also imposed in [28] to prove global well-posedness of homogeneous BBE equation with (slightly general than) hard sphere collisions.

For simplicity, let $\mathcal{M}\colonequals\mathcal{M}_{\rho},\mathcal{N}\colonequals\mathcal{N}_{\rho}$. With the expansion $F=\mathcal{M}+\mathcal{N}f$, the linearized quantum Boltzmann equation corresponding to (1.12) reads

$\displaystyle\left\{\begin{aligned} &\partial_{t}f+v\cdot\nabla_{x}f+\mathcal{L}^{\rho}f=\Gamma_{2}^{\rho}(f,f)+\Gamma_{3}^{\rho}(f,f,f),~{}~{}t>0,x\in\mathbb{T}^{3},v\in\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3};\\ &f|_{t=0}=f_{0}=\frac{(1-\rho\mu)F_{0}-\rho\mu}{\rho^{\frac{1}{2}}\mu^{\frac{1}{2}}}.\end{aligned}\right.$

(1.20)

Here the linearized quantum Boltzmann operator $\mathcal{L}^{\rho}$ is define by

$\displaystyle(\mathcal{L}^{\rho}f)(v)\colonequals\int B\mathcal{N}_{*}\mathcal{N}^{\prime}\mathcal{N}^{\prime}_{*}\mathrm{S}(\mathcal{N}^{-1}f)\mathrm{d}\sigma\mathrm{d}v_{*},$

(1.21)

where $\mathrm{S}(\cdot)$ is defined by

$\displaystyle\mathrm{S}(g)\colonequals g+g_{*}-g^{\prime}-g^{\prime}_{*}.$

(1.22)

The bilinear term $\Gamma_{2}^{\rho}(\cdot,\cdot)$ and the trilinear term $\Gamma_{3}^{\rho}(\cdot,\cdot,\cdot)$ are defined by

	$\displaystyle\Gamma_{2}^{\rho}(g,h)$	$\displaystyle\colonequals$	$\displaystyle\mathcal{N}^{-1}\int B\Pi_{2}(g,h)\mathrm{d}\sigma\mathrm{d}v_{*}.$		(1.23)
	$\displaystyle\Gamma_{3}^{\rho}(g,h,\varrho)$	$\displaystyle\colonequals$	$\displaystyle\mathcal{N}^{-1}\int B\mathrm{D}\big{(}(\mathcal{N}g)_{*}^{\prime}(\mathcal{N}h)^{\prime}((\mathcal{N}\varrho)_{*}+\mathcal{N}\varrho)\big{)}\mathrm{d}\sigma\mathrm{d}v_{*}.\quad\quad$		(1.24)

The notation $\Pi_{2}$ in (1.23) is defined by

	$\displaystyle\Pi_{2}(g,h)$	$\displaystyle\colonequals$	$\displaystyle\mathrm{D}\big{(}(\mathcal{N}g)^{\prime}_{*}(\mathcal{N}h)^{\prime}\big{)}$		(1.28)
			$\displaystyle+\mathrm{D}\big{(}(\mathcal{N}g)^{\prime}_{*}(\mathcal{N}h)^{\prime}(\mathcal{M}+\mathcal{M}_{*})\big{)}$
			$\displaystyle+\mathrm{D}\big{(}(\mathcal{N}g)_{*}(\mathcal{N}h)^{\prime}(\mathcal{M}^{\prime}_{*}-\mathcal{M})\big{)}$
			$\displaystyle+\big{(}(\mathcal{N}g)^{\prime}(\mathcal{N}h)\mathrm{D}(\mathcal{M}^{\prime}_{*})+(\mathcal{N}g)^{\prime}_{*}(\mathcal{N}h)_{*}\mathrm{D}(\mathcal{M}^{\prime})\big{)}.$

Remark that the three operators $\mathcal{L}^{\rho}$, $\Gamma_{2}^{\rho}(\cdot,\cdot)$ and $\Gamma_{3}^{\rho}(\cdot,\cdot,\cdot)$ depends on $\rho$ through $\mathcal{M}=\mathcal{M}_{\rho},\mathcal{N}=\mathcal{N}_{\rho}$.

