On Half-Space and Shock-Wave Problems for Discrete Velocity Models of the Boltzmann Equation

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THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

On Half-Space and Shock-Wave Problems for Discrete Velocity Models of the Boltzmann Equation

Niclas Bernho Department of Mathematics

Karlstad University Universitetsgatan 2 SE-65188 Karlstad, SWEDEN

Abstract

We study some questions related to general discrete velocity (with arbitrarily num-ber of velocities) models (DVMs) of the Boltzmann equation. In the case of plane sta-tionary problems the typical DVM reduces to a dynamical system (system of ODEs). Properties of such systems are studied in the most general case. In particular, a topological classication of their singular points is made and dimensions of the corre-sponding stable, unstable and center manifolds are computed.

These results are applied to typical half-space problems of rareed gas dynamics, including the problems of Milne and Kramer. A classication of well-posed half-space problems for linearized DVMs is made. Exact solutions of a (simplied) linearized kinetic model of BGK type are found as a limiting case of the corresponding discrete models.

Existence of solutions of weakly non-linear half-space problems for general DVMs are studied. The solutions are assumed to tend to an assigned Maxwellian at innity, and the data for the outgoing particles at the boundary are assigned, possibly depend-ing on the data for the incomdepend-ing particles. The conditions, on the data at the boundary, needed for the existence of a unique (in a neighborhood of the assigned Maxwellian) solution of the problem are investigated. Both implicit, in the non-degenerate cases, and sometimes, in both degenerate and non-degenerate cases, explicit conditions are found.

Shock-waves can be seen as heteroclinic orbits connecting two singular points (Maxwellians) for DVMs. We give a constructive proof for the existence of solutions of the shock-wave problem for the general DVM. This is worked out for shock speeds close to a typical speed, corresponding to the sound speed in the continuous case. We clarify how close the shock speed must be for our theorem to hold, and present an iteration scheme for obtaining the solution.

The main results of the paper can be used for DVMs for mixtures as well as for DVMs for one species.

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Acknowledgement

This thesis would have never become a reality without the valuable guidance and support from my supervisor, Professor Alexander Bobylev, to whom I am gratefully thanking.

My special thanks go to Mirela Vinerean for appreciated discussions and help with proofreading.

The support by NFR (Sweden) grant 20005092 and the Swedish Research Council grant 20035357 are gratefully acknowledged.

I acknowledge former and present colleagues (in particular, in the departments of mathematics and physics) for contributing to the scientic and social environment at the university. I am also grateful to the administrative sta for their support.

I thank the Kinetic Theory Group in Göteborg for their hospitality, and all nice moments at conferences.

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Chapter 1 Introduction

The basic equation in kinetic theory is the Boltzmann equation. The Boltzmann equation is an equation for the time-evolution of the particle densityi = i (x> >w) (x Rg> Rg andw R₊ are here the position, the particle velocity and the time respectively) in the phase-spaceRg× Rg(see Ref.[19]).

Discrete velocity models (DVMs) of the Boltzmann equation are models, where the velocity is discretizised, i.e. the velocity is assumed to be able to take only a nite number of dierent values. It is a well-known fact that the Boltzmann equation can be approximated by DVMs (see Refs.[10, 40, 41]), and that these approximations can be used for numerical methods. The study of DVMs can also give better concep-tual understanding and new ideas, which can be applied to the ”continuous” Boltz-mann equation. The rst DVMs (at least to our knowledge) are due to the Swedish mathematician Torsten Carleman (posthumously presented in 1957) and the American physicist James E. Broadwell in 1964 [13]. The French mathematicians Renée Gatignol [30] and Henri Cabannes [14] played a very important role in the development of the mathematical theory for DVMs (see also the reviews [6, 43] and references therein).

In this thesis we study some questions related to DVMs. The paper consists of three main parts. If the solutions depend on only one scalar variable the general DVM reduces to a system of ordinary dierential equations (dynamical system). This system is not very interesting in the spatially homogeneous case, however it is much less trivial in the case of planar stationary (or shock wave structure) problems. One can consider the Boltzmann equation (for such problems) as a limiting case of the dynamical system, when the number of discrete velocities tends to innity (see Ref.[7]). The DVM is considered as its nite-dimensional approximation, which in principle can explain many ”qualitative” features of the solution of the Boltzmann equation.

In Part I we study such systems in the most general case. The rst problem is always to clarify the nature of singular (stationary) points, which are Maxwellians in

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the case of DVMs. In Chapter 3, we show that main characteristics of such points (dimensions of corresponding stable, unstable and center manifolds) can be evaluated exactly.

Half-space problems for the Boltzmann equation are important in the study of corrections to Hilbert-Chapman-Enskog solutions near the boundaries (see Refs.[19, 20, 26, 35] and references therein). These boundary layers are usually called Knudsen layers and have a thickness of the order of the mean free part. Outside the Knudsen layer the usual hydrodynamic equations can be used, but these are not satisfactory inside the Knudsen layer. If the boundary of the domain is at, or if its radius of curvature is very large compared to the mean free path, then the Boltzmann equation in a half-space can be used inside the Knudsen layer.

In Ref.[18] Cercignani conjectured a general result for the well-posedness of kinetic layer problems with boundary conditions of Dirichlet type. This was later proved for the linearized Boltzmann equation in Ref.[27] and for the non-linear Boltzmann equation in Ref.[48] (in both cases for a hard sphere gas). Similar results for general (linear) DVMs were stated (in terms of the dimensions of the stable, unstable and center manifolds of the singular points) without proof in Ref.[7]. In this paper we continue the study of DVMs in the directions formulated in Refs.[7, 8].

For the Boltzmann equation in a linearized context, half-space problems have been extensively studied (see Ref.[5]). Half-space problems for the non-linear Boltzmann equation have also been investigated; with slightly perturbed specular reection in Ref.[32], and with boundary conditions of Dirichlet type in Ref.[48].

For the general non-linear DVM, half-space problems, with an assigned Maxwellian, which the solution shall tend to at innity, have been studied in Refs.[4, 33, 34, 47]. In Ref.[47], Ukai proved existence of solutions, for Dirichlet boundary data close to the Maxwellian, with assumptions valid just for the case corresponding to (in the con-tinuous context) supersonic condensation. This was generalized by Kawashima and Nishibata. First in Ref.[33], still with boundary conditions of Dirichlet type, and then in Ref.[34], with general boundary conditions, but still with some restrictive assump-tions (for an example, only non-degenerate cases are studied). In Ref.[4] Babovsky studied a (degenerate) case for the non-linear (and linear) DVM, with slightly per-turbed specular reection (cf. Ref.[32]), but with a quite restrictive condition on the non-linear part of the collision operator. All these results are generalized (for general boundary conditions) in this paper.

Stability of the boundary layers (cf. Refs.[23, 34, 49, 50]) have not been studied at all in this paper. We also mention that extensive numerical studies of the evaporation and condensation problems have been conducted by Aoki, Sone and their co-workers (see Ref.[46] and references therein), and that the rst rigorous results concerning

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1. Introduction 3

existence of solutions to half-space problems for the Boltzmann equation in the strongly nonlinear case were obtained by Arkeryd and Nouri in Ref.[2].

A main dierence between the present paper and above mentioned papers on DVMs is that we try to consider DVMs from a dynamical system point of view introduced in Ref.[7]. Especially, the results in Chapter 3 on the calculations of the dimensions of the stable, unstable and center manifolds of the singular points (Maxwellians for DVMs) are important tools in these studies.

