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''BGK-model''
 
''BGK-model''
  
 
Rarefied gas dynamics is described by the kinetic [[Boltzmann equation|Boltzmann equation]] ([[#References|[a13]]], [[#References|[a6]]])
 
Rarefied gas dynamics is described by the kinetic [[Boltzmann equation|Boltzmann equation]] ([[#References|[a13]]], [[#References|[a6]]])
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b1202201.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
\begin{equation} \tag{a1} \partial _ { t } f + v . \nabla _ { x } f = \frac { Q ( f ) } { \varepsilon }, \end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b1202202.png" /> is the particle density in the phase space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b1202203.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b1202204.png" /> is the mean free path and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b1202205.png" /> is the Boltzmann collision operator. This integral operator acts in the velocity variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b1202206.png" /> only, satisfies the moment relations
+
where $f ( t , x , v ) \geq 0$ is the particle density in the phase space $( x , v ) \in \mathbf{R} ^ { N } \times \mathbf{R} ^ { N }$, $\varepsilon$ is the mean free path and $Q$ is the Boltzmann collision operator. This integral operator acts in the velocity variable $v$ only, satisfies the moment relations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b1202207.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
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\begin{equation} \tag{a2} \int \phi ( v ) Q ( f ) ( v ) d v = 0, \end{equation}
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b1202208.png" />, and the entropy inequality
+
$\phi \in \operatorname { Span } ( 1 , v _ { j } , | v | ^ { 2 } )$, and the entropy inequality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b1202209.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
+
\begin{equation} \tag{a3} \int \operatorname { ln } f ( v ) Q ( f ) ( v ) d v \leq 0. \end{equation}
  
These properties ensure the local conservation of mass, momentum and energy by integrating (a1) with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022010.png" />,
+
These properties ensure the local conservation of mass, momentum and energy by integrating (a1) with respect to $v$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
+
\begin{equation} \tag{a4} \partial _ { t } \int \phi ( v )\, f d v + \operatorname { div } _ { x } \int v \phi ( v )\, f d v = 0, \end{equation}
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022012.png" />, and the decrease of entropy
+
$\phi \in \operatorname { Span } ( 1 , v _ { j } , | v | ^ { 2 } )$, and the decrease of entropy
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>
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\begin{equation} \tag{a5} \partial _ { t } \int f \operatorname { ln } f d v + \operatorname { div } _ { x } \int v f \operatorname { ln } f d v \leq 0. \end{equation}
  
Another striking property of the Boltzmann equation is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022014.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022015.png" /> is a Maxwellian, that is
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Another striking property of the Boltzmann equation is that $Q ( f ) = 0$ if and only if $f$ is a Maxwellian, that is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>
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\begin{equation} \tag{a6} f ( v ) = \frac { \rho } { ( 2 \pi T ) ^ { N / 2 } } e ^ { - |v - u| ^ { 2 } / 2 T }, \end{equation}
  
for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022019.png" />. When time and space dependence are allowed as in (a1), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022022.png" /> can depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022024.png" /> also. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022025.png" /> in (a1), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022026.png" /> therefore goes formally to a Maxwellian of parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022029.png" />, see [[#References|[a6]]], which satisfies the conservation laws (a4), and entropy inequality (a5), with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022030.png" /> given by (a6). This system is the Euler system of mono-atomic perfect gas dynamics.
+
for some $\rho \geq 0$, $T &gt; 0$, $u \in \mathbf{R} ^ { N }$. When time and space dependence are allowed as in (a1), $\rho$, $T$, $u$ can depend on $t$, $x$ also. When $\varepsilon \rightarrow 0$ in (a1), $f$ therefore goes formally to a Maxwellian of parameters $\rho ( t , x )$, $T ( t , x )$ and $u ( t , x )$, see [[#References|[a6]]], which satisfies the conservation laws (a4), and entropy inequality (a5), with $f$ given by (a6). This system is the Euler system of mono-atomic perfect gas dynamics.
  
 
In their paper [[#References|[a1]]], P.L. Bhatnagar, E.P. Gross, and M. Krook introduced a simplified Boltzmann-like model (called the BGK-model) which satisfies all the properties cited above. It is written in the form (a1) with
 
In their paper [[#References|[a1]]], P.L. Bhatnagar, E.P. Gross, and M. Krook introduced a simplified Boltzmann-like model (called the BGK-model) which satisfies all the properties cited above. It is written in the form (a1) with
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022031.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a7)</td></tr></table>
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\begin{equation} \tag{a7} Q ( f ) = M _ { f } - f, \end{equation}
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022032.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a8)</td></tr></table>
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\begin{equation} \tag{a8} M _ { f } ( v ) = \frac { \rho_{ f} } { ( 2 \pi T _ { f } ) ^ { N / 2 } } e ^ { -|\nu -u_{f} |^{2} / 2T_{f}} , \end{equation}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022033.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a9)</td></tr></table>
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\begin{equation} \tag{a9} \rho_f \left( 1 , u _ { f } , \frac { 1 } { 2 } | u_f | ^ { 2 } + \frac { N } { 2 } T _ { f } \right) = \int \left( 1 , v , \frac { | v |^ { 2 } } { 2 } \right) f ( v ) d v. \end{equation}
  
