Stationary solutions to the two-dimensional Broadwell model

TL;DR

This paper proves the existence of renormalized solutions for the 2D Broadwell model with L1 initial data, using L1-compactness and Kolmogorov-Riesz techniques due to the discrete velocities.

Contribution

It introduces a novel approach based on direct L1-compactness and Kolmogorov-Riesz arguments for the 2D Broadwell model with discrete velocities.

Findings

Existence of renormalized solutions established

Applicable to initial data in L1 space

Overcomes limitations of averaging techniques for discrete velocities

Abstract

Existence of renormalized solutions to the two-dimensional Broadwell model with given indata in L1 is proven. Averaging techniques from the continuous velocity case being unavailable when the velocities are discrete, the approach is based on direct L1-compactness arguments using the Kolmogorov-Riesz theoren.

Equations

$${\partial }_x{F}_1=F_3{F}_4-F_1{F}_2,$$

$$-{\partial }_x{F}_2=F_3{F}_4-F_1{F}_2,$$

$${\partial }_y{F}_3=F_1{F}_2-F_3{F}_4,$$

$$-{\partial }_y{F}_4=F_1{F}_2-F_3{F}_4,$$

$$v_1=(1,0),v_2=(-1,0),v_3=(0,1),v_4=(0,-1).$$

$$F_1(x,y)=f_{b1}(y)e^{-{\int }_0^x{F}_2(s,y)ds}+{\int }_0^x(F_3{F}_4)(s,y)e^{-{\int }_s^x{F}_2(\tau )d\tau }ds,\text{a.a. }(x,y)\in [0,1{]}^2,$$

$$F_1(x,y)=f_{b1}(y)+{\int }_0^x(F_3{F}_4-F_1{F}_2)(s,y)ds,\text{a.a. }(x,y)\in [0,1{]}^2,$$

$${\partial }_x\ln (1+F_1)=\frac{F_3{F}_4-F_1{F}_2}{1+F_1},$$

$${\partial }_x{F}_1^k=\frac{F_3^k}{1+\frac{F_3^k}{k}}\frac{F_4^k}{1+\frac{F_4^k}{k}}-\frac{F_1^k}{1+\frac{F_1^k}{k}}\frac{F_2^k}{1+\frac{F_2^k}{k}},$$

$$-{\partial }_x{F}_2^k=\frac{F_3^k}{1+\frac{F_3^k}{k}}\frac{F_4^k}{1+\frac{F_4^k}{k}}-\frac{F_1^k}{1+\frac{F_1^k}{k}}\frac{F_2^k}{1+\frac{F_2^k}{k}},$$

$${\partial }_y{F}_3^k=\frac{F_1^k}{1+\frac{F_1^k}{k}}\frac{F_2^k}{1+\frac{F_2^k}{k}}-\frac{F_3^k}{1+\frac{F_3^k}{k}}\frac{F_4^k}{1+\frac{F_4^k}{k}},$$

$$-{\partial }_y{F}_4^k=\frac{F_1^k}{1+\frac{F_1^k}{k}}\frac{F_2^k}{1+\frac{F_2^k}{k}}-\frac{F_3^k}{1+\frac{F_3^k}{k}}\frac{F_4^k}{1+\frac{F_4^k}{k}},(x,y)\in [0,1{]}^2,$$

$$F_1^k(0,y)=f_{b1}(y)\wedge \frac{k}{2},F_2^k(1,y)=f_{b2}(y)\wedge \frac{k}{2},y\in [0,1],$$

$$F_3^k(x,0)=f_{b3}(x)\wedge \frac{k}{2},F_4^k(x,1)=f_{b4}(x)\wedge \frac{k}{2},x\in [0,1].$$

$$\displaystyle c_{\alpha}=\frac{1}{\alpha}\int_{0}^{1}\sum_{i=1}^{4}f_{bi}(u)du,\quad K_{\alpha}=\{f\in\big{(}L_{+}^{1}([0,1]^{2})\big{)}^{4};\sum_{i=1}^{4}\int f_{i}(x,y)dxdy\leq c_{\alpha}\}.$$

$$\alpha F_1+{\partial }_x{F}_1=\frac{F_3}{1+\frac{F_3}{k}}\frac{f_4\ast {\mu }_{\alpha }}{1+\frac{f_4\ast {\mu }_{\alpha }}{k}}-\frac{F_1}{1+\frac{F_1}{k}}\frac{f_2\ast {\mu }_{\alpha }}{1+\frac{f_2\ast {\mu }_{\alpha }}{k}},$$

$$\alpha F_2-{\partial }_x{F}_2=\frac{f_3\ast {\mu }_{\alpha }}{1+\frac{f_3\ast {\mu }_{\alpha }}{k}}\frac{F_4}{1+\frac{F_4}{k}}-\frac{f_1\ast {\mu }_{\alpha }}{1+\frac{f_1\ast {\mu }_{\alpha }}{k}}\frac{F_2}{1+\frac{F_2}{k}},$$

$$\alpha F_3+{\partial }_y{F}_3=\frac{F_1}{1+\frac{F_1}{k}}\frac{f_2\ast {\mu }_{\alpha }}{1+\frac{f_2\ast {\mu }_{\alpha }}{k}}-\frac{F_3}{1+\frac{F_3}{k}}\frac{f_4\ast {\mu }_{\alpha }}{1+\frac{f_4\ast {\mu }_{\alpha }}{k}},$$

$$\alpha F_4-{\partial }_y{F}_4=\frac{f_1\ast {\mu }_{\alpha }}{1+\frac{f_1\ast {\mu }_{\alpha }}{k}}\frac{F_2}{1+\frac{F_2}{k}}-\frac{f_3\ast {\mu }_{\alpha }}{1+\frac{f_3\ast {\mu }_{\alpha }}{k}}\frac{F_4}{1+\frac{F_4}{k}},(x,y)\in [0,1{]}^2,$$

$$F_1(0,y)=f_{b1}(y)\wedge \frac{k}{2},F_2(1,y)=f_{b2}(y)\wedge \frac{k}{2},y\in [0,1],$$