In Part II the results in Chapter 3 are applied to typical half-space problems of rareed gas dynamics. Some few rst steps in these studies were done, for DVMs with boundary conditions of Dirichlet type, in Ref.[8] (appended as Part VI). A classi-cation of well-posed half-space problems for linearized DVMs (with general boundary conditions) is made. Furthermore, existence of solutions of weakly non-linear half-space problems for the general DVM are studied. In Ref.[22] (see also Ref.[29]), Cer-cignani et al. have shown that the solutions of the half-space problem for the general non-linear DVM (with boundary conditions of Dirichlet type) tend to Maxwellians at innity (without specifying the Maxwellians). In the present paper, the singular point (Maxwellian for DVMs) approached at innity is xed and small deviations of the solutions from the singular point is studied. The data for the outgoing particles at the boundary are assigned, possibly depending on the data for the incoming particles. The conditions, on the data at the boundary, needed for the existence of a unique (in a neighborhood of the assigned Maxwellian) solution of the problem are investigated. In the non-degenerate cases implicit conditions are found, using arguments by Ukai, Yang and Yu in Ref.[48] for the continuous Boltzmann equation. Explicit conditions are found, in some special (both degenerate and non-degenerate) cases. Our results in Sections 7.2-7.3 (Theorem 8 and Theorem 10) generalize results obtained for DVMs in Refs.[4, 33, 34, 47] and includes (for DVMs) results obtained in Ref.[48] for the continuous Boltzmann equation, with boundary conditions of Dirichlet type.

The analytically dicult problem of existence of shock-wave solutions are studied (for weak shocks) in Refs.[16, 38] for the continuous Boltzmann equation and Ref.[12] for DVMs. For DVMs the shock-waves can be seen as heteroclinic orbits connecting two singular points (Maxwellians). In Ref.[12], Bose, Illner and Ukai follow the ideas of Caisch and Nicolaenko in the pioneering work [16] for the continuous case. In contrast to Bose et al., in this paper we treat the problem from the dynamical system point of view, and can use our knowledge of the main characteristics (dimensions of corresponding stable, unstable and center manifolds) for singular points.

In Part III we study the weak shock wave problem for the general DVM of the Boltzmann equation. We give a constructive proof for the existence of solutions. We assume that a given Maxwellian is approached at innity, and consider shock speeds

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close to (and larger than) a typical speedf₀, corresponding to the sound speed in the continuous case. Then we transform the system in such a way that we can apply contraction mapping arguments to prove the existence of a non-negative locally unique (up to a shift in the independent variable) bounded solution. Arguments used by Cercignani et al. in Ref.[22] are used to show that the solution tends to a Maxwellian at minus innity. We clarify how close the shock speed must be to the typical speed f0 for our theorem to hold. Our proof is constructive, and we present an iteration scheme for obtaining the solution.

Evaporation and condensation problems (or shock wave problems) for mixtures are not as well investigated in the literature as corresponding problems for one species. In Ref.[1] half-space problems (both homogeneous and inhomogeneous) of a binary mix-ture, in the context of the linearized Boltzmann equation are investigated. Numerical studies of half-space problems for binary mixtures in the linearized context are also conducted (see Refs.[44, 51] and references therein). We stress that the main results of this paper can be used for DVMs for mixtures as well as for DVMs for one species (see Chapter 4).

This paper is (in more detail) organized as follows:

Part I: In Chapter 2, we introduce the general DVM of the Boltzmann equation (Section 2.1), and review some of its properties (some basic denitions and properties in the theory of the Boltzmann equation are briey reviewed in Appendix A). We discuss DVMs depending on one variable, (Section 2.2) viewed as systems of ordinary dierential equations (dynamical systems). We expand the general DVM around an equilibrium Maxwellian (Section 2.3), and note, using the Shoshitalshvili theorem, that in the non-degenerate case the linear and weakly non-linear systems are topologically equivalent.

In Chapter 3, we make a topological classication of the general dynamical system related to linearized DVMs, near the origin, and obtain a main result for the dimen-sions of the stable, unstable and center manifolds of the general dynamical system, stated in Theorem 1. Jordan normal forms, useful for applications, are obtained (Sec-tion 3.3). Calcula(Sec-tions of the above men(Sec-tioned dimensions are carried out for axially symmetric DVMs (in the shock-wave context), when we have expanded around an absolute Maxwellian (Section 3.4). The results are in accordance with the results for the "continuous" Boltzmann equation (cf. Ref.[27]).

We show that DVMs for mixtures also can be included in our general context in Chapter 4. Calculations of the degenerate values are carried out for axially symmetric DVMs for mixtures (in the shock-wave context), when we have expanded around an absolute Maxwellian (Section 4.3).

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1. Introduction 5

reviewed in Chapter 5.

Half-space problems for linearized DVMs, in our general context, are studied in Chapter 6. The problem is stated, and boundary conditions are discussed (Section 6.1). We use the results of Chapter 3 (Theorem 1) to investigate the number of additional conditions needed to obtain well-posedness of the homogeneous (Section 6.2) and inhomogeneous problems (Section 6.3). Solutions of the half-space problems (both the homogeneous and the inhomogeneous) are obtained (Sections 6.2-6.3). The asymptotic ow is discussed briey (Section 6.4), and the discrete version of the Albedo problem, as it is stated in [31], is solved (conditioned that we can nd the vector space spanned by all eigenvectors corresponding to the positive eigenvalues). The half-space problems (stated in Section 6.1) are solved exactly, for a (simplied) linearized kinetic model of BGK-type (Section 6.5.1). We use the solution of the discrete problem in Section 6.5.1, to nd an exact solution of a half-space problem, for a corresponding continuous linearized model (Section 6.5.2). The case when we allow velocities giving a singular "velocity-matrix" (for the boundary layer problem, that is, if we allow velocities that have zero as a rst component) is discussed (Section 6.6). We study a reduced plane 12-velocity DVM (Section 6.7) and solve (after expansion around an absolute Maxwellian) some half-space problems.

We stress that the main results of Chapters 3 and 6 are proved under very gen-eral restrictions, which are not directly related to DVMs. The results therefore have much wider area of applications than just classical DVMs of the Boltzmann equation. In particular, they can be also used for moment approximations of various kinetic equations.

In Chapter 7, half-space problems for weakly non-linear DVMs (in our general context) are studied. We state the problem (Section 7.1). Implicit conditions for the existence of a unique (in a neighborhood of the assigned Maxwellian) solution in the non-degenerate case and also for the degenerate case, but then with some restrictions on the non-linear part (of the collision operator), are obtained (Section 7.2). The results are in accordance with corresponding results for the continuous Boltzmann equation obtained in the non-degenerate case, with boundary conditions of Dirichlet type [48]. Arguments used in Refs.[34, 48] are used in our proof. We obtain explicit conditions for the existence of a unique (in a neighborhood of the assigned Maxwellian) solution (Section 7.3), but with more restrictions on the non-linear part (at least in the non-degenerate case). Some additional cases to the ones in Section 7.2 are covered in the degenerate cases. We show that our results (Theorem 6 and Theorem 10) in Sections 7.2-7.3 can be (trivially) applied for the spatially homogeneous problem (Section 7.4). We apply our results (Theorem 10) in Section 7.3 to a boundary layer problem of the type studied in Ref.[32] for the Boltzmann equation, and in Ref.[4] for

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DVMs, (Section 7.5). We consider a plane 12-velocity DVM (Section 7.5.1), a simplied model for the linearized collision operator (Section 7.5.2), and, nally, general axially symmetric DVMs (Section 7.5.3). The case when we allow velocities giving a singular "velocity-matrix" (for the boundary layer problem, that is, if we allow velocities that have zero as a rst component) is discussed (Section 7.6).

Part III: We study the weak shock wave problem for the general DVM of the Boltz-mann equation in Chapter 8. The shock wave problem for the BoltzBoltz-mann equation is reviewed (Section 8.1). We state the problem, and a result for existence of solu-tions (Theorem 12, Section 8.2). The problem is transformed (Section 8.3), and the proof is carried out in dierent steps (Sections 8.4-8.7): (1) (main part) existence of a non-trivial locally unique (up to a shift in the independent variable) bounded solution is proved by applying contraction mapping arguments (Section 8.4); (2) positivity of solutions (under suitable conditions) is proved (Section 8.5); (3) convergence of the solution to a Maxwellian at minus innity is proved, using arguments by Cercignani et al. in Ref.[22] (Section 8.6). We clarify how close the shock speed must be to the typical speedf₀for our theorem to hold, and present an iteration scheme for obtaining the solution (Section 8.8).

The key results of Chapters 2-3 are published in Ref.[8] (appended as Part VI) by Alexander Bobylev and the author. The rest of the material is in preparation for publication. Part III is a joint work by Bobylev and the author.