The existence of a global solution to the BGK-model has been proved by B. Perthame [[#References|[a14]]], and regularity properties are given in [[#References|[a16]]]. Variations of the model are possible, by taking <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022034.png" /> for some positive function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022035.png" />. The case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022036.png" /> is of interest because then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022037.png" /> is quadratic, as is the Boltzmann operator. However, there is no existence result in this case.
+
The existence of a global solution to the BGK-model has been proved by B. Perthame [[#References|[a14]]], and regularity properties are given in [[#References|[a16]]]. Variations of the model are possible, by taking $Q ( f ) = \psi ( \rho _ { f } , T _ { f } ) ( M _ { f } - f )$ for some positive function $\psi$. The case $\psi ( \rho _ { f } , T _ { f } ) = \rho _ { f }$ is of interest because then $Q$ is quadratic, as is the Boltzmann operator. However, there is no existence result in this case.
  
 
Many attempts have been done to generalize the BGK-formalism, in order to provide a natural kinetic description of hyperbolic systems of conservation laws, other than the Euler system of gas dynamics. Most of the known generalized BGK-models fit in the framework of [[#References|[a2]]]. According to [[#References|[a2]]], one writes
 
Many attempts have been done to generalize the BGK-formalism, in order to provide a natural kinetic description of hyperbolic systems of conservation laws, other than the Euler system of gas dynamics. Most of the known generalized BGK-models fit in the framework of [[#References|[a2]]]. According to [[#References|[a2]]], one writes
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022038.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a10)</td></tr></table>
+
\begin{equation} \tag{a10} \partial _ { t } f + a ( \xi ) . \nabla _ { x } f = \frac { M _ { f } - f } { \varepsilon }, \end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022042.png" /> a measure space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022043.png" />, and
+
where $f ( t , x , \xi ) \in \mathbf{R} ^ { p }$, $t &gt; 0$, $x \in \mathbf{R} ^ { N }$, $\xi \in \Xi$ a measure space, $a ( \xi ) \in \mathbf{R} ^ { N }$, and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022044.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a11)</td></tr></table>
+
\begin{equation} \tag{a11} M _ { f } ( t , x , \xi ) = M ( u ( t , x ) , \xi ), \end{equation}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022045.png" /></td> </tr></table>
+
\begin{equation*} u ( t , x ) = \int f ( t , x , \xi ) d \xi - k. \end{equation*}
  
The equilibrium state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022046.png" /> is assumed to satisfy
+
The equilibrium state $M ( u , \xi )$ is assumed to satisfy
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022047.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a12)</td></tr></table>
+
\begin{equation} \tag{a12} \int M ( u , \xi ) d \xi = u + k. \end{equation}
  
 
Then, defining
 
Then, defining
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022048.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a13)</td></tr></table>
+
\begin{equation} \tag{a13} F _ { j } ( u ) = \int a _ { j } ( \xi ) M ( u , \xi ) d \xi , \end{equation}
  
the system relaxes as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022049.png" /> to the system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022050.png" /> equations
+
the system relaxes as $\varepsilon \rightarrow 0$ to the system of $p$ equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022051.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a14)</td></tr></table>
+
\begin{equation} \tag{a14} \partial _ { t } u + \sum _ { j = 1 } ^ { N } \frac { \partial } { \partial x _ { j } } F _ { j } ( u ) = 0. \end{equation}
  
Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022052.png" /> remains in a convex domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022053.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022054.png" />. An interesting property of the kinetic equation (a10) is that it leaves invariant any family of convex sets indexed by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022055.png" />. Therefore if one chooses for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022056.png" /> a convex set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022057.png" /> such that
+
Assume that $u$ remains in a convex domain $\mathcal U$ of $\mathbf{R} ^ { p }$. An interesting property of the kinetic equation (a10) is that it leaves invariant any family of convex sets indexed by $\xi $. Therefore if one chooses for each $\xi $ a convex set $D _ { \xi } \subset \mathbf{R} ^ { p }$ such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022058.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a15)</td></tr></table>
+
\begin{equation} \tag{a15} \forall u \in \mathcal{U} : M ( u , \xi ) \in D _ { \xi }, \end{equation}
  