$$F_3(x,0)=f_{b3}(x)\wedge \frac{k}{2},F_4(x,1)=f_{b4}(x)\wedge \frac{k}{2},x\in [0,1].$$

$$\alpha F_1^{n+1}+{\partial }_x{F}_1^{n+1}=\frac{F_3^n}{1+\frac{F_3^n}{k}}\frac{f_4\ast {\mu }_{\alpha }}{1+\frac{f_4\ast {\mu }_{\alpha }}{k}}-\frac{F_1^{n+1}}{1+\frac{F_1^n}{k}}\frac{f_2\ast {\mu }_{\alpha }}{1+\frac{f_2\ast {\mu }_{\alpha }}{k}},$$

$$\alpha F_2^{n+1}-{\partial }_x{F}_2^{n+1}=\frac{f_3\ast {\mu }_{\alpha }}{1+\frac{f_3\ast {\mu }_{\alpha }}{k}}\frac{F_4^n}{1+\frac{F_4^n}{k}}-\frac{f_1\ast {\mu }_{\alpha }}{1+\frac{f_1\ast {\mu }_{\alpha }}{k}}\frac{F_2^{n+1}}{1+\frac{F_2^n}{k}},$$

$$\alpha F_3^{n+1}+{\partial }_y{F}_3^{n+1}=\frac{F_1^n}{1+\frac{F_1^n}{k}}\frac{f_2\ast {\mu }_{\alpha }}{1+\frac{f_2\ast {\mu }_{\alpha }}{k}}-\frac{F_3^{n+1}}{1+\frac{F_3^n}{k}}\frac{f_4\ast {\mu }_{\alpha }}{1+\frac{f_4\ast {\mu }_{\alpha }}{k}},$$

$$\alpha F_4^{n+1}-{\partial }_y{F}_4^{n+1}=\frac{f_1\ast {\mu }_{\alpha }}{1+\frac{f_1\ast {\mu }_{\alpha }}{k}}\frac{F_2^n}{1+\frac{F_2^n}{k}}-\frac{f_3\ast {\mu }_{\alpha }}{1+\frac{f_3\ast {\mu }_{\alpha }}{k}}\frac{F_4^{n+1}}{1+\frac{F_4^n}{k}},$$

$$F_1^{n+1}(0,y)=f_{b1}(y)\wedge \frac{k}{2},F_2^{n+1}(1,y)=f_{b2}(y)\wedge \frac{k}{2},y\in [0,1],$$

$$F_3^{n+1}(x,0)=f_{b3}(x)\wedge \frac{k}{2},F_4^{n+1}(x,1)=f_{b4}(x)\wedge \frac{k}{2},x\in [0,1],n\in \mathbb{N}.$$

$$\alpha \sum _{i=1}^4{F}_i^{n+1}+{\partial }_x(F_1^{n+1}-F_2^{n+1})+{\partial }_y(F_3^{n+1}-F_4^{n+1})$$

$$=\frac{f_1\ast {\mu }_{\alpha }}{1+\frac{f_1\ast {\mu }_{\alpha }}{k}}\frac{F_2^n-F_2^{n+1}}{1+\frac{F_2^n}{k}}+\frac{f_2\ast {\mu }_{\alpha }}{1+\frac{f_2\ast {\mu }_{\alpha }}{k}}\frac{F_1^n-F_1^{n+1}}{1+\frac{F_1^n}{k}}+\frac{f_3\ast {\mu }_{\alpha }}{1+\frac{f_3\ast {\mu }_{\alpha }}{k}}\frac{F_4^n-F_4^{n+1}}{1+\frac{F_4^n}{k}}+\frac{f_4\ast {\mu }_{\alpha }}{1+\frac{f_4\ast {\mu }_{\alpha }}{k}}\frac{F_3^n-F_3^{n+1}}{1+\frac{F_3^n}{k}}$$

$$\le 0,$$

$$\sum _{i=1}^4\int F_i^{n+1}(x,y)dxdy\le c_{\alpha }.$$

$$\forall n\in \mathbb{N},F_i^n\le G_i,1\le i\le 4.$$

$$F_1^{n+1}(x,y)$$

$$+{\int }_0^x\frac{F_3^n}{1+\frac{F_3^n}{k}}\frac{f_4\ast {\mu }_{\alpha }}{1+\frac{f_4\ast {\mu }_{\alpha }}{k}}(X,y)e^{-\alpha (x-X)-{\int }_X^x\frac{f_2\ast {\mu }_{\alpha }}{(1+\frac{F_1^n}{k})(1+\frac{f_2\ast {\mu }_{\alpha }}{k})}(r,y)dr}dX$$

$$\le (f_{b1}(y)\wedge \frac{k}{2})e^{-\alpha x-{\int }_0^x\frac{f_2\ast {\mu }_{\alpha }}{(1+\frac{G_1}{k})(1+\frac{f_2\ast {\mu }_{\alpha }}{k})}(X,y)dX}$$

$$+{\int }_0^x\frac{G_3}{1+\frac{G_3}{k}}\frac{f_4\ast {\mu }_{\alpha }}{1+\frac{f_4\ast {\mu }_{\alpha }}{k}}(X,y)e^{-\alpha (x-X)-{\int }_X^x\frac{f_2\ast {\mu }_{\alpha }}{(1+\frac{G_1}{k})(1+\frac{f_2\ast {\mu }_{\alpha }}{k})}(r,y)dr}dX$$

$$=G_1(x,y),(x,y)\in [0,1{]}^2.$$

$$F_i\le G_i,1\le i\le 4.$$

$$\alpha \sum _{i=1}^4\int (G_i-F_i)(x,y)dxdy$$

$$\displaystyle+\int_{0}^{1}\big{(}(G_{3}-F_{3})(x,1)+(G_{4}-F_{4})(x,0)\big{)}dx=0.$$

