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\markboth{\hfil Convergence of a continuous BGK model \hfil EJDE--2001/72}
{EJDE--2001/72\hfil Driss Seghir \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol. {\bf 2001}(2001), No.~72, pp. 1--18. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp)}
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Convergence of a continuous BGK model for initial boundary-value
problems \\ for conservation laws
%
\thanks{ {\em Mathematics Subject Classifications:} 35L65, 35B25, 82C40.
\hfil\break\indent
{\em Key words:} Conservation laws, boundary condition, BGK model.
\hfil\break\indent
\copyright 2001 Southwest Texas State University. \hfil\break\indent
Submitted October 15, 2001. Published November 26, 2001.} }
\date{}
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\author{Driss Seghir}
\maketitle
\begin{abstract}
We consider a scalar conservation law in the quarter plane.
This equation is approximated in a continuous kinetic
Bhatnagar-Gross-Krook (BGK) model. The convergence of the model
towards the unique entropy solution is established in the space
of functions of bounded variation, using kinetic entropy
inequalities, without special restriction on the flux nor on the
equilibrium problem's data.
As an application, we establish the hydrodynamic limit for a $2\times2$
relaxation system with general data. Also we construct a new family
of convergent continuous BGK models with simple maxwellians different
from the $\chi$ models.
\end{abstract}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
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\section{Introduction}
We consider the initial boundary-value problem, for a one-dimensional
scalar conservation law,
\begin{equation}
\partial_t u+\partial_x F(u)=0,\label{law}
\end{equation}
for $(x,t)\in\mathbb{R}\times(0,T)$ and $F$ a smooth flux function
with the initial condition
\begin{equation}
u(x,0)=u^0(x),\label{u initiale}
\end{equation}
for $x\in\mathbb{R}^+$. The boundary condition
$$
u(0,t)=u_b(t)\quad\mbox{for }t\geq 0,
$$
can not be assumed in the proper sense because this is not quite simply true.
Our boundary condition will be formulated as a compatibility \cite{BLN}
\begin{equation}
\sup\{\mathop{\rm sgn}(u(0,t)-u_b(t))(F(u(0,t))-F(k))\}=0,\label{BLN}
\end{equation}
for $t\in[0,T]$, where $\mathop{\rm sgn}(u)$ is the sign of $u$ and where the
$\sup$ is taken over $k$ lying between $u(0,t)$ and $u_b(t)$.
We recall that $u_b(t)$ is the prescribed boundary condition and
that (\ref{BLN}) means that $u(0,t)=u_b(t)$ whenever the flow is incoming,
i.e. $F'(u(0,t))>0$.
We will look at (\ref{law})-(\ref{BLN}) as an equilibrium for the
scalar Bhatnagar-Gross-Krook (BGK) model with (eventually) infinite set of
velocities,
\begin{equation}
\partial_t f+a(\xi)\partial_x f=\frac{M_f-f}{\epsilon},\label{model}
\end{equation}
where $(x,t)\in\mathbb{R}\times(0,T)$, $\xi$ is in a measure space $\Xi$
with measure $d\xi$, $f(x,t,\xi)$ is the unknown depending also on $\epsilon$, $a(\xi)$ is the velocity and where:
$$
M_f(x,t,\xi)=M(u^\epsilon(x,t),\xi),\quad u^\epsilon(x,t)=\int f(x,t,\xi)\,d\xi,
$$
are the maxwellian or equilibrium state, and the first momentum or density respectively.
In the next section, we will add some conditions on the maxwellian $M:\mathbb{R}\times\Xi\rightarrow\mathbb{R}$ so that (a subsequence of) $u^\epsilon$ converges to $u$, the
unique entropy solution of (\ref{law})-(\ref{BLN}), and $f$ approaches from $M(u,\xi)$
when $\epsilon$ goes to zero.
This model will be supplemented by the initial and boundary data
\begin{equation}
f(x,0,\xi)=M(u^0(x),\xi),\label{f initiale}
\end{equation}
for $(x,\xi)\in\mathbb{R}^+\times\Xi$ and
\begin{equation}
f(0,t,\xi)=M(u_b(t),\xi)\quad\mbox{if}\quad a(\xi)>0,\label{f bord}
\end{equation}
for $t\in[0,T]$. When $\epsilon\to 0$, $f(x,t,\xi)$ is intended to be near
$M(u(x,t),\xi)$, so we naturally assumed the initial-boundary data at equilibrium.