then one can start with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022059.png" />, for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022060.png" />, and then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022061.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022064.png" />. The kinetic entropy inequality is obtained by a convex function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022065.png" />, such that the following Gibbs minimization principle holds: for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022066.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022067.png" />-a.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022069.png" />,
+
then one can start with $f ^ { 0 } ( x , \xi ) = M ( u ^ { 0 } ( x ) , \xi )$, for some $u ^ { 0 }$, and then $f ( t , x , \xi ) \in D _ { \xi }$ for all $t \geq 0$, $x$, $\xi $. The kinetic entropy inequality is obtained by a convex function $H ( \cdot , \xi ) : D _ { \xi } \rightarrow R$, such that the following Gibbs minimization principle holds: for any $f : \Xi \rightarrow \mathbf{R} ^ { p }$ such that $\xi $-a.e. $f ( \xi ) \in D _ { \xi }$ and $u _ { f } \equiv \int f ( \xi ) d \xi - k \in \mathcal{U}$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022070.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a16)</td></tr></table>
+
\begin{equation} \tag{a16} \int H ( M ( u _ { f } , \xi ) , \xi ) d \xi \leq \int H ( f ( \xi ) , \xi ) d \xi. \end{equation}
  
This property ensures that in the limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022071.png" /> one obtains the entropy inequality
+
This property ensures that in the limit $\varepsilon \rightarrow 0$ one obtains the entropy inequality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022072.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a17)</td></tr></table>
+
\begin{equation} \tag{a17} \partial _ { t } \eta ( u ) + \operatorname { div } _ { x } G ( u ) \leq 0, \end{equation}
  
 
with
 
with
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022073.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a18)</td></tr></table>
+
\begin{equation} \tag{a18} \eta ( u ) = \int H ( M ( u , \xi ) , \xi ) d \xi, \end{equation}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022074.png" /></td> </tr></table>
+
\begin{equation*} G ( u ) = \int a ( \xi ) H ( M ( u , \xi ) , \xi ) d \xi. \end{equation*}
  
The original BGK-model (a1), (a7), (a8), (a9) enters this framework by taking <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022075.png" /> with Lebesgue measure, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022076.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022078.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022079.png" /> is the state and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022080.png" /> is the scalar physical Maxwellian given by (a6). One has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022081.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022082.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022083.png" />. Here, since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022084.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022085.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022086.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022087.png" />, the vector equation (a10) reduces to a scalar equation. It is also possible to treat polytropic gases by introducing internal energy, using the approach of [[#References|[a15]]]. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022088.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022089.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022090.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022091.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022092.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022093.png" />. See also [[#References|[a10]]] for related models.
+
The original BGK-model (a1), (a7), (a8), (a9) enters this framework by taking $\Xi = {\bf R} ^ { N }$ with Lebesgue measure, $\xi = v$, $a ( \xi ) = \xi$, $M ( \underline { u } , \xi ) = ( 1 , \xi _ { 1 } , \ldots , \xi _ { N } , | \xi | ^ { 2 } / 2 ) M _ { 0 } ( \underline { u } , \xi )$ where $u$ is the state and $M _ { 0 } ( \underline { u } , \xi )$ is the scalar physical Maxwellian given by (a6). One has $D _ { \xi } = ( 1 , \xi _ { 1 } , \dots , \xi _ { N } , | \xi | ^ { 2 } / 2 )\bf R _ { + }$, and $H ( f , \xi ) = f _ { 0 } \operatorname { ln } f _ { 0 }$ for any $f = ( 1 , \xi _ { 1 } , \ldots , \xi _ { N } , | \xi | ^ { 2 } / 2 ) f _ { 0 } \in D _ { \xi }$. Here, since $f ( t , x , \xi ) \in D _ { \xi }$ for all $t$, $x$, $\xi $, the vector equation (a10) reduces to a scalar equation. It is also possible to treat polytropic gases by introducing internal energy, using the approach of [[#References|[a15]]]. Then $ \Xi  = \mathbf{R} ^ { N } \times [ 0 , \infty [$, $\xi = ( v , I )$, $a ( \xi ) = v$, $d \xi = c d v I ^ { d- 1 } d I$, $N + d = 2 / ( \gamma - 1 )$, $D _ { \xi } = ( 1 , v _ { 1 } , \dots , v _ { N } , | v | ^ { 2 } / 2 + I ^ { 2 } / 2 ) \mathbf{R} _ { + }$. See also [[#References|[a10]]] for related models.
  
The success of such BGK-models has been revealed for scalar equations (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022094.png" />) by Y. Brenier [[#References|[a3]]], Y. Giga and T. Miyakawa [[#References|[a8]]], and later by B. Perthame and E. Tadmor [[#References|[a17]]], and by R. Natalini [[#References|[a12]]]. It appears that in this case there is a so-called  "kinetic formulation" , that is, an equation like (a10) but with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022095.png" /> and the right-hand side being replaced by a suitable term, see [[#References|[a11]]]. Another case where this holds can be found in [[#References|[a4]]] and [[#References|[a5]]]. More generally, BGK-models can be seen as a subclass of the general class of relaxation models, described for example in [[#References|[a7]]], [[#References|[a9]]].
+
The success of such BGK-models has been revealed for scalar equations ($p = 1$) by Y. Brenier [[#References|[a3]]], Y. Giga and T. Miyakawa [[#References|[a8]]], and later by B. Perthame and E. Tadmor [[#References|[a17]]], and by R. Natalini [[#References|[a12]]]. It appears that in this case there is a so-called  "kinetic formulation" , that is, an equation like (a10) but with $\varepsilon = 0$ and the right-hand side being replaced by a suitable term, see [[#References|[a11]]]. Another case where this holds can be found in [[#References|[a4]]] and [[#References|[a5]]]. More generally, BGK-models can be seen as a subclass of the general class of relaxation models, described for example in [[#References|[a7]]], [[#References|[a9]]].
  