Our goal is to prove global well-posedness of (1.20) in some weighted Sobolev space. More precisely, we use the following energy and dissipation functional

$\displaystyle\mathcal{E}_{N}(f)\colonequals\sum_{|\alpha|+|\beta|\leq N}\|W_{l_{|\alpha|,|\beta|}}\partial^{\alpha}_{\beta}f\|_{L^{2}_{x}L^{2}}^{2},\quad\mathcal{D}_{N}(f)\colonequals\sum_{|\alpha|+|\beta|\leq N}\|W_{l_{|\alpha|,|\beta|}}\partial^{\alpha}_{\beta}f\|_{L^{2}_{x}\mathcal{L}^{s}_{\gamma/2}}^{2},$

(1.29)

where $\partial^{\alpha}_{\beta}\colonequals\partial^{\alpha}_{x}\partial^{\beta}_{v}$. See subsection 1.8 for the definition of $\|\cdot\|_{L^{2}_{x}L^{2}}$ and $\|\cdot\|_{L^{2}_{x}\mathcal{L}^{s}_{\gamma/2}}$. See (1.51) for the definition of $|\cdot|_{\mathcal{L}^{s}_{\gamma/2}}$. For the moment, just keep in mind $\|\cdot\|_{L^{2}_{x}\mathcal{L}^{s}_{\gamma/2}}$ is the dissipation corresponding to the energy $\|\cdot\|_{L^{2}_{x}L^{2}}$. Here $W_{l}(v)\colonequals(1+|v|^{2})^{\frac{l}{2}}$ is a polynomial weight on the velocity variable $v$. The weight order $l_{|\alpha|,|\beta|}\geq 0$ depends on the derivative order $|\alpha|,|\beta|$ and the sequence $\{l_{|\alpha|,|\beta|}\}_{|\alpha|+|\beta|\leq N}$ verifies

	$\displaystyle l_{|\alpha|,|\beta|}+|\gamma+2s|\leq l_{|\alpha|+1,|\beta|-1},\quad\text{ for }|\alpha|\leq N-1,1\leq|\beta|\leq N,|\alpha|+|\beta|\leq N.$		(1.30)
	$\displaystyle l_{|\alpha|,0}\leq l_{0,|\alpha|-1},\quad\text{ for }1\leq|\alpha|\leq N.$		(1.31)

The condition (1.30) is used to deal with linear streaming term $v\cdot\nabla_{x}f$ as $\gamma+2s\leq 0$ (see (1.44) below). The two conditions (1.30) and (1.31) together ensure that $l_{|\alpha|,|\beta|}$ increases as $|\alpha|+|\beta|$ decreases (see (5.49)).

Now we are ready to give global well-posedness of (1.20) in the following theorem.

We give some explanations and comments on Theorem 1.1 in the rest of this subsection and the next two subsections.

We emphasize that the constants $\rho_{*},\delta_{*},C$ in Theorem 1.1 could depend on $N,l_{0,0},s,\gamma$. In Theorem 1.1 and the rest of the article, if a constant depends only on the given parameters $N,l_{0,0},s,\gamma$, we say it is a universal constant. In particular, the two constants $\delta_{*},C$ are independent of $0<\rho\leq\rho_{*}$.

The condition $N\geq 9$ owes to Theorem 5.8. Understandably, a certain high order is required to deal with the singular kernel (1.11) and the trilinear operator $\Gamma_{3}^{\rho}(\cdot,\cdot,\cdot)$. Recall that the hard sphere model in [44] needs $N\geq 8$.