Let us also recall about the boundary condition that, when
$\Xi=\{a_1,\dots ,a_N\}$ is finite, one must add $l$ linear boundary
conditions of the form
$$E(f_1(0,t),\dots ,f_N(0,t))^t=G(t),$$
where $E$ is a $l\times N$ matrix, $G$ is a $l$-component given function and
$l$ is the number of positive velocities $a_i$ \cite{H}. (\ref{f bord})
is a good way to express this fact in our circumstances as well as it is
nothing but (\ref{BLN}) for the scalar transport equation (\ref{model})
for fixed $\xi$ at the equilibrium $M_f=f$.
Our main task in this work is to show that the model (\ref{model})-(\ref{f bord}) describes
the problem (\ref{law})-(\ref{BLN}) when $\epsilon$ goes to zero. This will be made in a
bounded variation (BV) framework, for general flux $F$ and general BV-initial-boundary data $u^0$ and $u_b$
while comparing with \cite{NOV} and \cite{WX} respectively.
Our work appears in the more general setting of the relaxation which was deeply studied
last years in its theoretical and numerical aspects . We can cote in the Cauchy problem
case \cite{AN, C, CLL, CR, JX, KT, LW, N1, S, TT,TW} and one can see
\cite{N2} and references therein for more information.
For relaxation with boundary condition, there is an important $BV\cap L^\infty$ analysis
in \cite{WX}, especially when the initial boundary data are some small
perturbations of a constant non-transonic state. This rather restrictive
conditions will be removed in Example 7.1 where we show in fact
that the $2\times2$ system of \cite{WX}
describes the equilibrium law (\ref{law}) for general BV data.
In studying boundary value problem, the early relaxation stability conditions for Cauchy
problems may fail to imply the existence of the hydrodynamic limit, boundary layers can
appear and there are cases where the equilibrium system must be supplemented by proper
boundary conditions to determine the uniqueness in the limiting process. Such questions
are treated in \cite{LY, Ni, NY,Y}.
Relaxation schemes for conservation laws in the quarter plan can be found in
\cite{BCV, CS}.
Concerning kinetic BGK models with continuous cite of velocities with the maxwellian $\chi$,
the Cauchy problem is studied in \cite{PT}, see also \cite{B} and reference therein.
A weak entropy study of the initial boundary problem can be found in \cite{NOV} where the
authors established the hydrodynamic limit in several space dimensions but with a
restriction on the flux which must be convex, concave, non-increasing or non-decreasing.
Our technique in recovering boundary entropy condition allows us to remove this restriction.
Other works deal with BGK model in the quarter plan with finite cite of velocities.
In \cite{NT}, the authors treat BGK model with two velocities . An extension of their
techniques to more than two velocities is in \cite{M}. But the extension of this
techniques to continuous BGK model case seems difficult, especially in bounding the
variation in space variable $x$. We overcome this difficulty by using both \cite{NOV}
and \cite{NT} ideas.
Let us recall that boundary conditions carry on supplementary complications in such
approximation problems. We must not only impose correct conditions for the well-posedness
of the conservation law (\ref{law}) and the model (\ref{model}), but we must also try
to avoid the apparition of boundary layers. We chose the condition (\ref{BLN}) emanating
from parabolic viscosity approach of the approximated conservation law (\ref{BLN}). There
is another approach to exhibit correct boundary conditions by solving Riemann problems.
These two formulations are equivalent for linear systems and scalar conservation laws
\cite{DL, KSX}. Concerning the BGK model, we chose the simplest way to write boundary
conditions by respecting the ideas giving well-posedness of linear systems with finite
cite of velocities as in \cite{H, K}, on the one hand and by foreseeing the
equilibrium phenomenon on the boundary on the other hand.
We will see throughout this paper that the monotony and the momentum equations of the
maxwellian $M$ still give the BV compactness and the stability respectively, exactly
as in the Cauchy problem case. We also imitate the Cauchy situation in using an infinite
set of kinetic entropy inequalities \cite{B,N3,PT}.
The paper is organized as follows. In the next
section we specify some general and basic facts about equilibrium law, maxwellian and
kinetic entropies. We study the well-posedness of the BGK model in section
3. Sections 4 and 5 are devoted respectively to $L^\infty$ and BV stability estimates.
In section 6 we prove that our kinetic BGK model describes the initial
conservation law (\ref{law}) by $L^1$ compactness in BV and using kinetic entropy
$H$-functions with careful treatment of the calculus near the boundary. The last
section contains two examples, namely the convergence of the relaxation $2\times2$
system of \cite{WX} with general initial-boundary data and a continuous BGK model
with maxwellian distinct from the $\chi$ one.