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Part I

Discrete Velocity Models and

Dynamical Systems

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Chapter 2 Discrete Velocity Models

2.1 Denitions and properties

The general discrete velocity model (DVM) (see Refs.[6, 14, 30, 43] and references therein) reads

Cil

Cw +l· xil=T_l(i> i) , l = 1> ===> q, (2.1) where V = {₁> ===> _q} Rg is a nite set of velocities, i_l = i_l(x> w) = i (x> w> _l) forl = 1> ===> q, and i = i (x> w> ) represents the microscopic density of particles with velocity at time w R₊and positionx Rg.

Remark 1 For a function j = j() (possibly depending on more variables than ), we will identifyj with its restrictions to the points V, i.e.

j = (j1> ===> jq)> with j_l=j (_l) forl = 1> ===> q.

Theni = (i₁> ===> i_q) in Eq.(2=1). We say that j is non-negative, j 0, if, and only if,j_l 0 for all 1 l q.

The collision operatorsT_l(i> i) in (2=1) are given by Tl(i> i) =

q

X

m>n>o=1

no_lm(i_ni_o i_li_m) forl = 1> ===> q, (2.2) where it is assumed that the collision coecients no_lm, 1 l> m> n> o q, satisfy the relations

no_lm= no_ml= lm_no 0, (2.3)

with equality unless the conservation laws

_l+_m=_n+_oand |_l|2+¯¯_m¯¯2=|_n|2+|_o|2 9

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are satised. The collision operators (2=2) can be obtained from the bilinear expressions Tl(i> j) = 1₂ q X m>n>o=1 no_lm(i_nj_o+j_ni_o i_lj_m j_li_m) ,l = 1> ===> q= (2.4) Remark 2 The collision operatorT_l(i> j) is a function of _l, forl = 1> ===> q. Hence, for xed functionsi = i(x>w> ) and j = j(x>w> ), there is, in accordance with Remark 1> a function T(i> j) = T(i> j) (x>w> ) dened (for V) by Eqs.(2=4).

We will below, for arbitrary|> } Rq, denote byh|> }i the Euclidean scalar product inRq.

For arbitrary functionsi = i()> j = j() and k = k() T (i> j) = T (j> i) , and from the relations (2=3)

hk> T (i> j)i =1 8 q X l>m>n>o=1 no_lm(k_l+k_m k_n k_o) (i_nj_o+j_ni_o i_lj_m j_li_m) . (2.5) A function! = ! () is a collision invariant, if, and only if,

!_l+!_m=!_n+!_o, (2.6)

for all indices 1 l> m> n> o q such that no_lm 6= 0. By the relation (2=5) h!> T (i> i)i =1 4 q X l>m>n>o=1 no_lm¡!_l+!_m !_n !_o¢(i_ni_o i_li_m) , (2.7) which is zero, independently of our choice of non-negative functioni, if, and only if, ! is a collision invariant.

We consider below (even if this restriction is not necessary in our general reasoning) only normal DVMs. That is, DVMs such that any collision invariant is on the form

! = d + b · + f ||2 (2.8)

for some constantd> f R and b Rg (methods of their constructions are described in Refs.[9, 11]). In this case the equation

h!> T (i> i)i = 0, has the general solution (2=8).

A Maxwellian distribution (or just a Maxwellian) is a functionP = P(), such that

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2. Discrete Velocity Models 11

All Maxwellian distributions are on the form P = h!₌_Nhb·+f||2

> with N = hd_{A 0,} _(2.9)

where! is given in Eq.(2=8). To obtain this, we use the relation

(} |) log|_} 0, (2.10)

with equality if, and only if,| = }, which is valid for all |> } R₊. Assuming that a functioni is non-negative, we let ! = log i in Eq.(2=7), and obtain, by the relation (2=10), that

hlog i> T (i> i)i =1 4 q X l>m>n>o=1 no_lm(i_ni_o i_li_m) logilim inio 0, (2.11)

with equality if, and only if,

inio=i_li_m, (2.12)

for all indices 1 l> m> n> o q such that no_lm6= 0. Hence, there is equality in Eq.(2=11) if, and only if,i is a Maxwellian. Taking the logarithm of Eqs.(2=12), i is a Maxwellian if, and only if, logi is a collision invariant.

Maxwellians withb = 0 in Eq.(2=9) are called absolute Maxwellians.

2.2 DVMs as dynamical systems

We shall consider DVMs (2=1), where i = i (x> w> ) can be represented as a function of one scalar variable except . There are at least three interesting classes of such systems.

(A) The spatially homogeneous systems gi

gw =T (i> i) , with i = i (w> ) and w R+. (2.13) (B) The planar stationary systems

1 gi

g{1 =T (i> i) , with i = i({1> ) and {1 R+, (2.14)

where

x = ({1_{> ===> {}g_{) and}_{= (}1_{> ===>}g_).

(C) The shock wave systems. We assume that

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wherec =¡f1> 0> ===> 0¢is the velocity of the shock wave, and obtain (1 f1)gI

g| =T (I> I ) , where | R. (2.15) We assume below (if nothing else is stated) that the sets of velocities V ={₁> ===> _q} are chosen in such a way that in the case (B)1_l6= 0; and in the case (C) 1_l 6= f1; for l = 1> ===> q.

The above cases (A)-(C) can be considered in a unied form if we denote the independent variables (w> {1_and_{| in cases (A), (B) and (C) respectively) by w}

Ggi_gw=T (i> i) , with w R and G = diag(g₁> ===> g_q), (2.16) where

(A)g_l= 1, (B)g_l=1_l, (C)g_l=1_l f1_l> for l = 1> ===> q.

Remark 3 Eqs.(2=14) and (2=15) are equivalent for the continuous Boltzmann equa-tion, because of Galilei invariance (just shift the velocity variable). However, this is not the case for DVMs with a nite number of velocities (by shifting the velocities we end up, at least partly, outside the set of allowed velocities).

All solutions of the system (2=16), having physical meaning, can be represented by phase trajectories i = i(w> ) in the physical domain Rq₊ (i (w> ) 0) of the whole phase space Rq. The conservation laws (2=8) dene the invariant subspace parametrized byg + 2 numbers m₁> ===> m_g+2, such that

m1=hGi> 1i , (m₂> ===> m_g+1) =hGi> i and m_g+2= D

Gi> ||2E, (2.17) to which every phase trajectoryi (w> ) belongs for all w R. The g + 2 parame-ters (d> b> f) in a Maxwellian distribution (2=9) depend on the parameparame-ters m1> ===> mg+2, through the equations

m1=hGP> 1i , (m₂> ===> m_g+1) =hGP> i and m_g+2= D

GP> ||2E_. _(2.18)

Thus it is, at least implicitly, clear how to nd (under the assumptions above) all stationary points in the physical domain of the phase space of the dynamical system (2=16).

Remark 4 If we consider only such symmetric sets of velocities V, such that if_l= (1_l> ===> g_l) V> then (±1_l> ===> ±g_l) V (2.19)

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Figure 2.1: Velocity grid for the 32-velocity model

for any combinations of signs, then the equations (2=13)-(2=15) admit a class of solu-tions satisfying

il=i_l0 if1_l =_l10and |_l|2=|_l0|2.

This reduces the numberq of equations (2=13)-(2=15) to the number 2Q ? q of dierent combinations (1

l> |l|2) in the velocity set. However, the structure of the collision term

(2=2) (in slightly dierent notations) remains unchanged. We can, without loss of generality, assume that

(1

l+Q>¯¯l+Q¯¯2) = (1l> |l|2) and1l 0

forl = 1> ===> Q. Then, the Maxwellians are on the form

Pl=Nhe1l+f|l|2₌P_l+Qh2e1l_,l = 1> ===> Q> _(2.20) for some constants N> e> f R, with N A 0. Examples of (normal) such DVMs are the plane 12-velocity model [9] in Section 6.7, with velocities (±1> ±1), (±1> ±3) and (±3> ±1), and the innitely many (obvious, from the constructions in Ref.[9] - "with three corners of a square in the model, add the fourth") symmetric normal extensions of this model. These extensions include the plane square models, with (all combinations of ) coordinates from the set of all odd integers with absolute values less or equal than a maximal odd integer (these models are called Nicodin s-th squares in Ref.[24], but already, at least implicitly, constructed in Ref.[9]).

