A BGK kinetic model with local velocity alignment forces

1.   Introduction

In this paper, we are interested in the kinetic model which describes the particle system where the alignment effect of the ensemble and the relaxation process through binary collisions compete:

subject to initial data

Here $f = f(x, v, t)$ denotes the number density function on the phase point $(x, v) \in \mathbb T^d \times \mathbb R^d$ at time $t\in \mathbb R_+$. The local Maxwellian $\mathcal{M}(f) = \mathcal{M}(f)(x, v, t)$ is implicitly defined through the moments of $f$:

and

by the formula

The study of collective dynamics of self-propelled particle systems, in which simple local interaction laws between the particles eventually lead to the emergence of global collective behaviors, has received much attention recently. The most common such rules imposed on the constituent particles are rules of alignment phenomena and collisional interactions. As such, kinetic equations where the local alignment phenomena and collisional interactions are combined as in (1.1) arise in various physical or modelling contexts. For example in [6,16,17] a local alignment kinetic model with a Fokker-Planck type inter-particle interaction is considered, which arise as a coarse limit of a variant of Cucker-Smale flocking models [12,18]. To be more specific, the local particle velocity can be obtained by taking into account the following local averaged particle velocity:

where $K^r$ is called a communication weight and $r$ represents the radius of particle interactions, i.e., the support of $K^r$. This type of particle velocity is considered in [18] to describe the flocking behaviors. In [17], the rigorous justification of the limit

in the weak sense is investigated. At the formal level, we expect the kernel $K^r$ converges to the Dirac measure $\delta_0$, thus the limiting function will be the local particle velocity $u$. We refer to [4,11] for recent surveys on the Cucker-Smale type flocking models. Similar models also arise in the context of traffic model of Paveri-Fontana type where the evolution of the velocity distribution of the vehicles are explained by the adjustment of velocity with respect to the desired velocity (alignment), and the acceleration of the vehicles (traffic collision operator) [9,20]. The main motivation of our model (1.1) comes from the collisional alignment model of [16,17]. Instead of the Fokker-Planck type diffusive approximation of the collision operator as in [16,17], we are considering relaxational approximation of the collision operator.

Closely related models are kinetic-fluid equations with inter-particle interaction operators, in which a collisional kinetic equation and fluid equations are coupled through drag force terms. More precisely, $u$ in the local alignment force satisfies the fluid equations, for instance, incompressible Navier-Stokes system, see [8,10] where the existence theories for weak/strong solutions of the Navier-Stokes-BGK system are discussed. When the local velocity alignment force is ignored, our main equation (1.1) becomes the BGK model of Boltzmann equation [3], which is one of the most widely employed model equation of the Boltzmann equation providing reliable results on various problems in rarefied gas dynamics. Some of previous works on the BGK model can be summarized as follows. For the Cauchy problem in the framework of weak solutions, see [21,28,29]. The existence of strong solutions with uniqueness can be found in [19,22,25]. Behaviors of solutions that stays near global equilibriums are studied in [2,24,26,27]. For results on stationary solutions, see [1,23].

In the current work, we establish the global-in-time existence of weak solutions to the equation (1.1). The main difficulties are the nonlinear terms $fu$ in the local alignment force and the local Maxwellian $\mathcal{M}(f)$. In many previous works [5,7,14,15,18] on kinetic collective behavior models, the method of characteristics are crucially used to estimate the propagation of velocity support of solution $f$. However, in our case, due to the lack of regularity of $u$ and the BGK operator term, that types of support estimates of $f$ in velocity cannot be applied. Instead of that, we regularize the local particle velocity $u$, and linearize the equation (1.1). We then do some Cauchy estimates in the velocity-weighted $L^\infty( \mathbb T^d \times \mathbb R^d)$ space. Finally, we provide some uniform bound estimates in regularization parameters and pass to the limit which requires some weak and strong compactness arguments.

Let us now introduce a notion of weak solutions to the system (1.1).

Definition 1.1. We say that $f$ is a weak solution to the system (1.1) if the following conditions are satisfied:

(i) $f \in L^\infty(0, T;(L^1_+ \cap L^\infty)( \mathbb T^d \times \mathbb R^d))$¹,

¹ $L^1_+( \mathbb T^d \times \mathbb R^d)$ denotes the set of nonnegative $L^1( \mathbb T^d \times \mathbb R^d)$ functions.