\section{General setting}
Let us specify the meaning of (weak entropy) solutions of (\ref{law})-(\ref{BLN}) and
some needful technical assumptions.
\begin{definition}\label{solution} \rm
Let $u^0\in L^1(\mathbb{R}^+)$ and $u_b\in L^1(0,T)$. We say that a function
$u\in BV(\mathbb{R}^+\times(0,T))$ is a solution of (\ref{law})-(\ref{BLN})
if for all $k\in \mathbb{R}$
and all nonnegative test function $\phi \in C_{c}^{1}(\mathbb{R}^+\times[0,T))$
we have
\begin{eqnarray*}
&&\int|u - k|\partial_t\phi + \mathop{\rm sgn}(u - k)(F(u) - F(k))
\partial_x\phi\,dx\,dt \\
&&+ \int |u^0(x) - k|\phi(x,0)\,dx + \int \mathop{\rm sgn}
(u_{b}(t)-k)(F(u(0,t)-F(k)).
\phi(0,t)\,dt\\
&&\geq 0.\label{definition}
\end{eqnarray*}
\end{definition}
Here, and throughout this paper, BV stands for the space of the functions of bounded
variation, $u(0,t)$ for the trace of the function $u$ on the boundary $x=0$ and $u(x,0)$
for the trace of $u$ on $t=0$. Such traces are well defined whenever $u$ is of bounded
variation (see \cite{BLN}). Moreover, until opposite indication, we write
$$
\int q\,dm_1\dots dm_n
$$
instead of
$$
\int_\Omega q(x_1,\dots ,x_n)\,dm_1\dots dm_n=\int_{\omega_1} . . .\int_{\omega_n}
q(x_1,\dots ,x_n)dm_1(x_1)\dots dm_n(x_n),
$$
where the measure $dm_i$ is defined on the space $\omega_i$, $\Omega=\omega_1\times\dots \times\omega_n$ and $q\in L^1(\Omega)$. In the same way,
an integration on a subspace $\omega$ of $\Omega$ will be written as
$$
\int_\omega q\,dm_1\dots dm_n.
$$
It is well known that the initial boundary value problem
(\ref{law})-(\ref{BLN}) admits a unique solution described in definition \ref{solution},
see \cite{BLN, BCV}.
Concerning the BGK model, we used \cite{B} to construct ours. We will not go
back on Bouchut's technical details, but we just recall axioms for the equilibrium
state $M:\mathbb{R}\times\Xi\rightarrow\mathbb{R}$ and for kinetic entropies.
We postulate that $M=M(u,\xi)$ is smooth and monotone in $u\in\mathbb{R}$
for all $\xi\in\Xi$ and satisfies habitual moment equations, that is:
\begin{eqnarray}
&M(.,\xi)\mbox{ is nondecreasing for all }\xi,&\label{M monotone}\\
&\int M(u,\xi)\,d\xi=u\mbox{ \ for all }u\in\mathbb{R},&\label{M1}\\
&\int a(\xi)M(u,\xi)\,d\xi=F(u)\mbox{ \ for all }u\in\mathbb{R}&\label{M2}.
\end{eqnarray}
For technical reasons, we impose $a\in L^1(\Xi)$ and we prevent $t=0$ and $x=0$ to be
characteristics in (\ref{model}). This can be written as:
$$
a(\xi)\in[-a_\infty,0[\cup]0,a_\infty],\mbox{ for all }\xi\in\Xi,
$$
whit a positive real $a_\infty$.
Our infinite set of convex entropies will be the Kruzkov's one. Such an entropy is
written:
$$
\eta_k(u)=|u-k|,
$$
and its associated flux is:
$$
G_k(u)=\mathop{\rm sgn}(u-k)(F(u)-F(k)).
