View source for Bhatnagar-Gross-Krook model

''BGK-model''

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>

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

<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>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b1202208.png" />, 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>

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" />,

<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>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022012.png" />, 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>

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

<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>

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.

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>

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>

<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>

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.

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>

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

<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>

<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>

The equilibrium state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b12022046.png" /> 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>

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>

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

<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>

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

<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>

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" />,

<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>

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

<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>

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>

<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>

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 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 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

<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>

<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>

The new state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120220/b120220100.png" /> 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>

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>

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.

====References====
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