$$\alpha S_{l1}+{\partial }_x{S}_{l1}=\frac{S_{l3}}{1+\frac{S_{l3}}{k}}\frac{G_{l4}\ast {\mu }_{\alpha }}{1+\frac{G_{l4}\ast {\mu }_{\alpha }}{k}}-\frac{S_{l1}}{1+\frac{S_{l1}}{k}}\frac{g_{l2}\ast {\mu }_{\alpha }}{1+\frac{g_{l2}\ast {\mu }_{\alpha }}{k}},$$

$$\alpha S_{l2}-{\partial }_x{S}_{l2}=\frac{G_{l3}\ast {\mu }_{\alpha }}{1+\frac{G_{l3}\ast {\mu }_{\alpha }}{k}}\frac{S_{l4}}{1+\frac{S_{l4}}{k}}-\frac{g_{l1}\ast {\mu }_{\alpha }}{1+\frac{g_{l1}\ast {\mu }_{\alpha }}{k}}\frac{S_{l2}}{1+\frac{S_{l2}}{k}},$$

$$\alpha S_{l3}+{\partial }_y{S}_{l3}=\frac{S_{l1}}{1+\frac{S_{l1}}{k}}\frac{G_{l2}\ast {\mu }_{\alpha }}{1+\frac{G_{l2}\ast {\mu }_{\alpha }}{k}}-\frac{S_{l3}}{1+\frac{S_{l3}}{k}}\frac{g_{l4}\ast {\mu }_{\alpha }}{1+\frac{g_{l4}\ast {\mu }_{\alpha }}{k}},$$

$$\alpha S_{l4}-{\partial }_y{S}_{l4}=\frac{G_{l1}\ast {\mu }_{\alpha }}{1+\frac{G_{l1}\ast {\mu }_{\alpha }}{k}}\frac{S_{l2}}{1+\frac{S_{l2}}{k}}-\frac{g_{l3}\ast {\mu }_{\alpha }}{1+\frac{g_{l3}\ast {\mu }_{\alpha }}{k}}\frac{S_{l4}}{1+\frac{S_{l4}}{k}},$$

Full text

Stationary solutions to the two-dimensional Broadwell model.

Leif Arkeryd and Anne Nouri

Abstract          

Existence of renormalized solutions to the two-dimensional Broadwell model with given indata in $L^{1}$ is proven. Averaging techniques from the continuous velocity case being unavailable when the velocities are discrete, the approach is based on direct $L^{1}$-compactness arguments using the Kolmogorov-Riesz theorem. 11footnotetext: 2010 Mathematics Subject Classification 82C22, 82C40. 22footnotetext: Key words; Broadwell equation, discrete velocity Boltzmann equation, existence theory. 33footnotetext: Leif Arkeryd, Mathematical Sciences, Goteborg, Sweden. 44footnotetext: Anne Nouri, Aix-Marseille University, CNRS, Centrale Marseille, I2M, Marseille, France.

1 Introduction.

The two-dimensional stationary Broadwell model in a square is

[TABLE]

with unknown $(F_{i})_{1\leq i\leq 4}$ defined on $[0,1]^{2}$, and given $(f_{bi})_{1\leq i\leq 4}$ defined on $[0,1]$. It is a four velocity model for the Boltzmann equation, with $F_{i}(x,y)=f(x,y,v_{i})$,

[TABLE]

The boundary value problem (1.1) is considered in $L^{1}$ in one of the following equivalent forms,

the exponential multiplier form:

[TABLE]

and analogous equations for $F_{i}$, $2\leq i\leq 4$,

the mild form:

[TABLE]

and analogous equations for $F_{i}$, $2\leq i\leq 4$,

the renormalized form:

[TABLE]

in the sense of distributions, and analogous equations for $F_{i}$, $2\leq i\leq 4$.

The main result of the paper is the following.

Theorem 1.1

Given a non-negative boundary value $f_{b}$ with finite mass and entropy, there exists a stationary non-negative renormalized solution in $L^{1}$ with finite entropy-dissipation to the Broadwell model (1.1).

Most mathematical results for discrete velocity models of the Boltzmann equation have been performed in one space dimension. An overview is given in [8]. In two dimensions, special classes of solutions are given in [3] [4], and [9]. [3] contains a detailed study of the stationary Broadwell equation in a rectangle with comparison to a Carleman-like system, and a discussion of (in)compressibility aspects.

The existence of continuous solutions to the two-dimensional stationary Broadwell model with continuous boundary data for a rectangle, is proven in [6]. That proof starts by solving the problem with a given gain term, and uses the compactness of the corresponding twice iterated solution operator to conclude by Schaeffer’s fixed point theorem.

The present paper on the Broadwell model is set in a context of physically natural quantities. Mass and entropy flow at the boundary are given, and the solutions obtained, have finite mass and finite entropy dissipation. Averaging techniques from the continuous velocity case [7] being unavailable, a direct compactness approach is used, based on the Kolmogorov-Riesz theorem.

The plan of the paper is the following. An approximation procedure for the construction of solutions to (1.1) is introduced in Section 2. The passage to the limita in the approximations is performed in Section 3. Here a compactness property of the approximated gain terms in mild form is carried over to the corresponding solutions themselves, using a particular sequence of successive alternating approximations and the Kolmogorov-Riesz theorem [10], [11]. The approach also holds for domains which are strictly convex with $C^{1}$ boundary.

A common approach to existence for stationary Boltzmann like equations is based on the regularizing properties of the gain term. In the continuous velocity case an averaging propery is available to keep this study of the gain term within a weak $L^{1}$ frame as in [2]. However, in the discrete velocity case, averaging is not available. Instead strong convergence of an approximating sequence is here directly proved from the regularizing properties for the gain term (cf Lemma 3.5 below). But the technique in that proof is restricted to two dimensional velocities, whereas the averaging technique in the continuous velocity case is dimension independent.

2 Approximations.

Denote by $L^{1}_{+}([0,1]^{2})$ the set of non negative integrable functions on $[0,1]^{2}$, and by $a\wedge b$ the minimum of two real numbers $a$ and $b$. Approximations to (1.1) to be used in the proof of Theorem 1, are introduced in the following lemma.

