Bhatnagar-Gross-Krook model

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