Examples in three dimensions are the 32-velocity model, with velocities (±1> ±1> ±1), (±1> ±1> ±3), (±1> ±3> ±1) and (±3> ±1> ±1), (see g.2.1) and the innitely many (ob-vious) symmetric normal extensions of this model. These extensions include the cubic

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Figure 2.2: Velocity grid of the 9-velocity asymmetric model

models, with (all combinations of ) coordinates from the set of all odd integers with absolute values less or equal than a maximal odd integer. The 32-velocity model can be obtained by normal extensions, with the starting point in the 9-velocity asymmetric normal model with velocities (±1> ±1> ±1) and (3> 1> 1) (see g.2.2). The 24-velocity models, with velocities (±1> ±1> ±1), (±1> ±3> ±1) and (±3> ±1> ±1), and (±1> ±1> ±1), (±1> ±1> ±3) and (±3> ±1> ±1), are DVMs with fewer velocities (earlier in the "evolu-tion"), that can be constructed from the same asymmetric model.

2.3 Linearized collision operator

We make an expansion around a MaxwellianP (2=9), and obtain a linear part, called the linearized collision operator, and a quadratic part. We denote

i = P + P1@2_k, _(2.21)

in Eq.(2=1), and obtain Ckl

Cw +l· xkl= (Ok)l+Vl(k) , l = 1> ===> q,

whereO is the linearized collision operator (q × q matrix) given by

Ok = 2P1@2_{T(P> P}1@2_k), _(2.22)

andV is the quadratic part given by

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Given in more explicit forms, the operators (2=22) and (2=23) read (Ok)_l= q X m>n>o=1 no_lmP_m1@2 ³ P_n1@2ko+P_o1@2k_n P_l1@2k_m P_m1@2k_l ´ ,l = 1> ===> q> (2.24) and Vl(k) = q X m>n>o=1 no_lmP_m1@2(k_nk_o klkm)> l = 1> ===> q, respectively.

By Eqs.(2=24) and the relations (2=3), we obtain the equality hj> Oki = 1 4 q X l=1 q X m>n>o=1 no_lm ³ P_n1@2ko+P_o1@2k_n P_l1@2k_m P_m1@2k_l ´ × ×³P_n1@2jo+P_o1@2j_n P_l1@2j_m P_m1@2j_l ´ . Hence, the matrixO is symmetric, i.e.

hj> Oki = hOj> ki

for all functionsj = j() and k = k(), and semi-positive, i.e. hk> Oki 0

for all functionsk = k(). Furthermore, hk> Oki = 0 if, and only if,

P_n1@2ko+P_o1@2k_n=P_l1@2k_m+P_m1@2k_l (2.25) for all indices 1 l> m> n> o q satisfying no_lm6= 0. We let k = P1@2! in Eq.(2=25), and obtain Eq.(2=6), by the relations PlPm=P_nP_o6= 0. Hence, since O is semi-positive,

Ok = 0 if, and only if, k = P1@2_!, _(2.26)

where! is a collision invariant. In consequence, D

V (k) > P1@2!E=hT (i> i) > !i + D

k> OP1@2!E= 0 (2.27) for all collision invariants!.

The system (2=16) transforms in

Ggk_gw +Ok = V(k), (2.28)

where the operatorsO and V are given by Eqs.(2=22)-(2=23). The diagonal matrix G (2=16) (under our assumptions) has no zero diagonal elements and is non-singular. We denote k|_w=0=k₀. Then the formal solution of Eq.(2=28) reads

k(w) = hwG1_O k0+ w Z 0 h(w)G1_O [V (k)] () g.

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Moreover, the nonlinear term is not very important for the topological classication of the stationary pointk = 0 of the system (2=28). In fact, there is an invertible matrix S , such that

S1_G1_{OS = M + Q,}

whereM is a diagonal matrix M = diag(0> ===> 0_{| {z }}

s+o

> 1> ===> t)>

withq = s + o + t> ₁> ===> _p+A 0 and _p+₊₁> ===> _t? 0>

Q2_{= 0 and}_SW_{GS is a diagonal matrix with non-zero diagonal elements (see Chapter}

3).

We denote

j = S1_{k, with j = ({> |> }) > { R}s+o_{> | R}p+

and} Rp, wherep=t p+, in the system (2=28), and obtain the system

gj gw = (M + Q) j + S1G1V(S j) or, equivalently, g{ gw = eQ{ + I ({> |> }) g| gw = eM0| + J({> |> }) g} gw = eM1} + K({> |> }) , (2.29)

with (I> J> K) = S1_G1_{V(S j), e}_Q2_{= 0, e}_M₀_{= diag(}₁_{> ===>}

p+) and eM₁=

diag(_p+₊₁> ===> _t). The system (2=29) is, by the Shoshitalshvili theorem (see Ref.[42, p.161] and also Refs.[3, 28]), in a neighborhood of the equilibrium pointk = 0, topo-logically equivalent to the system

g{ gw = eQ{ + I ({> n1({) > n2({)) g| gw = eM0| g} gw = eM1} >

for someF1-functionsn1({) and n2({). If G1O is diagonalizable (the non-degenerate

caseo = 0, cf. Chapter 3), then the system (2=28) is topologically equivalent (in a neighborhood of the stationary pointk = 0) to the corresponding linear system

Ggk_gw +Ok = 0 sinceI 0 according to Eqs.(2=26)-(2=27))=

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Chapter 3 Topological Classication

3.1 General linear system

We study the general linear system corresponding to the system (2=28).

LetH be a real symmetric q × q matrix. We introduce a scalar product dened by

h{> |i_H=h{> H|i (3.1)

(we remind thath·> ·i denote the Euclidean scalar product in Rq) for{> | Rq, and denote the null-space ofH by Q(H), i.e.

Q(H) = {{ Rq_{| H{ = 0} .}

IfX is a subspace of Rq, then we denote the orthogonal complement, with respect to the scalar producth·> ·i_H, byXH_{, i.e.}

XH₌_{{{ R}q_{| h{> |i}

H= 0 for all| X } ,

and, in particular, ifH is the identity matrix, then we denote X₌_XH_.

We consider below any real symmetric matrices D and E, such that D is semi-positive and have a non-trivial null-spaceQ(D), and E is non-singular, i.e.

h{> {i_D 0 for all { Rq_{, det}_{E 6= 0 and dim(Q(D)) = s 1.} _(3.2)

The general linear system

Eg{_gw+D{ = 0, w R, (3.3)

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(cf. Eq.(2=28)) has the formal solution

{ (w) = hwE1_D { (0) .

In order to study the solutions of Eq.(3=3), we obtain a ”simple” form of the exponential matrixhwE1D. Our way to reach this goal is to nd a suitable canonical form of the matrix E1D. If the matrix E1D is diagonalizable, i.e. if E1D has q linearly independent eigenvectors, then the matrixhwE1D is diagonalizable. In our general case, the matrixE1D is not diagonalizable, but we will prove that we are, in some sense, not ”far” from being able to diagonalizeE1D. In order to nd the canonical form (Jordan normal form), we construct a basis of Rq, consisting of generalized eigenvectors (see Appendix B for the denition), with some special properties, of the matrix E1D. The main result of this section, presented in Theorem 1, is telling us how we can decide the number of positive and negative eigenvalues of the matrix E1_D.

The real eigenvalue problem reads

E1D{ = { or, equivalently, D{ = E{, (3.4)

with R, { Rqand{ 6= 0 (in general we have to consider C and { Cq, but in our case, by Lemma 2 below, we have a full set of real eigenvalues, and can consider only the real case).

We make the assumption that

dim(Q(D)) = s 1,

and denote the space of real generalized eigenvectors of the matrixE1D belonging to the eigenvalue = 0 by Y₀, i.e.

Y0=Y₀¡E1D¢=©{ Rq¯¯ p:¡E1D¢p{ = 0ª. (3.5) Our rst result is that all generalized eigenvectors of the matrixE1D belonging to the eigenvalue = 0 is in Q¡DE1D¢. In fact (see Lemma 2 below), if 6= 0, then all generalized eigenvectors are, in fact, eigenvectors.