(ii) for all $\phi \in \mathcal{C}^1_c( \mathbb T^d \times \mathbb R^d \times [0, T])$ with $\phi(x, v, T) = 0$,

We are now ready to state our main result:

Theorem 1.2. Let $T >0$. Suppose that the initial data $f_0$ satisfy

and

Then there exists at least one weak solution to the system (1.1) in the sense of Definition 1.1 satisfying the following estimates:

(i)

and

(ii)

for almost every $t \in (0, T)$ and $C>0$.

Remark 1.3. We can easily apply our strategy for the existence result for the equation (1.1) with $u^r$ appeared in (1.2) instead of $u$ under the assumption $K^r \in L^1( \mathbb T^d)$.

We introduce several notations used throughout the paper. For functions $f(x, v)$, $g(x)$, $\|f\|_{L^p}$ and $\|g\|_{L^p}$ denote the usual $L^p( \mathbb T^d \times \mathbb R^d)$-norm and $L^p( \mathbb T^d)$-norm, respectively. $\|f\|_{L_q^\infty}$ represents a weighted $L^{\infty}$-norm:

For any nonnegative integer $s$, $H^s$ denotes the $s$-th order $L^2$ Sobolev space. $\mathcal{C}^s([0, T]; E)$ is the set of $s$-times continuously differentiable functions from an interval $[0, T]\subset \mathbb R$ into a Banach space $E$, and $L^p(0, T;E)$ is the set of the $L^p$ functions from an interval $(0, T)$ to a Banach space $E$. We denote by $C$ a generic, not necessarily identical, positive constant, and $C = C_{\alpha, \beta, \dots}$ or $C = C(\alpha, \beta, \dots)$ stands for the positive constant depending on $\alpha, \beta, \dots$.

This paper is organized as follows: In the following section 2, we introduce a regularized equation of (1.1) and study the global-in-time existence of weak solutions to the regularized equation. For this, we construct the approximated solutions to the regularized equation, and provide that they are Cauchy sequences in the velocity-weighted $L^\infty( \mathbb T^d \times \mathbb R^d)$ space. We also establish uniform bound estimates in the regularizing parameter. In Section 3, we then use this uniform bounds to drive weak solution of (1.1) through the weak limit of the regularized distribution function and the strong limit of macroscopic fields.

2.   A regularized equation

In order to show the existence of weak solutions to (1.1), it is required to regularize the equation to remove the singularity in the local alignment force. In this section, we introduce a regularized equation and provide the global-in-time existence of weak solutions of that. Let us regularize the local particle velocity $u$ by using a mollifier $\theta_ \varepsilon(x) = \varepsilon^{-d}\theta(x/ \varepsilon)$ with $0 \leq \theta \in \mathcal{C}_0^\infty( \mathbb T^d)$ satisfying

Then our regularized equation is defined as follows:

subject to regularized initial data:

Here

with

and the regularized initial data $f_{0, \varepsilon}$ is defined by

where $1_A$ denotes the characteristic function on $A$ and $\eta$ is the standard mollifier. Note that $f_{0, \varepsilon}$ satisfies

(i) $f_{0, \varepsilon} \to f_0$ strongly in $L^p( \mathbb T^d \times \mathbb R^d)$ for all $p < \infty$ and weakly-$*$ in $L^\infty( \mathbb T^d \times \mathbb R^d)$,

(ii) $\| v \sqrt{f_{0, \varepsilon}}\|_{L^2}^2 \to \| v \sqrt{f_0}\|_{L^2}^2$ strongly.

In the following two sections, we prove the proposition below on the global-in-time existence of weak solutions and some uniform bound estimates of the regularized system (2.1).

Proposition 2.1. Let $T>0$. For any $\varepsilon >0$, there exists at least one weak solution $f_ \varepsilon$ of the regularized equation (2.1) on the interval $[0, T]$ in the sense of Definition 1.1. Furthermore we have

(i) kinetic energy estimate

(ii) third moment & entropy estimates:

where $C = C(f_0, T) > 0$ is independent of $\varepsilon$.