The BGK-model (a10) also exists in a time-discrete form, which appears in the literature as the transport-collapse method [[#References|[a3]]], kinetic or Boltzmann schemes [[#References|[a15]]], and which gives an approximate solution to (a14). It is an algorithm that gives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022096.png" /> from the knowledge of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022097.png" />, by solving
+
The BGK-model (a10) also exists in a time-discrete form, which appears in the literature as the transport-collapse method [[#References|[a3]]], kinetic or Boltzmann schemes [[#References|[a15]]], and which gives an approximate solution to (a14). It is an algorithm that gives $u ^ { n + 1 } ( x )$ from the knowledge of $u ^ { n } ( x )$, by solving
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022098.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a19)</td></tr></table>
+
\begin{equation} \tag{a19} \partial _ { t }\, f + a ( \xi ) \cdot \nabla _ { x }\, f = 0 \text { in } ] t _ { n } , t _ { n } + 1 [ \times \mathbf{R} ^ { N } \times \Xi, \end{equation}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022099.png" /></td> </tr></table>
+
\begin{equation*} f ( t _ { n } , x , \xi ) = M ( u ^ { n } ( x ) , \xi ). \end{equation*}
  
The new state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b120220100.png" /> is given by
+
The new state $u ^ { n + 1 }$ is given by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b120220101.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a20)</td></tr></table>
+
\begin{equation} \tag{a20} u ^ { n + 1 } ( x ) = \int f ( t _ { n + 1} ^ { - } , x , \xi ) d \xi - k. \end{equation}
  
 
Then,
 
Then,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b120220102.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a21)</td></tr></table>
+
\begin{equation} \tag{a21} \partial _ { t } f + a ( \xi ) . \nabla _ { x } f = \sum _ { n = 1 } ^ { \infty } \delta ( t - t _ { n } ) ( M _ { f ^{ n -}} - f ^ { n - } ), \end{equation}
  