Lemma 1 Y0=Q¡DE1D¢. Proof. Letn N. If ¡ E1_D¢n+1_{{ = 0,} then (D1@2E1D1@2)n(D1@2{) = 0,

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3. Topological Classication 19

sinceQ (D) = Q(D1@2) and detE 6= 0. But D1@2E1D1@2 is symmetric, and therefore (D1@2E1D1@2)(D1@2{) = 0.

Hence,

DE1_{D{ = 0,}

and the lemma is proved.

Remark 5 SinceQ (D) = Q(D1@2), we obtain dim

³

Q³D1@2_E1_D1@2´´_{= dim}¡_Q¡_DE1_D¢¢_. _(3.6)

Now we create a special basis of Y₀. We denote the image of the matrix D by Im (D), i.e.

Im (D) = {{ Rq| | Rq:{ = D| } . ThenRqis the direct sum ofQ(D) and Im (D), i.e.

Rq₌_{Q(D) Im (D) .}

We can choose an orthogonal, with respect to the scalar product h·> ·i_E, basis of

Q(D) _© |1> ===> |n> }01> ===> }o0 ª , withn + o = s, (3.7) such that h|l> |mi_E=_l_lm, with₁> ===> _n+A 0 and _n+₊₁> ===> _n? 0, and }0_u> |_l®_E=}_u0> }0_v®_E= 0, (3.8) forl> m = 1> ===> n> and v> u = 1> ===> o (see Appendix B). Then

}0

u Q(D)E,

or, equivalently,

E}0

u Q(D)= Im (D)

foru = 1> ===> o. Hence, for 1 u o, there is a vector y_u Rq such that E}0

u=Dyuor, equivalently,}0_u=E1Dy_u. (3.9) On the other hand,D}_u0 = 0, and therefore

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for u = 1> ===> o. If ₁y₁+=== + _oy_o = 0, then ₁}₁0 +=== + _o}_o0 = 0, and hence, 1==== = _o= 0. Therefore,

©

|1> ===> |n> }10> ===> }0o> y1> ===> yoª (3.10)

is a basis ofY₀.

We distinguish between two cases: the non-degenerate case when o = 0, and the degenerate case wheno 1. The next result tells us that if o = 0, then all generalized eigenvectors are eigenvectors, and hence, the matrixE1D is diagonalizable if, and only if,o = 0.

Lemma 2 The matrixE1D has t = q s o real orthogonal, with respect to the scalar producth·> ·i_E, eigenvectors

{x1> ===> xt} , with hxl> xmiE=llm, (3.11) inYE

0 , corresponding to non-zero real eigenvalues

{1> ===> t} , where 1> ===> p+A 0 and _p+₊₁> ===> _t? 0. (3.12) Proof. Multiplying Eq.(3=4) by D1@2and denoting| = D1@2{, we obtain the new eigenvalue problem

F| = |, (3.13)

where the matrixF = D1@2E1D1@2is symmetric. ThereforeF has q real orthonormal eigenvectors

{1> ===> {q, where h{_l> {_mi = _lmforl> m = 1> ===> q, (3.14) which, after possible reordering, have the corresponding real eigenvalues

1> ===> q, with₁> ===> _p+A 0, _p+₊₁> ===> _t? 0 and _t+1==== = _q= 0, (3.15) where 1 t q is still unknown. The number of zero eigenvalues is equal to dim¡Q(D1@2E1D1@2)¢, and hence, by Eq.(3=6), = 0 is an eigenvalue of multiplicity s + o to the problem (3=13). Therefore,

t = q s o.

If| is a solution of Eq.(3=13), then x = E1D1@2| is a solution of Eq.(3=4) for the same, since

Dx = D1@2_{F| = D}1@2_{| = Ex.}

Therefore,

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are eigenvectors ofE1D that correspond to the eigenvalues {₁> ===> _q} in (3=15). The orthogonality conditions (3=14) imply that

hxl> xmiE = D D1@2_{ l> E1D1@2{m E =h{_l> F{_mi = _l_lm, with_l 6= 0 for l> m = 1> ===> t. (3.17)

The vectors (3=16) are linearly independent (₁x₁+=== + _tx_t= 0 implies that₁= === = t= 0, by Eqs.(3=17)).

Forl = 1> ===> t, we obtain hxl> |iE=1

lhxl> |iD= 0 if| Q(D). (3.18)

On the other hand, ifz Y₀\ Q(D), then there exists | Q(D) such that Dz = E|, and then, by Eq.(3=18),

hxl> ziE=

1

lhxl> ziD=

1

lhxl> |iE= 0

forl = 1> ===> t. That is, {x₁> ===> x_t} YE

0 , and the lemma is proved.

Now we create a new basis ofY₀, to be used in the proof of Theorem 1. Lemma 3 There is a basis

B2={|₁> ===> |_n> }₁> ===> }_o> z₁> ===> z_o} (3.19) ofY₀, such that Eqs.(3=8) and

hzu> }viE=uv, hz_u> |_li_E=h}_u> |_li_E=hz_u> z_vi_E=h}_u> }_vi_E= 0,

E1_Dz_u₌_}_u_and_}_u_Q(D), _(3.20)

forl = 1> ===> n, and v> u = 1> ===> o, are fullled.

Proof. We start by substitutingy_u,u = 1> ===> o, in the basis (3=10) by y0 u=yu n X l=1 hyu> |liE h|l> |liE|l . By Eqs.(3=7)-(3=9), y0 u> |l®_E=}_u0> |_l®_E=}0_u> }_v0®_E= 0> E1Dy_u0=}_u0 and}_u0 Q(D), (3.21) foru = 1> ===> o> and l = 1> ===> n. We denote

z1= 1 1@2₁ y 0 1₂3@21 1 }0 1 }1= 1 1@2₁ } 0 1 ,

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where

1=y0₁> }0₁®_E=y0₁> y0₁®_DA 0 and ₁=y0₁> y₁0®_E, and obtain, by Eqs.(3=21),

hz1> |liE=h}1> |liE=hz1> z1iE=h}1> }1iE=

}0

u> }1®_E= 0, hz1> }1iE= 1,E1Dz1=}₁and}₁ Q(D) foru = 2> ===> o> and l = 1> ===> n.

Form = 2> ===> o, after constructing {}₁> ===> }_m1> z₁> ===> z_m1}, such that hzu> |liE=h}u> |liE=hzu> zviE=h}u> }viE=

}0> }v®_E= 0,

hzu> }viE=uv,E1Dzu=}uand}u Q(D) (3.22) forl = 1> ===> n, u> v = 1> ===> m 1> and = m> ===> o, we construct {}_m> z_m} by the following algorithm. We start by denoting

z0 m=ym0 m1_P u=1 ³D y0 m> zu E E}u+ D y0 m> }u E Ezu ´ }00 m =}m0 m1_P u=1 D y0 m> }u E E}u . By Eqs.(3=21)-(3=22), z0 m> }v®_E=z_m0> z_v®_E=z0_m> |_l®_E=}00_m> |_l®_E=}_m00> }_v®_E=}00_m> }00_m®_E= 0, E1_Dz0 m=}m00and}m00 Q(D) (3.23)

forl = 1> ===> n> and v = 1> ===> m 1> and moreover, for v = 1> ===> m 1, }m00> zv®_E=z_m0> z_v®_D=z_m0> }_v®_E= 0. (3.24) Next, we construct{}_m> z_m}, zm = 1 1@2_m z 0 m _m 23@2_m } 00 m }m= 1 1@2_m } 00 m , where m=z0_m> }_m00®_E=z_m0> z_m0®_DA 0 and _m=z_m0> z_m0®_E. Then, by Eqs.(3=22)-(3=24), hzm> |li_E=h}_m> |_li_E=hz_m> z_vi_E=h}_m> }_vi_E=}0_u> }_m®_E= 0, hzv> }miE=hzm> }viE=vm> E1Dzm=}_m and}_m Q(D), forl = 1> ===> n, v = 1> ===> m> and u = m> ===> o. The lemma follows.

The setB₁B₂, whereB₁={x₁> ===> x_t} is given in (3=11) and B₂is the basis (3=19), is a basis ofRq, consisting of generalized eigenvectors of the matrixE1D.