2.1.   A linearized equation

In order to obtain the existence of solutions to the regularized equation (2.1), we first construct the approximated solutions in the following way:

with the initial data and first iteration step:

and

Here

with

In the proposition below, we provide the global-in-time solvability of the regularized and linearized equation (2.2) and uniform-in-$n$ bound estimates of solutions.

Proposition 2.2. Let $T>0$ and $q >d+2$. For any $n \in \mathbb N$, there exists a unique solution $f^n_ \varepsilon$ of the regularized and linearized system (2.2) such that $f^n_ \varepsilon \in L^\infty(0, T; L^\infty_q( \mathbb T^d \times \mathbb R^d))$. Moreover, we have

and

where $C > 0$ is independent of both $n$ and $\varepsilon$, and $C_ \varepsilon>0$ is independent of $n$.

Before presenting the details of proof of Proposition 2.2, we list several technical lemmas showing some bound estimates related to the local Maxwellian. First two lemmas below are concerned with the upper bound estimates of local Maxwellian, and the third one is a type of Lipschitz continuity of the local Maxwellian.

Lemma 2.1. [22,p.291] Suppose $\|f\|_{L^{\infty}_q}<\infty$ for $q>d+2$. Then there exists a positive constant $C_q$, which depends only on $q$, such that

Lemma 2.2. [25,Proposition 4.1] Assume that $f$ satisfies

1. $\|f\|_{L^{\infty}_q}+\|\nabla_{x, v}f\|_{L^{\infty}_q}<C_{1}$,

2. $\rho+|u|+ T<C_{2}$,

3. $\rho, T>C_{3}$,

for some constants $C_{i}>0$ $(i = 1, 2, 3)$. Then, we have

where $C_T > 0$ depends only on $C_{1}$, $C_{2}$, $C_{3}$ and the final time $T$.

Lemma 2.3. [25,Proposition 6.1] Assume $f, g$ satisfy ($h$ denotes either $f$ or $g$)

1. $\|h\|_{L^{\infty}_q}<C_{1}$,

2. $\rho_{h}+|u_h|+ T_h<C_{2}$,

3. $\rho_{h}, T_{h}>C_{3}$,

for some constants $C_{i}>0$ $(i = 1, 2, 3)$. Then, we have

where $C_T>0$ depends only on $C_{1}$, $C_{2}$, $C_{3}$ and the final time $T$.

Proof of Proposition 2.2. For the proof, we first introduce the following backward characteristics:

defined by

with the terminal datum

Due to the regularization, the characteristics (2.3) is well-defined, thus global-in-time existence of solutions to (2.2) can be obtained by the Cauchy-Lipschitz existence theory. It follows from (2.3) that $V^{n+1}_ \varepsilon(s)$ satisfies

On the other hand, we get

for $q > d$ and

for $q > d+1$. Subsequently, these imply

and

where $C_ \varepsilon>0$ is independent of $n$. This together with (2.4) yields

Furthermore, we have

Along that characteristics we obtain from (2.2) that

Then it is easy to check from (2.9) that

due to Lemma 2.1. Applying the Gronwall's inequality, we conclude the first assertion. In order to prove the second estimate, we use (2.7) to obtain

Using the above estimate, we find

for $0 < q < \infty$. Similarly, with the aid of Lemma 2.1, we estimate

This together with (2.9) gives

This yields

and subsequently, we find

for all $n\geq 1$ due to (2.8). Then we also use similar argument as above to get

due to Lemma 2.2. Hence we obtain

Combining (2.10) and (2.11), we have

which concludes the desired result.

Corollary 2.1. Let $q >d+1$. For any $T>0$ and $n \in \mathbb N$, there exists a unique solution $f^n_ \varepsilon$ of the regularized and linearized system (2.2) such that $f^n_ \varepsilon \in L^\infty(0, T; L^\infty_q( \mathbb T^d \times \mathbb R^d))$. Moreover, we have

(i) lower bound estimates for the local particle density and temperature:

(ii) upper bound estimates for the macroscopic fields:

for all $0 \leq t \leq T$, where $C_ \varepsilon$ is independent of $n$.

