Bhatnagar-Gross-Krook model

BGK-model

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

(a1)

where is the particle density in the phase space , is the mean free path and is the Boltzmann collision operator. This integral operator acts in the velocity variable only, satisfies the moment relations

(a2)

, and the entropy inequality

(a3)

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

(a4)

, and the decrease of entropy

(a5)

Another striking property of the Boltzmann equation is that if and only if is a Maxwellian, that is

(a6)

for some , , . When time and space dependence are allowed as in (a1), , , can depend on , also. When in (a1), therefore goes formally to a Maxwellian of parameters , and , see [a6], which satisfies the conservation laws (a4), and entropy inequality (a5), with 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

(a7)

and

(a8)

(a9)

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 for some positive function . The case is of interest because then 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

(a10)

where , , , a measure space, , and

(a11)

The equilibrium state is assumed to satisfy

(a12)

Then, defining

(a13)

the system relaxes as to the system of equations

(a14)

Assume that remains in a convex domain of . An interesting property of the kinetic equation (a10) is that it leaves invariant any family of convex sets indexed by . Therefore if one chooses for each a convex set such that

(a15)

then one can start with , for some , and then for all , , . The kinetic entropy inequality is obtained by a convex function , such that the following Gibbs minimization principle holds: for any such that -a.e. and ,

(a16)

This property ensures that in the limit one obtains the entropy inequality

(a17)

with

(a18)

The original BGK-model (a1), (a7), (a8), (a9) enters this framework by taking with Lebesgue measure, , , where is the state and is the scalar physical Maxwellian given by (a6). One has , and for any . Here, since for all , , , 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 , , , , , . See also [a10] for related models.

The success of such BGK-models has been revealed for scalar equations () 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 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 from the knowledge of , by solving

(a19)

The new state is given by

(a20)

Then,

(a21)

which is similar to (a10) with . 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 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