which is similar to (a10) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b120220103.png" />. The minimization principle (a16) ensures that a discrete entropy inequality holds.
+
which is similar to (a10) with $t _ { n  + 1} - t _ { n } \sim \varepsilon$. The minimization principle (a16) ensures that a discrete entropy inequality holds.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.L. Bhatnagar,  E.P. Gross,  M. Krook,  "A model for collision processes in gases"  ''Phys. Rev.'' , '''94'''  (1954)  pp. 511</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Bouchut,  "Construction of BGK models with a family of kinetic entropies for a given system of conservation laws"  ''J. Statist. Phys.'' , '''95'''  (1999)  pp. 113–170</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  Y. Brenier,  "Averaged multivalued solutions for scalar conservation laws"  ''SIAM J. Numer. Anal.'' , '''21'''  (1984)  pp. 1013–1037</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  Y. Brenier,  L. Corrias,  "A kinetic formulation for multi-branch entropy solutions of scalar conservation laws"  ''Ann. Inst. H. Poincaré Anal. Non Lin.'' , '''15'''  (1998)  pp. 169–190</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  Y. Brenier,  L. Corrias,  R. Natalini,  "A relaxation approximation to a moment hierarchy of conservation laws with kinetic formulation"  ''preprint''  (1998)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  C. Cercignani,  R. Illner,  M. Pulvirenti,  "The mathematical theory of dilute gases"  ''Appl. Math. Sci.'' , '''106''' , Springer  (1994)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  G.Q. Chen,  C.D. Levermore,  T.-P. Liu,  "Hyperbolic conservation laws with stiff relaxation terms and entropy"  ''Commun. Pure Appl. Math.'' , '''47'''  (1994)  pp. 787–830</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  Y. Giga,  T. Miyakawa,  "A kinetic construction of global solutions of first order quasilinear equations"  ''Duke Math. J.'' , '''50'''  (1983)  pp. 505–515</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  S. Jin,  Z.-P. Xin,  "The relaxation schemes for systems of conservation laws in arbitrary space dimensions"  ''Commun. Pure Appl. Math.'' , '''48'''  (1995)  pp. 235–276</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  C.D. Levermore,  "Moment closure hierarchies for kinetic theories"  ''J. Statist. Phys.'' , '''83'''  (1996)  pp. 1021–1065</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  P.-L. Lions,  B. Perthame,  E. Tadmor,  "A kinetic formulation of multidimensional scalar conservation laws and related equations"  ''J. Amer. Math. Soc.'' , '''7'''  (1994)  pp. 169–191</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  R. Natalini,  "A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws"  ''J. Diff. Eq.'' , '''148'''  (1998)  pp. 292–317</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  C. Truesdell,  R.G. Muncaster,  "Fundamentals of Maxwell's kinetic theory of a simple monatomic gas, treated as a branch of rational mechanics" , ''Pure Appl. Math.'' , '''83''' , Acad. Press  (1980)</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  B. Perthame,  "Global existence to the BGK model of Boltzmann equation"  ''J. Diff. Eq.'' , '''82'''  (1989)  pp. 191–205</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  B. Perthame,  "Boltzmann type schemes for gas dynamics and the entropy property"  ''SIAM J. Numer. Anal.'' , '''27'''  (1990)  pp. 1405–1421</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  B. Perthame,  M. Pulvirenti,  "Weighted <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b120220104.png" /> bounds and uniqueness for the Boltzmann BGK model"  ''Arch. Rat. Mech. Anal.'' , '''125'''  (1993)  pp. 289–295</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  B. Perthame,  E. Tadmor,  "A kinetic equation with kinetic entropy functions for scalar conservation laws"  ''Comm. Math. Phys.'' , '''136'''  (1991)  pp. 501–517</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  P.L. Bhatnagar,  E.P. Gross,  M. Krook,  "A model for collision processes in gases"  ''Phys. Rev.'' , '''94'''  (1954)  pp. 511</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  F. Bouchut,  "Construction of BGK models with a family of kinetic entropies for a given system of conservation laws"  ''J. Statist. Phys.'' , '''95'''  (1999)  pp. 113–170</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  Y. Brenier,  "Averaged multivalued solutions for scalar conservation laws"  ''SIAM J. Numer. Anal.'' , '''21'''  (1984)  pp. 1013–1037</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  Y. Brenier,  L. Corrias,  "A kinetic formulation for multi-branch entropy solutions of scalar conservation laws"  ''Ann. Inst. H. Poincaré Anal. Non Lin.'' , '''15'''  (1998)  pp. 169–190</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  Y. Brenier,  L. Corrias,  R. Natalini,  "A relaxation approximation to a moment hierarchy of conservation laws with kinetic formulation"  ''preprint''  (1998)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  C. Cercignani,  R. Illner,  M. Pulvirenti,  "The mathematical theory of dilute gases"  ''Appl. Math. Sci.'' , '''106''' , Springer  (1994)</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  G.Q. Chen,  C.D. Levermore,  T.-P. Liu,  "Hyperbolic conservation laws with stiff relaxation terms and entropy"  ''Commun. Pure Appl. Math.'' , '''47'''  (1994)  pp. 787–830</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  Y. Giga,  T. Miyakawa,  "A kinetic construction of global solutions of first order quasilinear equations"  ''Duke Math. J.'' , '''50'''  (1983)  pp. 505–515</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  S. Jin,  Z.-P. Xin,  "The relaxation schemes for systems of conservation laws in arbitrary space dimensions"  ''Commun. Pure Appl. Math.'' , '''48'''  (1995)  pp. 235–276</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  C.D. Levermore,  "Moment closure hierarchies for kinetic theories"  ''J. Statist. Phys.'' , '''83'''  (1996)  pp. 1021–1065</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  P.-L. Lions,  B. Perthame,  E. Tadmor,  "A kinetic formulation of multidimensional scalar conservation laws and related equations"  ''J. Amer. Math. Soc.'' , '''7'''  (1994)  pp. 169–191</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  R. Natalini,  "A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws"  ''J. Diff. Eq.'' , '''148'''  (1998)  pp. 292–317</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  C. Truesdell,  R.G. Muncaster,  "Fundamentals of Maxwell's kinetic theory of a simple monatomic gas, treated as a branch of rational mechanics" , ''Pure Appl. Math.'' , '''83''' , Acad. Press  (1980)</td></tr><tr><td valign="top">[a14]</td> <td valign="top">  B. Perthame,  "Global existence to the BGK model of Boltzmann equation"  ''J. Diff. Eq.'' , '''82'''  (1989)  pp. 191–205</td></tr><tr><td valign="top">[a15]</td> <td valign="top">  B. Perthame,  "Boltzmann type schemes for gas dynamics and the entropy property"  ''SIAM J. Numer. Anal.'' , '''27'''  (1990)  pp. 1405–1421</td></tr><tr><td valign="top">[a16]</td> <td valign="top">  B. Perthame,  M. Pulvirenti,  "Weighted $L^{\infty}$ bounds and uniqueness for the Boltzmann BGK model"  ''Arch. Rat. Mech. Anal.'' , '''125'''  (1993)  pp. 289–295</td></tr><tr><td valign="top">[a17]</td> <td valign="top">  B. Perthame,  E. Tadmor,  "A kinetic equation with kinetic entropy functions for scalar conservation laws"  ''Comm. Math. Phys.'' , '''136'''  (1991)  pp. 501–517</td></tr></table>