Note that Theorem 1.1 gives global well-posedness under the condition

$\displaystyle 0<\rho\ll 1,\quad\mathcal{M}_{\rho}=\frac{\rho\mu}{1-\rho\mu}\sim\rho\mu,\quad\mathcal{N}_{\rho}=\frac{\rho^{\frac{1}{2}}\mu^{\frac{1}{2}}}{1-\rho\mu}\sim\rho^{\frac{1}{2}}\mu^{\frac{1}{2}},\quad\mathcal{E}_{N}\big{(}\frac{(1-\rho\mu)F_{0}-\rho\mu}{\rho^{\frac{1}{2}}\mu^{\frac{1}{2}}}\big{)}\ll\rho^{2N+1}.$

Roughly speaking, these conditions mean

$\displaystyle F_{0}\approx\rho\mu+\mathrm{o}(\rho^{N+1})\mu^{\frac{1}{2}}.$

Note that this $\rho$-related smallness did not appear in previous works where the parameter $\rho$ was regarded as a fixed constant. Other effects of viewing $\rho$ as a variable parameter will be pointed out later.

By (1.32) and (1.33), we have $\mathcal{E}_{N}(f^{\rho}(t))\lesssim\rho$ for any $t\geq 0$. This estimate is consistent with the trivial case $\rho=0$ where $F(t)\equiv 0$ is the solution to problem (1.12) starting with a zero initial datum $F_{0}\equiv 0$.

In terms of physical relevance of Theorem 5.8, we give the following two remarks.

In terms of condition and conclusion, Theorem 1.1 is the closest to the main result (Theorem 1.4) of [28]. More precisely, both of these two results validate global existence of anisotropic solutions under very high temperature condition, see (1.19) and Remark 1.6 of [28] for more details. The main differences of these two results are also obvious. To reiterate, [28] considers spatially homogeneous case and (slightly general than) hard sphere model, while we work with the inhomogeneous case and non-cutoff kernels.

In terms of mathematical methods, this article is closer to [5], [44] and [27] since all of these works fall into the close-to-equilibrium framework well established for the classical Boltzmann equation. There are many great works that contribute to this mathematically satisfactory theory for global well-posedness of the classical Boltzmann equation. For reference, we mention [48, 24] for angular cutoff kernels and [23, 4] for non-cutoff kernels.

Each of [5], [44] and [27] has its own features and focuses. The work [5] is the first to investigate both relativistic and quantum effects. The article [27] studies the hydro dynamic limit from BFD (but not BBE) to incompressible Navier-Stokes-Fourier equation. The work [44] includes three cases: torus near equilibrium, whole space near equilibrium and whole space near vacuum. These novel works contribute to the literature of quantum Boltzmann equation from different aspects.

Like the above works, our main result Theorem 1.1 also has some unique features that may better our understanding of quantum Boltzmann equation. In particular, this article may be the first in the spatially inhomogeneous case to

• •

  study the quantum Boltzmann equation with inverse power law potentials;

• •

  incorporate high temperature condition into global well-posedness;

• •

  rule out BEC globally in time.

Different from the three works ([5], [44] and [27]), we keep the parameter $\rho$ along our derivation throughout the article. This intentional choice enables us to relate the high temperature condition to the smallness of $\rho$ quantitatively in (1.19). As a result, BEC is rigorously ruled out globally.

Compared to the well-established global well-posedness theory for classical Boltzmann equations with inverse power law potentials, our Theorem 1.1 has much room to improve. In this subsection, some possible improvements of Theorem 1.1 are given based on the author’s limited knowledge. To keep the present article in a reasonable length, we leave these improvements in future works.

Recall $\mathcal{E}_{N}$ is defined in (1.29) with the weight order $\{l_{|\alpha|,|\beta|}\}_{|\alpha|+|\beta|\leq N}$ satisfying (1.30) and (1.31). Let $\mathcal{E}_{N}^{*}$ be the space with the minimal weight order $\{l^{*}_{|\alpha|,|\beta|}\}_{|\alpha|+|\beta|\leq N}$ where $l^{*}_{0,N}=0$ and $\{l^{*}_{|\alpha|,|\beta|}\}_{|\alpha|+|\beta|\leq N}$ verifies the two conditions (1.30) and (1.31) with identities. To illustrate, we give an example of the weight order $\{l^{*}_{|\alpha|,|\beta|}\}_{|\alpha|+|\beta|\leq N}$. Recall for the inverse power law $\phi(x)=|x|^{-p}$, $\gamma=2-p,s=2p-5$ and thus $\gamma+2s=-1$. Let us take $N=9$, the corresponding $\{l^{*}_{|\alpha|,|\beta|}\}_{|\alpha|+|\beta|\leq 9}$ with $\gamma+2s=-1$ is shown in Table 1. In Table 1, the column index represents derivative order $|\alpha|$ of space variable $x$ and the row index represents derivative order $|\beta|$ of velocity variable $v$. For example $l_{0,9}=0,l_{1,8}=1,l_{0,8}=9,l_{0,1}=44,l_{1,0}=l_{0,0}=45$.