Lemma 2.1

For any $k\in\mathbb{N}^{*}$, there exists a solution F^{k}\in\big{(}L^{1}_{+}([0,1]^{2})\big{)}^{4} to

[TABLE]

Proof of Lemma 2.1.

The sequence of approximations $(F^{k})_{k\in\mathbb{N}^{*}}$ is obtained in the limit of a further approximation with damping terms $\alpha F_{j}$ and convolutions in the collision operator.

                      

Step I. Approximations with damping and convolutions.

Take $\alpha>0$ and set

[TABLE]

Let $\mu_{\alpha}$ be a smooth mollifier in $(x,y)$ with support in the ball centered at the origin of radius $\alpha$. Let $\mathcal{T}$ be the map defined on $K_{\alpha}$ by $\mathcal{T}(f)=F$, where $F=(F_{i})_{1\leq i\leq 4}$ is the solution of

[TABLE]

$F=\mathcal{T}(f)$ is obtained as the limit in $L^{1}([0,1]^{2})$ of the sequence $(F^{n})_{n\in\mathbb{N}}$ defined by $F^{0}=0$ and

[TABLE]

The sequence $(F^{n})_{n\in\mathbb{N}}$ is monotone. Indeed, $F_{i}^{0}\leq F_{i}^{1}$, $1\leq i\leq 4$ by the exponential form of $F_{i}^{1}$. Moreover, assume $F_{i}^{n}\leq F_{i}^{n+1}$, $1\leq i\leq 4$. It follows from the exponential form that $F_{1}^{n+1}-F_{1}^{n+2}\leq 0$. The inequalities $F_{i}^{n+1}-F_{i}^{n+2}\leq 0$, $2\leq i\leq 4$ can be reached in a similar way. Moreover,

[TABLE]

so that

[TABLE]

By the monotone convergence theorem, $(F^{n})_{n\in\mathbb{N}}$ converges in $L^{1}([0,1]^{2})$ to some solution $F$ of (2.8)-(2.13). The solution of (2.8)-(2.13) is unique in the set of non negative functions. Indeed, let $G=(G_{i})_{1\leq i\leq 4}$ be a solution of (2.8)-(2.13) with $G_{i}\geq 0$, $1\leq i\leq 4$. Let us prove by induction that

[TABLE]

(2.15) holds for $n=0$, since $G_{i}\geq 0$, $1\leq i\leq 4$. Assume (2.15) holds for $n$. Using the exponential form of $F_{1}^{n+1}$ implies

[TABLE]

The same argument can be applied to prove that $F_{i}^{n+1}\leq G_{i}$, $2\leq i\leq 4$. Consequently,

[TABLE]

Moreover, substracting the partial differential equations satisfied by $G_{i}$ from the partial differential equations satisfied by $F_{i}$, $1\leq i\leq 4$, and integrating the resulting equation on $[0,1]^{2}$, it results

[TABLE]

It results from (2.16)-(2) that $G=F$.

The map $\mathcal{T}$ is continuous in the $L^{1}$-norm topology (cf [1] pages 124-5). Namely, let a sequence $(f_{l})_{l\in\mathbb{N}}$ in $K_{\alpha}$ converge in $L^{1}([0,1]^{2})$ to $f\in K_{\alpha}$. Set $F_{l}=T(f_{l})$. Because of the uniqueness of the solution to (2.8)-(2.13), it is enough to prove that there is a subsequence of $(F_{l})$ converging to $F=T(f)$. Now there is a subsequence of $(f_{l})$, still denoted $(f_{l})$, such that decreasingly (resp. increasingly) $(G_{l})=(\sup_{m\geq l}f_{m})$ (resp. $(g_{l})=(\inf_{m\geq l}f_{m})$) converges to $f$ in $L^{1}$. Let $(S_{l})$ (resp. $(s_{l})$) be the sequence of solutions to

[TABLE]

(resp.

[TABLE]

$(S_{l})$ is a non-increasing sequence, since that holds for the successive iterates defining the sequence. Then $(S_{l})$ decreasingly converges in $L^{1}$ to some $S$. Similarly $(s_{l})$ increasingly converges in $L^{1}$ to some $s$. The limits $S$ and $s$ satisfy (2.8)-(2.13). It follows by uniqueness that $s=F=S$, hence that $(F_{l})$ converges in $L^{1}$ to $F$.

The map $\mathcal{T}$ is also compact in the $L^{1}$-norm topology. Indeed, let $(f_{l})_{l\in\mathbb{N}}$ be a sequence in $K_{\alpha}$ and $(F_{l})_{l\in\mathbb{N}}=(\mathcal{T}(f_{l}))_{l\in\mathbb{N}}$. For any $\lvert h\rvert<1$, denote by $G_{l1}(x,y)=F_{l1}(x,y+h)-F_{l1}(x,y)$ and

[TABLE]

They satisfy

[TABLE]

so that

[TABLE]

The boundedness by $k^{2}$ of the integrands in the r.h.s. of (2.8) and (2.10) induces uniform $L^{1}$-equicontinuity of $(F_{l1})_{l\in\mathbb{N}}$ (resp. $(F_{l3})_{l\in\mathbb{N}}$) w.r.t. the $x$ (resp. $y$) variable. Together with the $L^{1}$-compactness of $(f_{l}\ast\mu_{\alpha})_{l\in\mathbb{N}}$, this implies uniform $L^{1}$-equicontinuity w.r.t. the $y$ variable of $(H_{l1})_{l\in\mathbb{N}}$, then of $(F_{l1})_{l\in\mathbb{N}}$. This proves the $L^{1}$ compactness of $(F_{l1})_{l\in\mathbb{N}}$. The $L^{1}$ compactness of $(F_{li})_{l\in\mathbb{N}}$, $2\leq i\leq 4$ can be proven similarly.

Hence by the Schauder fixed point theorem there is a fixed point $\mathcal{T}(F)=F$, i.e. a solution $F$ to

[TABLE]

Step II. Removal of the damping and the convolutions in (2.30)-(2.35).