Latest revision as of 16:58, 1 July 2020

BGK-model

Rarefied gas dynamics is described by the kinetic Boltzmann equation ([a13], [a6])

\begin{equation} \tag{a1} \partial _ { t } f + v . \nabla _ { x } f = \frac { Q ( f ) } { \varepsilon }, \end{equation}

where $f ( t , x , v ) \geq 0$ is the particle density in the phase space $( x , v ) \in \mathbf{R} ^ { N } \times \mathbf{R} ^ { N }$, $\varepsilon$ is the mean free path and $Q$ is the Boltzmann collision operator. This integral operator acts in the velocity variable $v$ only, satisfies the moment relations

\begin{equation} \tag{a2} \int \phi ( v ) Q ( f ) ( v ) d v = 0, \end{equation}

$\phi \in \operatorname { Span } ( 1 , v _ { j } , | v | ^ { 2 } )$, and the entropy inequality

\begin{equation} \tag{a3} \int \operatorname { ln } f ( v ) Q ( f ) ( v ) d v \leq 0. \end{equation}

These properties ensure the local conservation of mass, momentum and energy by integrating (a1) with respect to $v$,

\begin{equation} \tag{a4} \partial _ { t } \int \phi ( v )\, f d v + \operatorname { div } _ { x } \int v \phi ( v )\, f d v = 0, \end{equation}

$\phi \in \operatorname { Span } ( 1 , v _ { j } , | v | ^ { 2 } )$, and the decrease of entropy

\begin{equation} \tag{a5} \partial _ { t } \int f \operatorname { ln } f d v + \operatorname { div } _ { x } \int v f \operatorname { ln } f d v \leq 0. \end{equation}

Another striking property of the Boltzmann equation is that $Q ( f ) = 0$ if and only if $f$ is a Maxwellian, that is

\begin{equation} \tag{a6} f ( v ) = \frac { \rho } { ( 2 \pi T ) ^ { N / 2 } } e ^ { - |v - u| ^ { 2 } / 2 T }, \end{equation}

for some $\rho \geq 0$, $T > 0$, $u \in \mathbf{R} ^ { N }$. When time and space dependence are allowed as in (a1), $\rho$, $T$, $u$ can depend on $t$, $x$ also. When $\varepsilon \rightarrow 0$ in (a1), $f$ therefore goes formally to a Maxwellian of parameters $\rho ( t , x )$, $T ( t , x )$ and $u ( t , x )$, see [a6], which satisfies the conservation laws (a4), and entropy inequality (a5), with $f$ given by (a6). This system is the Euler system of mono-atomic perfect gas dynamics.

In their paper [a1], P.L. Bhatnagar, E.P. Gross, and M. Krook introduced a simplified Boltzmann-like model (called the BGK-model) which satisfies all the properties cited above. It is written in the form (a1) with

\begin{equation} \tag{a7} Q ( f ) = M _ { f } - f, \end{equation}

and

\begin{equation} \tag{a8} M _ { f } ( v ) = \frac { \rho_{ f} } { ( 2 \pi T _ { f } ) ^ { N / 2 } } e ^ { -|\nu -u_{f} |^{2} / 2T_{f}} , \end{equation}

\begin{equation} \tag{a9} \rho_f \left( 1 , u _ { f } , \frac { 1 } { 2 } | u_f | ^ { 2 } + \frac { N } { 2 } T _ { f } \right) = \int \left( 1 , v , \frac { | v |^ { 2 } } { 2 } \right) f ( v ) d v. \end{equation}

The existence of a global solution to the BGK-model has been proved by B. Perthame [a14], and regularity properties are given in [a16]. Variations of the model are possible, by taking $Q ( f ) = \psi ( \rho _ { f } , T _ { f } ) ( M _ { f } - f )$ for some positive function $\psi$. The case $\psi ( \rho _ { f } , T _ { f } ) = \rho _ { f }$ is of interest because then $Q$ is quadratic, as is the Boltzmann operator. However, there is no existence result in this case.

Many attempts have been done to generalize the BGK-formalism, in order to provide a natural kinetic description of hyperbolic systems of conservation laws, other than the Euler system of gas dynamics. Most of the known generalized BGK-models fit in the framework of [a2]. According to [a2], one writes

\begin{equation} \tag{a10} \partial _ { t } f + a ( \xi ) . \nabla _ { x } f = \frac { M _ { f } - f } { \varepsilon }, \end{equation}

where $f ( t , x , \xi ) \in \mathbf{R} ^ { p }$, $t > 0$, $x \in \mathbf{R} ^ { N }$, $\xi \in \Xi$ a measure space, $a ( \xi ) \in \mathbf{R} ^ { N }$, and

\begin{equation} \tag{a11} M _ { f } ( t , x , \xi ) = M ( u ( t , x ) , \xi ), \end{equation}

\begin{equation*} u ( t , x ) = \int f ( t , x , \xi ) d \xi - k. \end{equation*}