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3.2 Dimensions of the stable, unstable and center

mani-folds

We develop a method for nding the number of positive and negative eigenvalues of the matrixE1D.

Remark 6 We denote

u = (x1> ===> xv) andv = (y₁> ===> y_v), (3.25) where y_l> x_m Rq are arbitrary vectors for l> m = 1> ===> v, and v q. Let H be a symmetricq × q matrix and bX₁and bX₂twov × v matrices (hats are used for matrices of lower dimensions). Then we introduce anv × v matrix

hu vi_H, with entries (hu vi_H)_lm=hx_l> y_mi_H, and consider the change of variables

eu = bX1u and ev = bX2v,

whereu, eu, v and ev are formally understood as v-dimensional vectors (3=25). Then it is easy to verify the following formulas

heu vi_H= bX₁hu vi_H and hu evi_H=hu vi_HXb₂W, and hence,

heu evi_H= bX₁hu vi_HXb₂W.

Remark 7 Lety = (|₁> ===> |_s), whereQ (D) = span(|₁> ===> |_s). The matrixhy yi_E is symmetric and hence, there is an orthogonal matrix bX (i.e. with bX1= bXW), such that

b

X hy yi_EXb1_{= b}_{X hy yi}

EXbW =hey eyiE, whereey = bXy, (3.26)

is a diagonal matrix, with the eigenvalues ofhy yi_Eon the diagonal. Hence,{e|₁> ===> e|_s}, where e|_l = bX|_l for l = 1> ===> s, is an orthogonal, with respect to the scalar product h·> ·i_E> basis of Q(D). But, by Sylvester’s theorem (and its corollary in Appendix B), for any orthogonal basis {}₁> ===> }_s} of Q(D), the number of positive, negative and zero diagonal elements of the matrixhey eyi_Eare the same as of the diagonal matrix hz zi_E. Hence, thes × s matrix N = hy yi_E, with entriesn_lm=h|_l> |_mi_E, has the same number of positive, negative and zero eigenvalues, independently of the choice of the basis{|₁> ===> |_s} of Q (D). That is, the matrix N = hy yi_E hasn+positive, n₌_{n n}+_{negative and}_{o zero eigenvalues (with n}+₊_n₊_{o = s), independently of}

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Example 1 A typical case for the Boltzmann equation (if we have made the expansion (2=21) around an absolute Maxwellian and we have an orthogonal basis, with respect to the scalar producth·> ·i_E, ofQ(D)) is (here g = 3)

N = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , with A 0. Accordingly,n+=n= 1 ando = 3.

We shall denote byq±, whereq++q=q, and p±, withp++p=t, the num-bers of positive and negative eigenvalues (counted with multiplicity) of the matrices E and E1_{D respectively, and by p}0_{the number of zero eigenvalues of}_E1_{D. Then}

the numbersp+,pand p0 are the dimensions of the stable, unstable and center manifolds of the system (3=3) (and corresponding non-linear systems) respectively.

In applied problems, the numbers of collision invariants is relatively small com-pared toq (note that formally q = for the continuous Boltzmann equation whenas s 5). Therefore the numbers n+_{> n}_and_{o are usually known in (almost for DVMs)}

explicit form. On the other hand, the matrixE is usually diagonal and therefore all its eigenvalues are also known. This explains the importance of the following result. Theorem 1 The numbers of positive, negative and zero eigenvalues ofE1D are given

by p+₌_q+_n+_o p₌_q_n_o p0₌_{s + o} . (3.27)

Proof. The matrixE is symmetric and non-singular, and therefore there is an orthonormal basis

{h1> ===> hq} , with hhl> hmi = lm forl> m = 1> ===> q, (3.28) ofRq, consisting of eigenvectors ofE. After possible reordering,

Ehl=e_lh_lfor 1 l q, where

e1> ===> eq+A 0 and e_q+₊₁> ===> e_q? 0. Hence,

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for l> m = 1> ===> q, and the basis (3=28) is an orthogonal, with respect to the scalar producth·> ·i_E> basis of Rq.

We denote z+ u =zu+}u 2 andz u =zu}₂u,

and substitute{}_u> z_u} by {z+_u> z_u} in the basis (3=19) for u = 1> ===> o. By Eqs.(3=20), z+ u> z+v®E=uv, z u> zv®E=uvand z+ u> zv®E= 0 (3.30)

foru> v = 1> ===> o. Adding {x₁> ===> x_t} in (3=11) to our new basis of Y₀, we obtain another orthogonal, with respect to the scalar producth·> ·i_E, basis ofRq. By Eqs.(3 =8),(3=11)-(3=12),(3=30) we know that the elements of this basis are such that

z+

u> z+u®E= 1A 0, hxl> xliE=lA 0 and h|m> |mi_E=_mA 0 (3.31) foru = 1> ===> o, l = 1> ===> p+, andm = 1> ===> n+, and

z u> zu ® E=1 ? 0, hxl+p+> xl+p+iE=l+p+? 0 and |m+n+> |_m+n+®_E=_m+n+? 0 (3.32) foru = 1> ===> o, l => ===> p, andm = 1> ===> n, wheren=n n+, witht + n + 2o = q and p = t p+. By Sylvester’s theorem (and its corollary in Appendix B) and Eqs.(3=29),(3=31)-(3=32), we obtain Eqs.(3=27). The theorem is proved.

3.3 Jordan normal form

We have constructed a basis

{x1> ===> xt> |1> ===> |n> }1> ===> }o> z1> ===> zo} (3.33) ofRq, such that |l> }u Q(D)> E1Dzu=}_uandE1Dx=x> (3.34) and hx> xiE=, with₁> ===> _p+A 0 and _p+₊₁> ===> _t? 0, h|l> |mi_E=_l_lm, with₁> ===> _n+A 0 and _n+₊₁> ===> _n? 0> hx> }uiE=hx> zuiE=hx> |liE=hzu> |liE=h}u> |liE= 0> hzu> zviE=h}u> }viE= 0 and hzu> }viE=uv> (3.35) for> = 1> ===> t, l> m = 1> ===> n, and v> u = 1> ===> o=

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Therefore, the Jordan normal form ofE1D is 1 . .. t 0 . .. 0 0 1 0 0 . .. 0 1 0 0 >

where there areo blocks of the type Ã

0 1

0 0

!

. In consequence, the Jordan normal form ofhwE1Dis hw1 . .. hwt 1 . .. 1 1 w 0 1 . .. 1 w 0 1 >

and hence, for any{0 Rq, we obtain hwE1_D {0= n X l=1 l|l+ o X m=1 ¡¡ m wm¢}m+_mz_m¢+ t X u=1 uhuwxu, (3.36) where _l=h{0> |liE _l ,u=h{0> xuiE u ,m=h{0> }miE andm=h{0> zmiE, withl=h|l> |liE, forl = 1> ===> n, u = 1> ===> t, and m = 1> ===> o.

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3.4 Axially symmetric DVMs

We assume that (i) we have a symmetric set (2=19) of 2Q velocities; (ii) the reduction in Remark 4 is done; and (iii) we have made an expansion (cf. Eq.(2=21)) around an absolute MaxwellianP (with e = 0 in Eq.(2=20)). Let (cf. Eq.(2=15))

E = E(f) = diag(1

1 f> ===> 1Q f> 11 f> ===> 1Q f), (3.37)

and let the matrixD be such that the null-space of D is given by Q(D) = span (!₁> !₂> !₃) , where !1=P1@2· (1> ===> 1) !₂=P1@2· (1₁> ===> 1_Q> 1₁> ===> 1_Q) !₃=P1@2· (|₁|2> ===> |_Q|2> |₁|2> ===> |_Q|2) . Then, N = f"₁ "₂ f"₃ "₂ f"₂ "₄ f"₃ "₄ f"₅ >

whereN = (!_l> !_m®_E(f)),"₁=h!₁> !₁i, "₂=h!₂> !₂i, "₃=h!₁> !₃i, "₄=h!₂> !₃i_E(0) and"₅=h!₃> !₃i. Hence,

det(N) = f("₁"2₄+"2₂"₅ 2"₂"₃"₄) f3("₁"₂"₅ "₂"2₃), and the degenerate values off (the values of f for which o 1) are

f0= 0 andf_±=± s "1"24+"22"5 2"2"3"4 "₂("₁"₅ "2 3) . (3.38) Moreover, by denoting *1="2("2"5 "3"4)!1+!2+"2("1"4 "2"3)!3 *2="2("2"5 "3"4)!1!2+"2("1"4 "2"3)!3 *3="4!1 "2!3 , with = ("₁"2 4+"22"5 2"2"3"4)("1"5 "23), we obtain e N = "1"5 "23 2p"₂ f2"₂("₁"₅ "2₃) 0 0 0 2p"₂ f2"₂("₁"₅ "2₃) 0 0 0 f >

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where eN = (P₊*_l> *_m®_E(f)). We obtain the following table for the values ofn+,n ando: f ? f f = f f? f ? 0 f = 0 0 ? f ? f+ f = f+ f+? f n+ ₃ ₂ ₂ ₁ ₁ ₀ ₀ n ₀ ₀ ₁ ₁ ₂ ₂ ₃ o 0 1 0 1 0 1 0 . (3.39) If the assumptions (i)-(iii), given above, is fullled (as usual we assume thatf 6= ±1_l forl = 1> ===> Q), then we obtain the following theorem.