Let $k>1$ be fixed. Denote by $F^{\alpha}$ the solution to (2.30)-(2.35) defined in Step I. Each component of $F^{\alpha}$ being bounded by a multiple of $k^{2}$, $(F^{\alpha})_{\alpha\in]0,1[}$ is weakly compact in $L^{1}([0,1]^{2})$. Denote by $F^{k}$ a limit of a subsequence for the weak topology of $L^{1}([0,1]^{2})$. Let us prove that the convergence is strong in $L^{1}([0,1]^{2})$. Consider the approximation scheme $(f^{\alpha,l}_{1},f^{\alpha,l}_{2})_{l\in\mathbb{N}}$ of $(F^{\alpha}_{1},F^{\alpha}_{2})_{\alpha\in]0,1[}$,

[TABLE]

By induction on $l$ it holds that

[TABLE]

For every $l\in\mathbb{N}$, $(f^{\alpha,l}_{1})_{\alpha\in]0,1[}$ (resp. $(f^{\alpha,l}_{2})_{\alpha\in]0,1[}$) is translationnaly equicontinuous in the $x$-direction, since all integrands in its exponential form are bounded. It is translationnaly $L^{1}$-equicontinuous in the $y$-direction by induction on $l$. Indeed, it is so for $(F^{\alpha}_{3})$ (resp. $(F^{\alpha}_{4})$) since $\partial_{y}(e^{\alpha y}F^{\alpha}_{3})$ ( resp. $\partial_{y}(e^{\alpha y}F^{\alpha}_{4})$) is bounded by $ek^{2}$, and $(\frac{F^{\alpha}_{i}}{1+\frac{F^{\alpha}_{i}}{k}})_{\alpha\in]0,1[}$, $i\in\{3,4\}$, is bounded by $k$. Consequently, it is so for $(\frac{F^{\alpha}_{3}}{1+\frac{F^{\alpha}_{3}}{k}}\frac{F^{\alpha}_{4}\ast\mu_{\alpha}}{1+\frac{F^{\alpha}_{4}\ast\mu_{\alpha}}{k}})_{\alpha\in]0,1[}$. There is a limit sequence $(g^{l}_{1},g^{l}_{2})$ in $(L^{1}([0,1]^{2}))^{2}$ such that up to subsequences $(f^{\alpha,l}_{1})$ (resp. $(f^{\alpha,l}_{2})$) converges to $g^{l}_{1}$ (resp. $g^{l}_{2}$) in $L^{1}([0,1]^{2})$ when $\alpha\rightarrow 0$. They satisfy

[TABLE]

where $G$ is the weak $L^{1}$ limit of $(\frac{F^{\alpha}_{3}}{1+\frac{F^{\alpha}_{3}}{k}}\frac{F^{\alpha}_{4}\ast\mu_{\alpha}}{1+\frac{F^{\alpha}_{4}\ast\mu_{\alpha}}{k}})_{\alpha\in]0,1[}$ when $\alpha\rightarrow 0$. In particular, $(g^{2l}_{1})_{l\in\mathbb{N}}$ and $(g^{2l}_{2})_{l\in\mathbb{N}}$ (resp $(g^{2l+1}_{1})_{l\in\mathbb{N}}$ and $(g^{2l+1}_{2})_{l\in\mathbb{N}}$) non decreasingly (resp. non increasingly) converge in $L^{1}$ to some $g_{1}$ and $g_{2}$ (resp. $h_{1}$ and $h_{2}$) when $l\rightarrow+\infty$. The limits satisfy

[TABLE]

Hence,

[TABLE]

and

[TABLE]

The non negativity of $h_{1}-g_{1}$, $g_{1}$, $g_{2}$, $h_{1}$ and $h_{2}$ implies that $h_{1}-g_{1}=0$. The same holds for $h_{2}-g_{2}$. Consequently

[TABLE]

$(F^{\alpha}_{1})_{\alpha\in]0,1[}$ converges to $F^{k}_{1}$ in $L^{1}([0,1]^{2})$ when $\alpha\rightarrow 0$. Indeed, given $\eta>0$, choose $l_{0}$ big enough so that $\parallel g^{2l_{0}+1}_{1}-g^{2l_{0}}_{1}\parallel_{L^{1}}<\eta$ and $\parallel g^{2l_{0}}_{1}-F^{k}_{1}\parallel_{L^{1}}<\eta$, then $\alpha_{0}$ small enough so that

[TABLE]

Then split $\parallel F^{\alpha}_{1}-F^{k}_{1}\parallel_{L^{1}}$ as follows

[TABLE]

The $L^{1}$ convergence of $(F^{\alpha}_{i})_{k\in\mathbb{N}}$ to $F^{k}_{i}$, $2\leq i\leq 4$, can be proven similarly. Passing to the limit when $\alpha\rightarrow 0$ in (2.30)-(2.35) is straightforward. And so, $F^{k}$ is a solution to (2.1)-(2.6).

3 Passage to the limit when $k\rightarrow+\infty$.

The study of the passage to the limit is split into six lemmas. In Lemma 3.1, uniform bounds are obtained for mass, entropy and entropy production term of the approximations. Lemma 3.2 splits $[0,1]^{2}$ into ‘large’ sets of type $0\leq x\leq 1$ times a ’large’ set in $y$ for $(F^{k}_{1},F^{k}_{2})$ (resp. a ’large’ set in $x$ times $0\leq y\leq 1$ for $(F^{k}_{3},F^{k}_{4})$), where the approximations are uniformly bounded in $L^{\infty}$, and their complements where the mass of the approximations is small. Lemma 3.3 proves uniform equicontinuity with respect to the $x$ (resp. $y$) variable of the two first (resp. last) components of the approximations. In Lemma 3.4, $L^{1}$-compactness of a truncated gain term of the approximations is proven. Lemma 3.5 proves that the approximations form a Cauchy sequence in $L^{1}([0,1]^{2})$. Their limit is proven to be a renormalized solution to the Broadwell model in Lemma 3.6 .