Proof. (i) We take into account the integration of (2.9) and recall how we regularized $f_0$ to see

where we used

for all $0 \leq s \leq T$. This gives the lower bound for $\rho^n_ \varepsilon$. On the other hand, it follows from [21,Proposition 2.1] that

for $q > d$ or $q = 0$. This combined with the above lower bound estimate for $\rho^n_ \varepsilon$ asserts $T^n_ \varepsilon > C_ \varepsilon$ for some $C_ \varepsilon$, which is independent of $n$. Note that this is essential for the local Maxwellian to be well-defined.

(ii) Straightforward computations give

2.2.   Cauchy estimate for $f^n$

In this part, we show that $f^n_ \varepsilon$ is a Cauchy sequence in $L^\infty(0, T; L^\infty_q( \mathbb T^d \times \mathbb R^d))$. It follows from (2.9) that

Here $I_1$ is readily estimated as

where $C>0$ is independent of $n$ and $\varepsilon$. For the estimate of $I_2$, we use Lemma 2.2 to find

It follows from Lemma 2.3 that

Putting these estimates into (2.13) gives

We next present Cauchy estimate for the characteristic $Z^{n+1}_ \varepsilon$. For this, we first notice that

for $q > d$ and

for $q > d+1$. This yields

where $C_ \varepsilon>0$ is independent of $n$. Then, by using the above combined with (2.6), we obtain from (2.3) that

Thus we get

Applying Gronwall inequality to the above, we have

for all $0 \leq s \leq t \leq T$. We then combine this with (2.14) to conclude

2.3.   Proof of Proposition 1

We now provide the details of proof of the global-in-time existence of weak solutions to the regularized equation (2.1).

2.3.1.   Existence of weak solutions

It follows from (2.15) that $f^n_ \varepsilon$ is a Cauchy sequence in $L^\infty(0, T; L^\infty_q( \mathbb T^d \times \mathbb R^d))$ from which, for a fixed $\varepsilon > 0$, there exists a limiting function $f_ \varepsilon$ such that

as $n\to\infty$. Then it remains to show that the liming function $f_ \varepsilon$ solves the regularized equation (2.1). Since there is no singularity in the local alignment force, it suffices to have

as $n \to \infty$. Since $q > d+2$, for $\phi(v) = 1, v$, and $|v|^2$, we estimate

This together with (2.16) yields

and

in $L^\infty( \mathbb T^d \times (0, T))$ as $n \to \infty$. Here $(\rho_ \varepsilon, u_ \varepsilon, T_ \varepsilon)$ denote the macroscopic fields constructed from $f_ \varepsilon$. On the other hand, the lower bound estimate of $\rho^n_ \varepsilon$ is obtained in (2.12), and this gives

as $n \to \infty$, uniformly in $x$ and $t$. The convergence of $f^n_ \varepsilon$ in $\|\cdot\|_{L^\infty(0, T;L^{\infty}_q)}$ and the uniform convergence of $(\rho^n_ \varepsilon, u^n_ \varepsilon, T^n_ \varepsilon)$ to $(\rho_ \varepsilon, u_ \varepsilon, T_ \varepsilon)$ imply that $f_ \varepsilon$ and $(\rho_ \varepsilon, u_ \varepsilon, T_ \varepsilon)$ also satisfy the assumptions of Lemma 2.3. This concludes from Lemma 2.3 and (2.16) that

as $n \to \infty$. This completes the proof of the existence part.

2.3.2.   Uniform-in-$\varepsilon$ bound estimates

In this part, we show the uniform-in-$\varepsilon$ bound estimates of solutions obtained in the above. Since the third moment and entropy estimates can be obtained by using almost the same arguments as in [10,Section 5], we only provide the kinetic energy estimate (i). We first notice from [16,Lemma 2.5] that

where $C>0$ is independent of $\varepsilon$. On the other hand, a straightforward computation yields

Note that

and this together with (2.17) gives

where $C>0$ is independent of $\varepsilon$. Hence we have

i.e.,

where $C>0$ is independent of $\varepsilon$. This concludes the desired result.

