Stability of Maxwellian states for discrete velocity models of the extended Boltzmann equation

Milan Miklavčič^{∗} and Giampiero Spiga^{†}

1 Abstract

Extended kinetic theory has been introduced a few years ago in the literature in order to study a gas in which, together with the usual interactions of elastic scattering, other effects that are important for real world applications are taken into account [1]. Such effects include presence of a background medium, absorptions, inelastic scattering, chemical or nuclear reactions, and self-generating processes, like fission for neutrons and ionization for electrons. The resulting nonconservativity of the governing Boltzmann-like equation, coupled to its nonlinearity, gives rise to interesting behaviors (like Hopf bifurcations, limit cycles, chaotic dynamics) which are absent in standard kinetic theory and in linear transport.

This paper deals with the simplified version of the extended kinetic equation obtained in the frame of the so called discrete velocity models [2] [3], where the collision integral is approximated by quadratic polynomials with respect to the distribution functions relevant to the fixed prescribed velocities. Indeed, discrete kinetic theory has become in the past few years an useful mathematical playground for the actual integrodifferential Boltzmann equation, because of the possibility of pursuing analytical investigations (often leading even to exact solutions), and of obtaining reliable predictions in several (not in all, of course) respects [4].

The same kind of analysis was started more recently also for extended kinetic theory. In such a new frame, it has immediately become clear that even small nonconservative changes with respect to the standard formulation may cause dramatic changes in the mathematical structure of the evolution problem [5] [6].

This paper deals with the extended kinetic equations in one space dimension relevant to the most popular discrete schemes: the six-velocity Broadwell model [7], and/or the two-velocity Carleman and Mc Kean models, which all satisfy an H-theorem [8]. Stability of space independent equilibria (Maxwellian states) to space dependent perturbations is investigated both on the real line (infinite medium problem), and on a finite interval, as a boundary value problem with extended reflection boundary conditions [9]. The physical models considered include, along with elastic scattering between test particles, absorption and self-generation interactions with the field particles of a given background medium, and presence of external sources. Absorbing and self-generating processes may occur at the boundary walls as well, and boundary sources or sinks are also allowed. Extensive analytical and numerical calculations are in progress in order to show possible occurrence of regions of instability in parameter space, in comparison to the corresponding unextended cases. The role played by symmetry in boundary conditions and isotropy in self-generation interactions will be investigated with respect to their stabilizing effects.

∗Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, USA (milan@math.msu.edu)

†Department of Mathematics, University of Parma, Via M. D’Azeglio 85, 43100 Parma, Italy (spiga@prmat.math.unipr.it)

The two velocity extended Carleman model in dimensionless form is given by

\frac{∂N_{1}}{∂t} + \frac{∂N_{1}}{∂x} = −N_{1}^{2} + N_{2}^{2} − εN_{1} + χηρ + S_{1}  (1)
\frac{∂N_{2}}{∂t} − \frac{∂N_{2}}{∂x} = N_{1}^{2} − N_{2}^{2} − εN_{2} + (1 − χ)ηρ + S_{2}  (2)

where ρ = N_{1} + N_{2}. ε > 0, η > 0 and 0 ≤ χ ≤ 1 are constants. S_{i} ≥ 0 are sources and S = S_{1} + S_{2}. ε represents an absorption coefficient, and η takes generation processes (like fission for neutrons, or ionization for electrons) into account. The parameter χ describes the angular distribution of generated particles.

The system (1-2) will be studied on the real line (i.e. −∞ < x < ∞) and as a boundary value problem on [−1, 1], with boundary conditions

N_{1}(−1, t) = α_{1}N_{2}(−1, t) + γ_{1}, N_{2}(1, t) = α_{2}N_{1}(1, t) + γ_{2}  (3)

where α_{1}, α_{2}, γ_{1}, γ_{2} are some nonnegative constants. Again the γ_{i} represent possible boundary sources (sinks), depending on their positive (negative) sign, and the α_{i} stand for nonnegative reflection (necessarily specular) coefficients, which might be allowed to be less or greater than unity in the presence of absorption or generation processes at the walls. Negative inhomogeneous terms would model extraction of material through the boundaries.

Maxwellian states are constants N_{1}, N_{2} that satisfy (1-2) and which are physically relevant, i.e. N_{i} ≥ 0 and ρ > 0. The boundary conditions (3) can be satisfied by the above Maxwellian states, restricted to the interval (−1, 1), by assuming that the four boundary parameters α_{i} and γ_{i} satisfy the set (3) itself, considered as a set of two algebraic linear equations, with N_{1} and N_{2} known constants.