Theorem 2 LetQ₊ =Q₊(f) and Q= Q(f) denote the number of 1 l 2Q, such that f A 1_l and f ? 1_l respectively. Then the numbersp+, p andp0 (the numbers of positive, negative and zero eigenvalues, counted with multiplicity, of the matrixE(f)1D ) are given by the following table:

f ? f f = f f? f ? 0 f = 0 0 ? f ? f+ f = f+ f+? f p+ _Q₊_{3 Q}₊₃ _Q₊₂ _{Q 2} _Q₊₁ _Q₊₁ _Q₊ p _Q _Q₁ _Q₁ _{Q 2} _Q₂ _Q_{3 Q}₃ p0 ₃ ₄ ₃ ₄ ₃ ₄ ₃ . (3.40) Remark 8 The degenerate values off are the same before the reduction, just that f0 = 0 is of order g. In fact, if we do not make the reduction (but still have the same symmetric set of velocities), then we obtain g 1 extra collision invariants, !2+l= P1@2· (l1> ===> l2Q), l = 2> ===> g, which all are orthogonal, with respect to the

scalar producth·> ·i_E(f), to!₁,!₂and!₃.

The modied table 3.39 for the values ofn+,nando reads:

f ? f f = f f? f ? 0 f = 0 0 ? f ? f+ f = f+ f+? f

n+ _{g + 2} _{g + 1} _{g + 1} ₁ ₁ ₀ ₀

n ₀ ₀ ₁ ₁ _{g + 1} _{g + 1} _{g + 2}

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and the modied table 3.40 for the values ofp+,pandp0 reads:

f ? f f = f f? f ? 0 f = 0 p+ _Q₊_{(g + 2) Q}₊_{(g + 2) Q}₊_{(g + 1) Q (g + 1)} p _Q _Q_g _Q_g _{Q (g + 1)} p0 _{g + 2} _{g + 3} _{g + 2} _{g + 3} 0? f ? f+ f = f+ f+? f Q+ g Q+ g Q+ Q (g + 1) Q (g + 2) Q (g + 2) g + 2 g + 3 g + 2 _.

Remark 9 For the continuous Boltzmann equation (withg = 3) the numbers "₁> ===> "₅ are given by

"₁=, "₂=W , "₃= 3W , "₄= 5W2and"₅= 15W2,

(where and W denote the density and the temperature respectively), if we have made an expansion around the absolute Maxwellian P =

(2W )3@2h||

2_@2W

on the form (2=21). Therefore, for the Boltzmann equation (with g = 3) the values (3=38) are (cf. Ref.[27])

f0= 0 andf_±=± r

5W 3 .

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Chapter 4 Discrete Velocity Models for

Mixtures

4.1 Denitions and properties

The general discrete velocity model (DVM) for a binary mixture of the gasesD and E (we hope that the fact that D and E denotes gases here, and not matrices as in the rest of the thesis, do not lead to any confusion) reads

CiD l Cw +Dl · xilD=TDDl (iD> iD) +TEDl (iE> iD),l = 1> ===> qD, CiE m Cw +Em · ximE=TDEm (iD> iE) +TEEm (iE> iE),m = 1> ===> qE, (4.1)

whereY={₁> ===> _q} Rgis a nite set of velocities,i_l=i_l(x> w) = i(x> w> _l) forl = 1> ===> q, andi=i(x> w> ) represents the microscopic density of particles (of the gas) with velocity at time w R₊and position x Rg. Here and below, > > {D> E}. We denote by p_{the mass of a molecule of the gas}_.

Remark 10 For a function j =j() (possibly depending on more variables than ), we identify j_{with its restrictions to the points}_Y_{, i.e.}

j_{= (j}

1> ===> jq)> with j_l =j(_l) forl = 1> ===> q. Theni= (i₁> ===> i_q) in Eq.(4=1).

The collision operatorsT_l (i> i) in (4=1) are given by T_l (i> i) = q X n=1 q X m>o=1 no_lm(> ) ³ i nio ilim ´ forl = 1> ===> q, 31

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where it is assumed that the collision coecients no_lm(> ), with 1 l> n q and 1 m> o q, satisfy the relations

no_lm(> ) = no_ml(> ) and no_lm(> ) = lm_no(> ) = on_ml(> ) 0, (4.2) with equality unless the conservation laws

p

l +pm =pn+poandp|l|2+p¯¯¯m¯¯¯ 2

=p|_n|2+p¯¯¯_o¯¯¯2 are satised. We denote

i = Ã iD iE ! = Ã iD₍₎ iE₍₎ ! ,j = Ã jD jE ! = Ã jD₍₎ jE₍₎ ! , andT(i> i) = Ã TDD_(iD_{> i}D_{) +}_TED_(iE_{> i}D₎ TDE_(iD_{> i}E_{) +}_TEE_(iE_{> i}E₎ ! =

Then the collision operatorT(i> i) can be obtained from the bilinear expressions Tl(i> j) = 1₂ qD X m>n>o=1 no_lm(D> D)¡i_nDj_oD+jD_ni_oD i_lDjD_m jD_li_mD¢+ +1 2 qD X n=1 qE X m>o=1 no_lm(E> D)¡i_nDjE_o +j_nDi_oE i_lDjE_m j_lDi_mE¢,l = 1> ===> qD, and T_qD_+l(i> j) =1 2 qE X n=1 qD X m>o=1 no_lm(D> E)¡i_nEjD_o +jE_ni_oD i_lEjD_m jE_li_mD¢+ +1 2 qE X m>n>o=1 no_lm(E> E)¡i_nEjE_o +j_nEi_oE i_lEjE_m j_lEi_mE¢,l = 1> ===> qE,

DenotingT(i> j) = (T₁(i> j) > ===> T_qD_+qE(i> j)), we see that, for arbitrary i, j and k = Ã kD kE ! = Ã kD₍₎ kE₍₎ ! , T (i> j) = T (j> i) , and by the relations (4=2),

hk> T (i> j)i = = 1 8 qD X l>m>n>o=1 no_lm(D> D)¡kD_l +k_mD kD_n kD_o¢ ¡i_nDjD_o +j_nDi_oD i_lDjD_m j_lDi_mD¢+ +1 4 qD X l>n=1 qE X m>o=1 no_lm(E> D)¡kD_l +kE_m kD_n kE_o ¢ ¡i_nDjE_o +jD_ni_oE i_lDj_mE jD_li_mE¢+ +1 8 qE X l>m>n>o=1 no_lm(E> E)¡kE_l +k_mE kE_n kE_o¢ ¡i_nEjE_o +jE_ni_oE i_lEjE_m jE_li_mE¢. (4.3)

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4. Discrete Velocity Models for Mixtures 33 A vector! = Ã !D !E !

is a collision invariant if, and only if, !

l +!m =!n+!o, (4.4)

for all indices 1 l> n q, 1 m> o q and> {D> E}, such that no_lm(> ) 6= 0. By the relation (4=3) h!> T (i> i)i = 1 4 qD X l>m>n>o=1 no_lm(D> D)¡!_lD+!D_m !D_n !D_o¢ ¡i_nDi_oD i_lDi_mD¢+ +1 2 qD X l>n=1 qE X m>o=1 no_lm(E> D)¡!D_l +!E_m !D_n !E_o ¢ ¡i_nDi_oE i_lDi_mE¢+ +1 4 qE X l>m>n>o=1 no_lm(E> E)¡!_lE+!E_m !E_n !E_o ¢ ¡i_nEi_oE i_lEi_mE¢. (4.5) which is zero, independently of our choice of non-negative vectori (i_l 0 for all 1 l q), if, and only if,! is a collision invariant.