In this section, $c_{b}$ denotes constant that only depend on the given boundary value $f_{b}$.

Lemma 3.1

There are constants $c_{b}$ such that

[TABLE]

Proof of Lemma 3.1.

Adding (2.1)-(2.4), integrating the resulting equation on $[0,1]^{2}$ and taking (2.5)-(2.6) into account, implies that total outflow equals total inflow. Also using $\partial_{x}(F_{1}^{k}+F_{2}^{k})=\partial_{y}(F_{3}^{k}+F_{4}^{k})=0$ implies boundedness of the total mass $\sum_{i=1}^{4}\int F^{k}_{i}(x,y)dxdy$. Multiply (2.1) (resp. (2.2), resp. (2.3), resp. (2.4)) by $\ln\frac{F^{k}_{1}}{1+\frac{F^{k}_{1}}{k}}$ (resp. $\ln\frac{F^{k}_{2}}{1+\frac{F^{k}_{2}}{k}}$, resp. $\ln\frac{F^{k}_{3}}{1+\frac{F^{k}_{3}}{k}}$, resp. $\ln\frac{F^{k}_{4}}{1+\frac{F^{k}_{1}}{4}}$), add the corresponding equations, and integrate the resulting equation on $[0,1]^{2}$. Denoting by $D^{k}$ the entropy production term for the approximation $F^{k}$,

[TABLE]

leads to

[TABLE]

Moreover,

[TABLE]

Hence

[TABLE]

Consequently,

[TABLE]

And so, (3.3) holds. Moreover, for any $\Lambda>2$ and $k>2$,

[TABLE]

In particular,

[TABLE]

Since

[TABLE]

it holds that

[TABLE]

Consequently, for some subset $\omega_{k}$ of $[0,1]$ such that $\lvert\omega_{k}\rvert<\frac{c}{k}$,

[TABLE]

by (3) and the boundedness of the $f_{b2}$ entropy.

Lemma 3.2

For $\epsilon>0$, $\Lambda\geq\exp(\frac{2c_{b}}{\epsilon})$ and $k\geq\exp(\frac{3c_{b}}{\epsilon})$, there is a subset $\Omega^{\epsilon\Lambda}_{k1}$ of $[0,1]$ with measure smaller than $\frac{c_{b}\epsilon}{\Lambda}$ such that

[TABLE]

Proof of Lemma 3.2.

Since $f_{b2}\in L^{1}([0,1])$ and

[TABLE]

the measure of the set

[TABLE]

is smaller than $\frac{c_{b}\epsilon}{\Lambda}$. $(F^{k}_{1},F^{k}_{2})$ is uniformly bounded on $[0,1]\times([0,1]\setminus\Omega^{\epsilon\Lambda}_{k1})$, since

[TABLE]

and

[TABLE]

Moreover, for any $\Lambda\geq\exp(\frac{2c_{b}}{\epsilon})$ and $k\geq\exp(\frac{3c_{b}}{\epsilon})$,

[TABLE]

Lemma 3.3

There is $c_{b}>0$, and for $\epsilon>0$ given there is $\delta>0$ such that for $\lvert h\rvert<\delta$, uniformly in $k\in\mathbb{N}^{*}$,

[TABLE]

Proof of Lemma 3.3.

The four cases $F^{k}_{1}$,…, $F^{k}_{4}$ are analogous. The detailed estimates are carried out for $F^{k}_{1}$. The translational $L^{1}$ equicontinuity in the $x$-direction for $\ln(1+F^{k}_{1})$ is obtained as follows from the $\partial_{x}$-term in the renormalized equation. Consider $h\in[0,1[$. Write the equation for $F^{k}_{1}$ in renormalized form (1.4) integrated on $[x,x+h]$, where the integration from $x+h>1$ tending to zero with $h$ uniformly in $k$, is being omitted from the following computations;

[TABLE]

Denote by $\rm{sgn}$ the sign function, $\rm{sgn}(r)=1$ if $r>0$, $\rm{sgn}(r)=-1$ if $r<0$. Multiply the previous equation by \rm{sgn}\big{(}\ln(1+F^{k}_{1}(x+h,y))-\ln(1+F^{k}_{1}(x,y))\big{)} and integrate on $[0,1]^{2}$. Uniformly w.r.t. $k\in\mathbb{N}^{*}$,

[TABLE]

Recall that for any non negative real numbers $x_{1}>x_{2}$, there is $\theta\in]0,1[$ such that

[TABLE]

And so the $L^{1}$-norms of the translation differences of $F^{k}_{1}$ and $\ln(1+F^{k}_{1})$, are equivalent on $[0,1]\times([0,1]\setminus\Omega^{\epsilon\Lambda}_{k1})$ since $F^{k}_{1}$ and $(x,y)\rightarrow F^{k}_{1}(x+h,y)$ are bounded in $L^{\infty}([0,1]\times([0,1]\setminus\Omega^{\epsilon\Lambda}_{k1}))$. There is also the small set with mass bounded by $\epsilon$, where $(x,y)\rightarrow F^{k}_{1}(x+h,y)$ is not in $\Omega^{\epsilon\Lambda}_{k1}$. Together with (3) this proves the translational equicontinuity in the $x$-direction for $k\geq\exp(\frac{3c_{b}}{\epsilon})$. The proof for $h\in]-1,0[$ is similar.  

Given $\epsilon>0$, $\Lambda\geq\exp(\frac{2c_{b}}{\epsilon})$ and $k\geq\exp(\frac{3c_{b}}{\epsilon})$, let $\Omega^{\epsilon\Lambda}_{k1}\subset[0,1]$ as defined in Lemma 3.2, and take $\chi^{\epsilon\Lambda}_{k1}$ as the corresponding cutoff function,

[TABLE]

Lemma 3.4

Let $(\alpha^{k})_{k\in\mathbb{N}}$ be a non negative sequence bounded in $L^{\infty}$ and compact in $L^{1}$ . The sequences

[TABLE]

are compact in $L^{1}([0,1]^{2})$ (resp. in $L^{1}([0,1]))$.