The equilibrium state $M ( u , \xi )$ is assumed to satisfy

\begin{equation} \tag{a12} \int M ( u , \xi ) d \xi = u + k. \end{equation}

Then, defining

\begin{equation} \tag{a13} F _ { j } ( u ) = \int a _ { j } ( \xi ) M ( u , \xi ) d \xi , \end{equation}

the system relaxes as $\varepsilon \rightarrow 0$ to the system of $p$ equations

\begin{equation} \tag{a14} \partial _ { t } u + \sum _ { j = 1 } ^ { N } \frac { \partial } { \partial x _ { j } } F _ { j } ( u ) = 0. \end{equation}

Assume that $u$ remains in a convex domain $\mathcal U$ of $\mathbf{R} ^ { p }$. An interesting property of the kinetic equation (a10) is that it leaves invariant any family of convex sets indexed by $\xi $. Therefore if one chooses for each $\xi $ a convex set $D _ { \xi } \subset \mathbf{R} ^ { p }$ such that

\begin{equation} \tag{a15} \forall u \in \mathcal{U} : M ( u , \xi ) \in D _ { \xi }, \end{equation}

then one can start with $f ^ { 0 } ( x , \xi ) = M ( u ^ { 0 } ( x ) , \xi )$, for some $u ^ { 0 }$, and then $f ( t , x , \xi ) \in D _ { \xi }$ for all $t \geq 0$, $x$, $\xi $. The kinetic entropy inequality is obtained by a convex function $H ( \cdot , \xi ) : D _ { \xi } \rightarrow R$, such that the following Gibbs minimization principle holds: for any $f : \Xi \rightarrow \mathbf{R} ^ { p }$ such that $\xi $-a.e. $f ( \xi ) \in D _ { \xi }$ and $u _ { f } \equiv \int f ( \xi ) d \xi - k \in \mathcal{U}$,

\begin{equation} \tag{a16} \int H ( M ( u _ { f } , \xi ) , \xi ) d \xi \leq \int H ( f ( \xi ) , \xi ) d \xi. \end{equation}

This property ensures that in the limit $\varepsilon \rightarrow 0$ one obtains the entropy inequality

\begin{equation} \tag{a17} \partial _ { t } \eta ( u ) + \operatorname { div } _ { x } G ( u ) \leq 0, \end{equation}

with

\begin{equation} \tag{a18} \eta ( u ) = \int H ( M ( u , \xi ) , \xi ) d \xi, \end{equation}

\begin{equation*} G ( u ) = \int a ( \xi ) H ( M ( u , \xi ) , \xi ) d \xi. \end{equation*}

The original BGK-model (a1), (a7), (a8), (a9) enters this framework by taking $\Xi = {\bf R} ^ { N }$ with Lebesgue measure, $\xi = v$, $a ( \xi ) = \xi$, $M ( \underline { u } , \xi ) = ( 1 , \xi _ { 1 } , \ldots , \xi _ { N } , | \xi | ^ { 2 } / 2 ) M _ { 0 } ( \underline { u } , \xi )$ where $u$ is the state and $M _ { 0 } ( \underline { u } , \xi )$ is the scalar physical Maxwellian given by (a6). One has $D _ { \xi } = ( 1 , \xi _ { 1 } , \dots , \xi _ { N } , | \xi | ^ { 2 } / 2 )\bf R _ { + }$, and $H ( f , \xi ) = f _ { 0 } \operatorname { ln } f _ { 0 }$ for any $f = ( 1 , \xi _ { 1 } , \ldots , \xi _ { N } , | \xi | ^ { 2 } / 2 ) f _ { 0 } \in D _ { \xi }$. Here, since $f ( t , x , \xi ) \in D _ { \xi }$ for all $t$, $x$, $\xi $, the vector equation (a10) reduces to a scalar equation. It is also possible to treat polytropic gases by introducing internal energy, using the approach of [a15]. Then $ \Xi = \mathbf{R} ^ { N } \times [ 0 , \infty [$, $\xi = ( v , I )$, $a ( \xi ) = v$, $d \xi = c d v I ^ { d- 1 } d I$, $N + d = 2 / ( \gamma - 1 )$, $D _ { \xi } = ( 1 , v _ { 1 } , \dots , v _ { N } , | v | ^ { 2 } / 2 + I ^ { 2 } / 2 ) \mathbf{R} _ { + }$. See also [a10] for related models.

The success of such BGK-models has been revealed for scalar equations ($p = 1$) by Y. Brenier [a3], Y. Giga and T. Miyakawa [a8], and later by B. Perthame and E. Tadmor [a17], and by R. Natalini [a12]. It appears that in this case there is a so-called "kinetic formulation" , that is, an equation like (a10) but with $\varepsilon = 0$ and the right-hand side being replaced by a suitable term, see [a11]. Another case where this holds can be found in [a4] and [a5]. More generally, BGK-models can be seen as a subclass of the general class of relaxation models, described for example in [a7], [a9].