We consider below (even if this restriction is not necessary in our general reasoning) only DVMs, such that any collision invariant is of the form

! = Ã !D !E ! , with!=!() = d+pb · + fp||2, (4.6) for some constantdD> dE> f R and b Rg. Examples of such DVMs can be found in Refs.[9, 25]. In this case the equation

h!> T (i> i)i = 0 has the general solution (4=6).

A binary Maxwellian distribution (or just a bi-Maxwellian) is a function P = Ã PD PE ! , such that T(P> P) = 0 and P l 0 for all 1 l q.

All bi-Maxwellians are of the form P = h!_{, i.e.} _{P =} Ã PD PE ! , withP=h!, (4.7) where! = Ã !D !E !

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! = log i in Eq.(4=5) and obtain by the relation (2=10) that hlog i> T (i> i)i =1

4 qD X l>m>n>o=1 _lmno(D> D)¡i_nDi_oD i_lDi_mD¢logi D l imD iD nioD + +1 2 qD X l>n=1 qE X m>o=1 no_lm(E> D)¡i_nDi_oE i_lDi_mE¢logi D limE iD nioE + +1 4 qE X l>m>n>o=1 _lmno(E> E)¡i_nEi_oE i_lEi_mE¢logi E l imE iE nioE 0, (4.8)

with equality if, and only if,

i

nio=ilim (4.9)

for all indices 1 l> n q, 1 m> o qand> {D> E}, such that no_lm(> ) 6= 0. Hence, there is equality in Eq.(4=8) if, and only if, i is a bi-Maxwellian. Taking the logarithm of Eqs.(4=9), i is a bi-Maxwellian if, and only if, log i is a collision invariant.

4.2 Linearized collision operator

For a bi-MaxwellianP = Ã PD PE ! (4=7), by denoting i = P +Pk, (4.10)

in Eqs.(4=1), we obtain the new system Ck

Cw+ · xk = Ok + V (k) ,

where · _xk = (D₁ · _xk₁D> ===> D_qD· xkD_qD> E1 · xkE1> ===> EqE· xkE_qE). Furthermore, O is the linearized collision operator (qDE_{× q}DE_{matrix, with}_qDE₌_qD₊_qE_{) given}

by Ok = 2 PT(P> Pk), (4.11) andV is given by V (k) = 1 PT( Pk>Pk). (4.12)

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4. Discrete Velocity Models for Mixtures 35

By Eqs.(4=3) and the relations P_lP_m=P_nP_o6= 0, we obtain the equality hj> Oki = 2 ¿ j P> T(P> Pk) À = =1 4 qD X l>m>n>o=1 no_lm(D> D) μq PD o jDn + q PD njoD q PD m jlD q PD l jmD ¶ × ×μqPD o kDn + q PD nkDo q PD m kDl q PD l kDm ¶ + +1 2 qD X l>n=1 qE X m>o=1 no_lm(E> D) μq PE o jnD+ q PD njoE q PE m jDl q PD ljEm ¶ × ×μqPE o kDn + q PD nkEo q PE m kDl q PD l kEm ¶ + +1 4 qE X l>m>n>o=1 no_lm(E> E) μq PE o jnE+ q PE njEo q PE m jEl q PE l jEm ¶ × ×μqPE o kEn + q PE nkEo q PE m kEl q PE l kEm ¶ . Hence, the matrixO is symmetric, i.e.

hj> Oki = hOj> ki for allj and k, and semi-positive, i.e.

hk> Oki 0 for allk. Also hk> Oki = 0 if, and only if,

p P nko + q P_ok n = p P lkm + q P_mk l (4.13)

for all indices 1 l> n q, 1 m> o q, and> {D> E}, satisfying no_lm(> ) 6= 0. We let k =P! in Eq.(4=13), and obtain Eq.(4=4), by the relations P_lP_m = P

nPo6= 0. Hence, since O is semi-positive,

Ok = 0 if, and only, if k =P!, where! is a collision invariant. In consequence,

D

V (k) >P!E=hT (i> i) > !i + D

k> OP!E= 0 for all collision invariants!.

The system for mixtures corresponding to Eqs.(2=28) reads Ggk

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whereG = Ã

GD ₀

0 GE

!

, withG = diag(g₁> ===> g_q), and the operators O and V are given by Eqs.(4=11)-(4=12). In the spatially homogeneous case G = L and w actually stands for time, in the planar stationary caseg_l =>1_l and w = {1, where

l = (>1l > ===> >gl ) andx = ({1> ===> {g), and in the shock wave casegl =>1l f1l and

w = {1_f1_{w, where c =}¡_f1_{> 0> ===> 0}¢_{is the velocity of the shock wave.}

4.3 Axially symmetric DVMs for mixtures

We assume that (i) we have two symmetric sets (2=19) of velocities; (ii) that the reduction in Remark 4 is done; and (iii) we have made an expansion (cf. Eq.(4=10)) around an absolute bi-MaxwellianP (with b = 0 in Eqs.(4=6)-(4=7)). Let

G = G(f) = Ã GD₍_f) ₀ 0 GE(f) ! > with G₍_{f) = diag(}>1 1 f> ===> >1Q f> >1₁ f> ===> >1_Q f> ), wheref @ n ±D>1₁ > ===> ±D>1_QD> ±E>11 > ===> ±E>1QE o

. The null-space ofO is given by Q(O) = span (!0> !1> !2> !3) , where !0=P1@2· (1> ===> 1_{| {z }} 2QD > 0> ===> 0_{| {z }} 2QE ) !₁=P1@2· (0> ===> 0_{| {z }} 2QD > 1> ===> 1_{| {z }} 2QE ) !2=P1@2· (!D2> !E2)> with !2 = (>11 > ===> >1Q> >1₁ > ===> >1_Q) !3=P1@2· (!D3> !E3)> with !3 = (|1|2> ===> |Q|2> |₁|2> ===> |_Q|2) . Then, N = f"D 1 0 "D2 f"D3 0 f"E₁ "E₂ f"E₃ "D 2 "E2 f"2 "4 f"D 3 f"E3 "4 f"5 > whereN = (!_l+1> !_m+1®_G(f)),"D₁ =h!₀> !₀i, "₂D=h!₀> !₂i_G(0),"D₃ =h!₀> !₃i, "E₁ = h!₁> !₁i, "E 2 =h!1> !2iG(0),"E3 =h!1> !3i, "2="D2 +"E2 =h!2> !2i, "4=h!2> !3iG(0) and"₅=h!₃> !₃i. Hence, det(N) = f4_("D 1"E1"2"5 "D1"2("E3)2 "E1"2("D3)2)+ +f2(2"₄("₁D"E₂"E₃ +"E₁"D₂"D₃) + ("D₂"E₃ "E₂"D₃)2 ("D₁("E₂)2+"E₁("D₂)2)"₅ "D₁"E₁"2₄),

On Half-Space and Shock-Wave Problems for Discrete Velocity Models of the Boltzmann Equation

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Contents

Chapter 1

Introduction

Part I

Discrete Velocity Models and

Dynamical Systems

Chapter 2

Discrete Velocity Models

2.1

Denitions and properties

2.2

DVMs as dynamical systems

2.3

Linearized collision operator

Chapter 3

Topological Classication

3.1

General linear system

3.2

Dimensions of the stable, unstable and center

mani-folds

3.3

Jordan normal form

3.4

Axially symmetric DVMs

Chapter 4

Discrete Velocity Models for

Mixtures

4.1

Denitions and properties

4.2

Linearized collision operator

4.3

Axially symmetric DVMs for mixtures

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