3.   Proof of Theorem 1.2

We are now ready to send the regularization parameter $\varepsilon \to \infty$ in the regularized equation (2.1). We will use several compactness arguments based on the Dunford-Pettis theorem and the velocity averaging lemma. Since the compactness of the BGK relaxation operator and the macroscopic fields are already discussed in [10,Section 6], we sketch the idea of proofs here. In the previous section, we obtained

where $C>0$ is independent of $\varepsilon$. Then by Dunford-Pettis theorem, we find that $f_{ \varepsilon}$, $f_{ \varepsilon}v$ and $f_{ \varepsilon}|v|^2$ are weakly compact in $L^1( \mathbb T^d \times \mathbb R^d \times (0, T))$, and thus there exists $f\in L^1( \mathbb T^d\times \mathbb R^d \times (0, {T}))$ such that $f_{ \varepsilon}$, $f_{ \varepsilon}v$, $f_{ \varepsilon}|v|^2$ converge to $f$, $fv$, $f|v|^2$ weakly in $L^1( \mathbb T^d\times \mathbb R^d \times (0, {T}))$ respectively, which also implies

and

in $L^1( \mathbb T^d \times (0, T))$. Then with the aid of the velocity averaging lemma [13], the above convergence actually is strong, which gives the almost everywhere convergence of the macroscopic fields:

where

Concerning the compactness of the local Maxwellian, we use the entropy bound estimate in (3.1) together with Dunford-Pettis theorem to have the weak compactness of $\mathcal{M}(f)$ in $L^1( \mathbb T^d\times \mathbb R^d \times (0, T))$. In fact, we can also show that $\mathcal{M}(f_{ \varepsilon})$ converges weakly to $\mathcal{M}(f)$ in $L^1( \mathbb T^d \times \mathbb R^d \times (0, T))$, see [10,Section 6.3] for details.

Employing the previous convergence estimates, we can pass to the limit $\varepsilon \to 0$ in the regularized equation (2.1) except the term $f_ \varepsilon u^ \varepsilon_ \varepsilon$ in the local alignment force. Thus in the rest of this section, we will show that

For this, we use a similar argument as in [16]. By the strong convergence (3.2), we get

up to a subsequence, for all $p \in (1, (d+2)/(d+1))$. Let

for a given test function $\varphi(v)$. Consider a test function $\psi(x, v, t) : = \phi(x, t)\varphi(v)$ with $\phi \in \mathcal{C}^\infty_c( \mathbb T^d \times (0, T))$ and $\varphi \in \mathcal{C}^\infty_c( \mathbb R^d)$. Then we obtain

Note that if $p \in (1, (d+2)/(d+1))$, then $p/(2-p) \in (1, (d+2)/d)$, and we use Hölder's inequality and Proposition 2.1 (i) to deduce

Thus there exists a function $m \in L^\infty(0, T; L^p( \mathbb T^d))$ such that

up to a subsequence. We then claim that

By using the set $E$ appeared in (3.3), we first easily estimate

This implies that it suffices to check

For this, we introduce a set

By the compactness of $\rho_ \varepsilon$ and $\rho_ \varepsilon \star \theta_ \varepsilon$ and Egorov's theorem, for any $\eta> 0$, there exists a set $C_\eta \subset E^\delta$ with $|E^\delta \setminus C_\eta | < \eta$ on which both $\rho_ \varepsilon$ and $\rho_ \varepsilon \star \theta_ \varepsilon$ uniformly converge to $\rho$ as $\varepsilon \to 0$. This yields that $\rho_ \varepsilon \star \theta_ \varepsilon > \delta/2$ in $C_\eta$ for sufficiently small $\varepsilon>0$. This yields

and subsequently, this asserts

since $\eta > 0$ and $\delta > 0$ were arbitrary. Hence we have

for all test functions of the form $\psi(x, v, t) = \phi(x, t)\varphi(v)$.

Incorporating all of the above observations allows us to send $\varepsilon \to 0$ in (2.1) to conclude that the limiting function $f$ is the weak solution to our main equation (1.1) in the sense of Definition 1.1. $L^\infty$ bound and kinetic energy estimates in Theorem 1.2 can be easily obtained.