Proof of Lemma 3.4. For any $\gamma>1$, using Lemmas 3.1-3.2,

[TABLE]

Choosing $\gamma$ big enough, then $h$ small enough, proves the translational $L^{1}$ equicontinuity in the $x$ direction of \Big{(}\chi_{k1}^{\epsilon\Lambda}(y)\int_{0}^{x}\frac{F^{k}_{3}}{1+\frac{F^{k}_{3}}{k}}\frac{F^{k}_{4}}{1+\frac{F^{k}_{4}}{k}}(X,y)e^{-\int_{X}^{x}\alpha^{k}(u,y)du}dX\Big{)}_{k\in\mathbb{N}^{*}}. Let us prove its translational $L^{1}$ equicontinuity in the $y$ direction. Given $\tilde{\epsilon}>0$, let

[TABLE]

Let $\Omega^{\epsilon_{3}\Lambda_{3}}_{k3}\subset[0,1]$ as defined in Lemma 3.2 for $(F^{k}_{3},F^{k}_{4})$, and $\chi^{\epsilon_{3}\Lambda_{3}}_{k3}$ the corresponding cutoff function,

[TABLE]

First,

[TABLE]

Moreover,

[TABLE]

Given the boundedness of $(F^{k}_{3},F^{k}_{4})_{k\geq\exp(\frac{3c_{b}}{\epsilon_{3}})}$ on \big{(}\Omega_{k3}^{\epsilon_{3}\Lambda_{3}}\big{)}^{c}\times[0,1], and the statements of Lemmas 3.2-3.3 for $(F^{k}_{3},F^{k}_{4})$, there is $h_{3}>0$ such that

[TABLE]

for $h\in]0,h_{3}[$, uniformly with respect to $k\geq\exp(\frac{3c_{b}}{\epsilon_{3}})$.

The proofs of the $L^{1}([0,1]^{2})$ (resp. $L^{1}([0,1])$) compactness of

[TABLE]

are similar.

Lemma 3.5

$(F^{k})_{k\in\mathbb{N}^{*}}$* is compact in $L^{1}([0,1]^{2})$.

Proof of Lemma 3.5.

By (3.1)-(3.2), $(F^{k})_{k\in\mathbb{N}^{*}}$ is weakly compact in $(L^{1}([0,1]^{2}))^{4}$. Denote by $F$ the weak limit of a subsequence, still denote $(F^{k})$. Let us prove that $(F^{k}_{1})_{k\in\mathbb{N}^{*}}$ is strongly compact in $L^{1}([0,1]^{2})$. It is by (3.8) enough to prove that up to a subsequence, given $\epsilon>0$, for $\Lambda\geq e^{\frac{2c_{b}}{\epsilon}}$, $k\geq e^{\frac{3c_{b}}{\epsilon}}$ and $\Omega^{\epsilon\Lambda}_{k1}$ as defined in Lemma 3.2, $(\chi^{\epsilon\Lambda}_{k1}F^{k}_{1})_{k\in\mathbb{N}^{*}}$ is strongly compact in $L^{1}([0,1]^{2})$. For every $F^{k}$ in the subsequence, consider the approximation scheme $(f^{k,l}_{1},f^{k,l}_{2})_{l\in\mathbb{N}}$ of $(F^{k}_{1},F^{k}_{2})$, defined by

[TABLE]

By induction on $l$, and using an exponential form of $(f_{1}^{k,l+1},f_{2}^{k,l+1})$, it holds that

[TABLE]

and

[TABLE]

The sequence $(\chi^{\epsilon\Lambda}_{k1}f^{k,2l}_{1})_{k\geq e^{\frac{3c_{b}}{\epsilon}}}$ (resp. $(\chi^{\epsilon\Lambda}_{k1}f^{k,2l}_{2})_{k\geq e^{\frac{3c_{b}}{\epsilon}}}$) is bounded from above by $(\chi^{\epsilon\Lambda}_{k1}F^{k}_{1})_{k\geq e^{\frac{3c_{b}}{\epsilon}}}$ (resp. $(\chi^{\epsilon\Lambda}_{k1}F^{k}_{2})_{k\geq e^{\frac{3c_{b}}{\epsilon}}}$), hence by $\frac{2\Lambda}{\epsilon}\exp(\frac{2\Lambda}{\epsilon})$. The sequence $(\chi^{\epsilon\Lambda}_{k1}f^{k,2l+1}_{1})_{k\geq e^{\frac{3c_{b}}{\epsilon}}}$ (resp. $(\chi^{\epsilon\Lambda}_{k1}f^{k,2l+1}_{2})_{k\geq e^{\frac{3c_{b}}{\epsilon}}}$) is bounded by $\frac{2\Lambda}{\epsilon}\exp(\frac{2\Lambda}{\epsilon})(1+\frac{2\Lambda}{\epsilon}\exp(\frac{2\Lambda}{\epsilon}))$, since

[TABLE]

By Lemma 3.4, there is a subsequence of \big{(}\chi_{k1}^{\epsilon\Lambda}(y)\int_{0}^{1}\frac{F^{k}_{3}}{1+\frac{F^{k}_{3}}{k}}\frac{F^{k}_{4}}{1+\frac{F^{k}_{4}}{k}}(X,y)dX\big{)}_{k\in\mathbb{N}^{*}}, still denoted by