Let N_{1}, N_{2} be a Maxwellian state. Assuming that small perturbations N_{j} + z_{j}(x, t) also satisfy (1-2) and that the terms of the form z_{j}z_{i} are negligible we obtain

\frac{∂z_{1}}{∂t} + \frac{∂z_{1}}{∂x} = (−2N_{1} − ε + χη)z_{1} + (2N_{2} + χη)z_{2}  (4)

\frac{∂z_{2}}{∂t} − \frac{∂z_{2}}{∂x} = (2N_{1} + (1 − χ)η)z_{1} + (−2N_{2} − ε + (1 − χ)η)z_{2}.  (5)

The boundary conditions (3) become

z_{1}(−1, t) = α_{1}z_{2}(−1, t), z_{2}(1, t) = α_{2}z_{1}(1, t).  (6)

The real line problem is studied by the ansatz z_{j} = A_{j}e^{λt+iωx}, which yields, as dispersion relation, a quadratic equation for λ (depending on ω, parameters and N_{j}). Available results seem to indicate stability for all values of parameters.

Up to now, we were also able to demonstrate decay of

∫_{−∞}^{∞}(w_{1}z_{1}^{2} + w_{2}z_{2}^{2})dx  (7)

where z_{1}, z_{2} are the solutions of the fully nonlinear version of (4-5) on the real line and w_{1}, w_{2} are appropriate weights. When w_{1} = w_{2} = 1 our approach closely resembles to the energy method, however, other weights have to be chosen to obtain the decay of the ’energy’ (7) in all cases. This result confirms the result obtained by the much simpler linear theory in the case when sources are present. When there are no sources present the linear theory predicts marginal stability (one solution is λ = 0) and our nonlinear theory does not predict anything.

As regards the boundary value problem, separation of variables technique suggests that we look first for solutions of (4-6) in the form

z_{j}(x, t) = e^{λt}ζ_{j}(x).  (8)

Using (8) in (4-6) we obtain, after some algebra, a single transcendental equation in order to determine the eigenvalues λ. Such an equation allows, to a great extent, an analytical investigation in the physically meaningful case of no boundary sources, i.e. γ_{1} = γ_{2} = 0, and globally conservative walls, i.e. α_{1}α_{2} = 1 (excess generation on one side is compensated by excess absorption on the other). Preliminary results prove that there are actually regions of positive measure in the parameter space for which one positive eigenvalue exists, implying instability.

Analogous steps will be repeated for the two velocity extended McKean model, given by

\frac{∂N_{1}}{∂t} + \frac{∂N_{1}}{∂x} = −N_{1}N_{2} + N_{2}^{2} − εN_{1} + χηρ + S_{1}  (9)
\frac{∂N_{2}}{∂t} − \frac{∂N_{2}}{∂x} = N_{1}N_{2} − N_{2}^{2} − εN_{2} + (1 − χ)ηρ + S_{2},  (10)

with the same boundary conditions. This second model shows, for special values of parameters, the peculiar feature of two simultaneous Maxwellian states.

Attention will be possibly devoted also to more physical but more complicated six velocity Broadwell model in three space dimensions with axial symmetry about the xaxis. Boundary conditions would be given again by (3).

References

[1] G. Spiga, 1991, Rigorous solution to the extended kinetic equations for homogeneous gas mixteures, in Lecture Notes in Mathematics, Vol.1460, 203-221, Springer, Berlin.

[2] H. Cabannes, 1980, The discrete Boltzmann equation, Lecture Notes, University of California, Berkeley.

[3] R. Gatignol, 1975, Théorie cinétique d’un gaz à répartition discrète des vitesses, Lecture Notes in Physics, 36, Springer, Berlin.

[4] H. Cornille, 1990, in Partially Integrable Evolution Equations in Physics, 50-90, Kluwer, Dordrecht

[5] T. Platkowski and G. Spiga, 1994, Similarity solutions of 2-velocity models of the extended Boltzmann equation, Phys. A, 210, 155-164.

[6] H. Cornille, 1997, Exact solutions for nonconservative discrete Boltzmann models, preprint.

[7] J. Broadwell, 1964, Shock structure in a simple discrete velocity gas, Phys. Fluids, 7, 1243-1247.

[8] R. Illner, 1979, Global existence for two-velocity models of the Boltzmann equation, Math. Meth. Appl. Sci., 1, 187-193.

[9] T. Platkowski and G. Spiga, 1992, Exact solutions to stationary problems in discrete extended kinetic theory, Eur. J. Mech., B/Fluids, 11, 349-362.