Note that in Theorem 1.1, smallness of $\mathcal{E}_{N}(f_{0})$ is imposed to prove well-posedness in $L^{\infty}([0,\infty);\mathcal{E}_{N})$. We may try to prove well-posedness in $L^{\infty}([0,\infty);\mathcal{E}_{N})$ under the condition $\mathcal{E}_{9}^{*}(f_{0})\ll\rho^{19},\mathcal{E}_{N}(f_{0})<\infty$. That is, smallness assumption is only imposed on $\mathcal{E}_{9}^{*}$. Such kind better result was derived by Guo in [25] for the Vlasov-Possion-Landau system.

Theorem 1.1 does not consider relaxation to equilibrium. Using the methods in [46] and [16] for classical Boltzmann equations, it is very promising to derive similar long time behaviors for the solution $f^{\rho}$ obtained in Theorem 1.1. For example, we can derive almost exponential decay like in [46] by recalling (1.51) $|\cdot|_{\mathcal{L}^{s}_{\gamma/2}}\geq|\cdot|_{L^{2}_{s+\gamma/2}}$ and using some interpolation inequality to deal with soft potential $\gamma+2s<0$. In this regard, a possible result may be as follows. Let $N\geq 9,l_{2}>l_{1}\geq 0$ and assume $\mathcal{E}_{9}^{*}(f_{0})\ll\rho^{19},\mathcal{E}_{N}(W_{l_{2}}f_{0})<\infty$. Try to prove for any $t\geq 0$,

$\displaystyle\mathcal{E}_{N}(W_{l_{2}}f^{\rho}(t))\lesssim\rho^{-2N}C(\mathcal{E}_{N}(W_{l_{2}}f_{0})),\quad\mathcal{E}_{N}(W_{l_{1}}f^{\rho}(t))\lesssim(1+t)^{-q}\rho^{-2N}\mathcal{E}_{N}(W_{l_{1}}f_{0}).$

The first upper bound estimate on $\mathcal{E}_{N}(W_{l_{2}}f^{\rho}(t))$ is used together with interpolation method to derive the polynomial decay of $\mathcal{E}_{N}(W_{l_{1}}f^{\rho}(t))$. By interpolation, we should have $q=-\frac{l_{2}-l_{1}}{s+\gamma/2}$.

We may try to derive sub-exponential decay rate under more assumption on initial data like in [16]. Roughly speaking, we could get something as follows. Let $\lambda>0$ be small enough and assume $\mathcal{E}_{N}(e^{2\lambda\langle v\rangle}f_{0})\ll\rho^{2N+1}$. Try to prove for any $t\geq 0$,

$\displaystyle\mathcal{E}_{N}(e^{2\lambda\langle v\rangle}f^{\rho}(t))\lesssim\rho^{-2N}\mathcal{E}_{N}(e^{2\lambda\langle v\rangle}f_{0}),\quad\mathcal{E}_{N}(f^{\rho}(t))\lesssim e^{-\lambda t^{\kappa}}\rho^{-2N}\mathcal{E}_{N}(e^{2\lambda\langle v\rangle}f_{0}),$

where $\kappa=\frac{1}{1+|\gamma+2s|}$.

Another topic is to prove global well-posedness in a larger space than $\mathcal{E}_{9}^{*}$. Note that for classical Boltzmann equation with inverse power law potential, see [16] for global well-posedness in the up-to-date largest space $L^{1}_{k}L^{2}$ (containing $\supset H^{\frac{3}{2}+\delta}_{x}L^{2}$ for any $\delta>0$). One may try to establish global well-posedness in such kind low regularity space.