\big{(}\chi_{k1}^{\epsilon\Lambda}(y)\int_{0}^{1}\frac{F^{k}_{3}}{1+\frac{F^{k}_{3}}{k}}\frac{F^{k}_{4}}{1+\frac{F^{k}_{4}}{k}}(X,y)dX\big{)}_{k\in\mathbb{N}^{*}}, converging in $L^{1}([0,1])$ to some $\tilde{F}_{1}$. Given $\eta>0$, there is a subset $\omega_{\eta}$ of $[0,1]$ with measure smaller than $\eta$ such that on $\omega_{\eta}^{c}$ the convergence of this sequence is uniform and $(\tilde{F}_{1},f_{b1},f_{b2})$ is bounded. It follows from (3.14)-(3.15) and the non-negativity of $(f_{1}^{k,2l},f_{2}^{k,2l})_{(k,l)\in\mathbb{N}^{2}}$ that $(f_{1}^{k,2l},f_{2}^{k,2l})_{(k,l)\in\mathbb{N}^{2}}$ is bounded on $[0,1]\times\omega_{\eta}^{c}$. Given these bounds, Lemma 3.4 and the expression of $(f_{1}^{k,l},f_{2}^{k,l})$ in exponential form, it holds by induction that for each $l\in\mathbb{N}$, the sequence $(f^{k,l}_{1},f^{k,l}_{2})_{k\geq e^{\frac{3c_{b}}{\epsilon}}}$ is strongly compact in $L^{1}([0,1]\times\omega_{\eta}^{c})$. Denote by $(g^{l}_{1},g^{l}_{2})$ its limit up to a subsequence. By Lemma 3.4, let $G$ (resp. $H$) with $\partial_{x}G=-\partial_{x}H$, be the limit in $L^{1}$ when $k\rightarrow+\infty$ of

[TABLE]

$(g^{2l}_{1},g^{2l}_{2},g^{2l+1}_{1},g^{2l+1}_{2})$ satisfies

[TABLE]

By induction on $l$ it holds that

[TABLE]

Moreover,

[TABLE]

By the monotone convergence theorem, $(g^{2l})_{l\in\mathbb{N}}$ (resp. $(g^{2l+1})_{l\in\mathbb{N}}$) increasingly (resp. decreasingly) converges in $L^{1}([0,1]\times\omega_{\eta}^{c})$ and almost everywhere on $[0,1]\times\omega_{\eta}^{c}$ to some $g$ (resp. $h$). By the dominated convergence theorem,

[TABLE]

Consequently,

[TABLE]

and

[TABLE]

Hence

[TABLE]

so that, by (3.20),

[TABLE]

Consequently, $h_{1}-g_{1}=h_{2}-g_{2}$, $g_{1}h_{2}-h_{1}g_{2}=(g_{1}-g_{2})(h_{1}-g_{1})$ and

[TABLE]

It follows from $(h_{1}-g_{1})(0,y)=0$ and the boundedness of $(g_{1},g_{2})$ on $[0,1]\times\omega_{\eta}^{c}$ that $h_{1}-g_{1}=0$ and $(g_{1},g_{2})=(h_{1},h_{2})$ on $[0,1]\times\omega_{\eta}^{c}$. Hence the whole sequence $(g_{1}^{l},g_{2}^{l})_{l\in\mathbb{N}}$ converges to $(g_{1},g_{2})$ in $L^{1}([0,1]\times\omega_{\eta}^{c})$. Letting $\eta\rightarrow 0$ and using (2.16), the convergence holds in $L^{1}([0,1]^{2})$.

Given $\bar{\epsilon}>0$, choose $l_{0}$ big enough so that $\parallel g^{2l_{0}}_{1}-g^{2l_{0}+1}_{1}\parallel_{L^{1}}<\bar{\epsilon}$, then $k_{0}$ big enough so that

[TABLE]

Hence $\parallel f_{1}^{k,2l_{0}+1}-f_{1}^{k,2l_{0}}\parallel_{L^{1}}\leq 3\bar{\epsilon}$  for $k\geq k_{0}$. Then

[TABLE]

And so $(F_{1}^{k})$ is a Cauchy sequence in $L^{1}([0,1]^{2})$ with the limit equal to the weak limit $F_{1}$. Similarly, $(F_{j}^{k})_{2\leq j\leq 4}$ is a Cauchy sequence in $(L^{1}([0,1]^{2}))^{3}$ with the limit equal to the weak limit $(F_{j})_{2\leq j\leq 4}$.

Lemma 3.6

The limit $F$ of $(F^{k})_{k\in\mathbb{N}^{*}}$ in $L^{1}([0,1]^{2})$ is a renormalized solution to the Broadwell model (1.1).

Proof of Lemma 3.6.

Start from a renormalized formulation of (2.1),

[TABLE]

for test functions $\varphi\in(C^{1}([0,1]^{2}))^{4}$. Using the strong $L^{1}$ convergence of the sequence $(F^{k})$ to pass to the limit when $k\rightarrow+\infty$ in the left hand side of (3), gives in the limit,

[TABLE]

For the passage to the limit when $k\rightarrow+\infty$ in the right hand side of (3), given $\eta>0$ there is a subset $A_{\eta}$ of $[0,1]^{2}$ with $\lvert A_{\eta}^{c}\rvert<\eta$, such that up to a subsequence, $(F^{k})_{k\in\mathbb{N}^{*}}$ uniformly converges to $F$ on $A_{\eta}$ and $F\in L^{\infty}(A_{\eta})$. Passing to the limit when $k\rightarrow+\infty$ on $A_{\eta}$ is straightforward. Moreover,

[TABLE]

uniformly with respect to $k$, since

[TABLE]

uniformly with respect to $k$.

The gain term can be estimated as follows. The uniform boundedness of the entropy production term of $(F^{k})$ is given in Lemma 3.1. A convexity argument together with the $L^{1}$ convergence of $(F^{k})$ to $F$ (see [7]), imply that

[TABLE]

It follows that, for any $\gamma>1$,

[TABLE]

which tends to zero when $\eta\rightarrow 0$. Similarly, using (3.3),

[TABLE]

which tends to zero when $\eta\rightarrow 0$, uniformly in $k$. It follows that the right hand side of (3) converges to

[TABLE]

when $k\rightarrow+\infty$. Consequently, $F_{1}$ satisfies the first equation of (1.1) in renormalized form. It can be similarly proven that $(F_{j})_{2\leq j\leq 4}$ is solution to the last equations of (1.1).  

This completes the proof of Theorem 2.1.