$$
Consider now, for any $k\in \mathbb{R}$, the kinetic entropy given by:
$$
H_k(f,\xi)=|f-M(k,\xi)|,
$$
for $f\in\mathbb{R}$ and $\xi\in\Xi$. This kinetic entropy is of course convex in $f$ and one
can easily check, using (\ref{M monotone})-(\ref{M2}), that:
\begin{eqnarray}
&&\int H_k(M(u,\xi),\xi)\,d\xi=\eta_k(u),\label{H1}\\
&&\int a(\xi)H_k(M(u,\xi),\xi)\,d\xi=G_k(u),\label{H2}\\
&&\int H_k(M(u^f,\xi),\xi)\,d\xi\leq \int H_k(f(\xi),\xi)\,d\xi,\label{H3}
\end{eqnarray}
for all $u\in\mathbb{R}$, for $f:\Xi\rightarrow \mathbb{R}$ and for
$$u^f=\int f(\xi)\,d\xi.$$
These properties will allow us to obtain the Lax entropy inequalities in the
hydrodynamic limit. Indeed, multiplying (\ref{model}) by $\mathop{\rm sgn}(f(x,t,\xi)-M(k,\xi))$
and using the convexity of $H_k(.,\xi)$, yields:
$$
\partial_t H_k(f,\xi)+a(\xi)\partial_x H_k(f,\xi)\leq\frac{H_k(M_f,\xi)-H_k(f,\xi)}{\epsilon}.
$$
Then integrating with respect to $\xi$ and using (\ref{H3}), we obtain
\begin{equation}
\partial_t\int H_k(f(x,t,\xi),\xi)\,d\xi+\partial_x\int a(\xi)H_k(f(x,t,\xi),\xi)\,d\xi\leq0.
\label{E1}
\end{equation}
Suppose that $u^\epsilon$ converges to $u\in BV$, and remember that $f=f^\epsilon$ is devoted
to be close to $M(u,\xi)$ at equilibrium. So, let $\epsilon\to 0$ in (\ref{E1}) and use (\ref{H1})-(\ref{H2}) to end up with
$$
\partial_t\eta_k(u)+\partial_x G(u)\leq0.
$$
That is $u$ is a Lax-entropy solution of (\ref{law}) disregarding the boundary condition.
But this is not sufficient to give uniqueness in our framework. Later, we will deeply
develop (\ref{E1}) to reach the boundary entropy inequality (\ref{BLN}).
%
% The BGK model %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section{The BGK model}
Let us show first that the kinetic problem defined by (\ref{model})-(\ref{f bord}) is
well-posed in
$L^{\infty}((0,T);L^1(\mathbb{R}^+\times\Xi))$. To do this, we rewrite (\ref{law}) in an
equivalent integral form by using Duhamel's principle; and use a Banach fixed point
argument. Because of the boundary data, the quarter plan is divided into two zones for
positive $a(\xi)$. We are brought to consider the sets:
\begin{eqnarray*}
&Q_-=\{(x,t,\xi)\in\mathbb{R}^+\times(0,T)\times\Xi;\ x0} a(\xi)|\partial_t f|\,d\xi(0,s)ds.
\end{eqnarray*}
But we can easily see that
$$
\partial_t f(x,0,\xi)=-a(\xi) \partial_x M(u^0(x),\xi)
$$
and
$\partial_t f(0,t,\xi)=\partial_t M(u_b(t),\xi)$
when $a(\xi)>0$. We conclude by using (\ref{M monotone})-(\ref{M1}).
%
% lemma BV in x %%%%%%%%%%%%%%%%%%
%
\begin{lemma}\label{bvx}
If $M$ satisfies (\ref{M monotone})-(\ref{M1}) and if $u^0$ and $u_b$ are
of bounded variation, then the solution $f$ of the model satisfies, for every $t$
$$
\int|\partial_x f(x,t,\xi)|\,d\xi\,dx\leq K,
$$
where $K$ is a constant independent of $\epsilon$. That is to say that its density satisfies
$$
\int|\partial_x u^\epsilon(x,t)|\,dx\leq K,
$$
for every $t$.
\end{lemma}
\paragraph{Proof.}
Let us reconsider (\ref{f+})-(\ref{f-}) to write
$$
\int|\partial_x f(x,t,\xi)|\,d\xi\,dx=I+J
$$
with
\begin{eqnarray*}
I&=&\int_{xat}|\partial_x f|\,dx\,d\xi\leq J_1+J_2
\end{eqnarray*}
where $a=a(\xi)$ and
\begin{eqnarray*}
I_1&=&\int_{x0}a(\xi) H_k(M(u_b(t),\xi),\xi)\phi(0,t)\,d\xi\,dt\\
&&+\int_{a(\xi)<0}a(\xi) H_k(f(0,t,\xi),\xi)\phi(0,t)\,d\xi\,dt\ \geq\ 0.
\end{eqnarray*}
Make so that the integral on $\Xi$ appears in the third line and use the definition of the
subdifferential of a convex function to conclude.