The BGK-model (a10) also exists in a time-discrete form, which appears in the literature as the transport-collapse method [a3], kinetic or Boltzmann schemes [a15], and which gives an approximate solution to (a14). It is an algorithm that gives $u ^ { n + 1 } ( x )$ from the knowledge of $u ^ { n } ( x )$, by solving

\begin{equation} \tag{a19} \partial _ { t }\, f + a ( \xi ) \cdot \nabla _ { x }\, f = 0 \text { in } ] t _ { n } , t _ { n } + 1 [ \times \mathbf{R} ^ { N } \times \Xi, \end{equation}

\begin{equation*} f ( t _ { n } , x , \xi ) = M ( u ^ { n } ( x ) , \xi ). \end{equation*}

The new state $u ^ { n + 1 }$ is given by

\begin{equation} \tag{a20} u ^ { n + 1 } ( x ) = \int f ( t _ { n + 1} ^ { - } , x , \xi ) d \xi - k. \end{equation}

Then,

\begin{equation} \tag{a21} \partial _ { t } f + a ( \xi ) . \nabla _ { x } f = \sum _ { n = 1 } ^ { \infty } \delta ( t - t _ { n } ) ( M _ { f ^{ n -}} - f ^ { n - } ), \end{equation}

which is similar to (a10) with $t _ { n + 1} - t _ { n } \sim \varepsilon$. The minimization principle (a16) ensures that a discrete entropy inequality holds.

References

[a1] P.L. Bhatnagar, E.P. Gross, M. Krook, "A model for collision processes in gases" Phys. Rev. , 94 (1954) pp. 511
[a2] F. Bouchut, "Construction of BGK models with a family of kinetic entropies for a given system of conservation laws" J. Statist. Phys. , 95 (1999) pp. 113–170
[a3] Y. Brenier, "Averaged multivalued solutions for scalar conservation laws" SIAM J. Numer. Anal. , 21 (1984) pp. 1013–1037
[a4] Y. Brenier, L. Corrias, "A kinetic formulation for multi-branch entropy solutions of scalar conservation laws" Ann. Inst. H. Poincaré Anal. Non Lin. , 15 (1998) pp. 169–190
[a5] Y. Brenier, L. Corrias, R. Natalini, "A relaxation approximation to a moment hierarchy of conservation laws with kinetic formulation" preprint (1998)
[a6] C. Cercignani, R. Illner, M. Pulvirenti, "The mathematical theory of dilute gases" Appl. Math. Sci. , 106 , Springer (1994)
[a7] G.Q. Chen, C.D. Levermore, T.-P. Liu, "Hyperbolic conservation laws with stiff relaxation terms and entropy" Commun. Pure Appl. Math. , 47 (1994) pp. 787–830
[a8] Y. Giga, T. Miyakawa, "A kinetic construction of global solutions of first order quasilinear equations" Duke Math. J. , 50 (1983) pp. 505–515
[a9] S. Jin, Z.-P. Xin, "The relaxation schemes for systems of conservation laws in arbitrary space dimensions" Commun. Pure Appl. Math. , 48 (1995) pp. 235–276
[a10] C.D. Levermore, "Moment closure hierarchies for kinetic theories" J. Statist. Phys. , 83 (1996) pp. 1021–1065
[a11] P.-L. Lions, B. Perthame, E. Tadmor, "A kinetic formulation of multidimensional scalar conservation laws and related equations" J. Amer. Math. Soc. , 7 (1994) pp. 169–191
[a12] R. Natalini, "A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws" J. Diff. Eq. , 148 (1998) pp. 292–317
[a13] C. Truesdell, R.G. Muncaster, "Fundamentals of Maxwell's kinetic theory of a simple monatomic gas, treated as a branch of rational mechanics" , Pure Appl. Math. , 83 , Acad. Press (1980)
[a14] B. Perthame, "Global existence to the BGK model of Boltzmann equation" J. Diff. Eq. , 82 (1989) pp. 191–205
[a15] B. Perthame, "Boltzmann type schemes for gas dynamics and the entropy property" SIAM J. Numer. Anal. , 27 (1990) pp. 1405–1421
[a16] B. Perthame, M. Pulvirenti, "Weighted $L^{\infty}$ bounds and uniqueness for the Boltzmann BGK model" Arch. Rat. Mech. Anal. , 125 (1993) pp. 289–295
[a17] B. Perthame, E. Tadmor, "A kinetic equation with kinetic entropy functions for scalar conservation laws" Comm. Math. Phys. , 136 (1991) pp. 501–517
How to Cite This Entry:
Bhatnagar-Gross-Krook model. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bhatnagar-Gross-Krook_model&oldid=50259
This article was adapted from an original article by François Bouchut (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article