These calculations are valid for general
convex entropies $H$, in our Kruskove's entropies $H_k$ case, we have:
%
% corollaire entropie %%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\begin{corollary}\label{corentrop}
Under the assumptions of lemma \ref{entro. cin.} and (\ref{M2}), we have:
\begin{eqnarray*}
\lefteqn{\int H_k(f,\xi)(\partial_t\phi+a\partial_x\phi)\,d\xi\,dx\,dt
+\int H_k(M^0,\xi)\phi(x,0)\,d\xi\,dx}\\
&\geq&\int \mathop{\rm sgn}(u_b-k)[F(k)-\int_{a>0} a M_b\,d\xi
-\int_{a<0}a f(0,t,\xi)\,d\xi]\phi(0,t)\,dt.
\end{eqnarray*}
with $a=a(\xi),\ \ M^0=M(u^0(x),\xi)$ and $M_b=M(u_b(t),\xi)$.
\end{corollary}
\paragraph{Proof.}
Use $\mathop{\rm sgn}(f-M(k,\xi))\in\partial H_k(f,\xi)$ and (\ref{M2}).
We are now able to demonstrate our main theorem
%
% theorem main %%%%%%%%%%%%%%%%%%%%%%
%
\begin{theorem}\label{aim}
Let the maxwellian $M$ satisfies (\ref{M monotone})-(\ref{M2}). If $u^0$ and
$u_b$ are of bounded variation, then the sequence of first momentums $u^\epsilon$ arising
from the solutions $f$ of the BGK model (\ref{model})-(\ref{f bord}) converges to the
unique entropy solution $u$ of the initial-boundary value problem (\ref{law})-(\ref{BLN})
described in definition \ref{solution}.
\end{theorem}
%
% proof
%
\paragraph{Proof.}
The momentums $u^\epsilon$ are uniformly bounded in $L^\infty(\mathbb{R}^+\times(0,T))$ by theorem
\ref{exist. born.} as they are uniformly bounded in $BV(\mathbb{R}^+\times(0,T))$ by theorem
\ref{bv}. We can then extract a subsequence, which we denote also by $u^\epsilon$, converging
in $L^1$ and almost every where to $u\in L^\infty\cap BV(\mathbb{R}^+\times(0,T))$. In addition,
$$
|\int_{a(\xi)<0}a(\xi) f(0,t,\xi)\,d\xi|\leq K\int|a(\xi)|\,d\xi,
$$
that is we can extract from
$$
v^\epsilon=\int_{a>0} a M_b\,d\xi+\int_{a<0}a f(0,t,\xi)\,d\xi
$$
a subsequence, indexed also by $\epsilon$, converging in the weak* $L^\infty$-topology toward
$h\in L^\infty(0,T)$. Therefore, we pass to the limit on $\epsilon$ in corollary \ref{corentrop},
up to a subsequence, using lemma \ref{disteq} and (\ref{H1})-(\ref{H2}) to obtain
\begin{eqnarray}
\lefteqn{\int |u-k|\partial_t\phi+\mathop{\rm sgn}(u-k)(F(u)-F(k))\partial_x\phi\,dx\,dt\nonumber}\\
&&+\int|u^0-k|\phi(x,0)\,dx\ \geq\ \int \mathop{\rm sgn}(u_b-k)(F(k)-h(t))\phi(0,t)\,dt.\label{E2}
\end{eqnarray}
Choosing $\phi=\rho_\delta(x)\psi(t)$ with $\psi\in C_c(]0,T[)$ and
$$
\rho_\delta(0)=1,\quad \rho_\delta(x)=0\mbox{ if }x\geq\delta,\quad0\leq\rho_\delta\leq1,
$$
and tending $\delta$ toward zero in (\ref{E2}) yields
$$
sg(u(0,t)-k)(F(k)-F(u(0,t))\geq \mathop{\rm sgn}(u_b(t)-k)(F(k)-h(t)),
$$
for all $k\in\mathbb{R}$. Choose now
$$
k<\min\{\inf(u^0),\inf (u_b)\}\mbox{ \ and \ }k>\max\{\sup(u^0),\sup (u_b)\}
$$
to get $h(t)=F(u(0,t))$ , which ends this proof.
\section{Examples}
\paragraph{Example 7.1: Relaxation.}
Let us consider the so called relaxation system introduced by Jin and Xin \cite{JX} to
approximate the conservation law (\ref{law}):
\begin{eqnarray}
&\partial_t u^\epsilon+\partial_x v^\epsilon=0\quad 0