Boltzmann Equation and Knudsen Group - Boundary Shape and Boundary Conditions

A Boltzmann type equation is considered where both, the physical and the velocity space, are bounded. It is assumed that the boundary conditions consist of a reflective as well as a diffusive component. Existence of positive bounds from below and above has been proved. It has been demonstrated that under conditions on the shape of the boundary, the underlying Knudsen type transport semigroup can be embedded in a not necessarily positive group for time t ∈ ℝ. As a consequence, the Boltzmann type equation considered has a unique global solution for time t ∈ [τ 0, ∞) for some τ 0 < 0. The time τ 0 does not depend on the initial value at time 0. It can be arbitrarily small depending on the intensity of the collisions.


Introduction 1.Contents
The paper deals with a class of Knudsen type (semi)groups which refer to collisionless gases and related Boltzmann type equations which refer to dilute gases.The dynamics takes place in a container of dimension d = 2 or d = 3 while the modulus of the velocity is positively bounded from above and below.In the paper it is demonstrated that certain shapes of the container allow the gas densities to be traced back in time, to some extent.In particular certain classes of convex polygons if d = 2 or convex polyhedrons if d = 3 are eligible for those containers.
In the situation considered, the Knudsen type semigroup can be embedded in a group on time t ∈ R. However there may exist a negative time such that, for even smaller times, the Knudsen type group may no longer consist of positive densities, even if the dynamics at all times t ≥ 0 is observed with strictly positive densities.Here we refer to Theorem 6 and Corollary 7 below.
If the transport in a Boltzmann type equation is associated with such a Knudsen type (semi)group then, under suitable conditions on the collisions there exists a time τ 0 < 0 such that, for a given strictly positive density p 0 at time t = 0, there is a solution p t (p 0 ) to the Boltzmann type equation for all t ≥ τ 0 .As soon as the density p • (p 0 ) is strictly positive, say at some (negative) time s it remains strictly positive for all times in [s, ∞).See Theorem 8.
Besides results on this phenomenon, Theorem 8 is also a global existence and uniqueness theorem for Boltzmann type equations which, in particular, refers to positive boundedness from below and above.All Lemmas, Propositions, Corollaries, and Theorems provided in this proceedings article are comprehensively proved in [6].The present paper may be used to get access to the subject without coping with the vast amount of technicalities in [6], but still be made familiar with the basic mathematical approach to the topic, see Sections 4 and 5 below.It should be mentioned that in the case of stationary solutions to Boltzmann type equations, uniform positive lower and upper bounds on p • (p 0 ) can be established under much more relaxed assumptions, with much easier proofs, see also [5].

Historical roots and related research
Without claiming completeness, a few related works should be referred to.
Boundary conditions Boundary conditions which contain a specular reflective component as well as a diffusive component have already been emphasized by C. Maxwell, cf.Appendix to [7].Such boundary conditions are also called Maxwell boundary conditions.They are physically relevant and have therefore attracted physicists and mathematicians.For the current state of the art, see e.g.[13], Theorem 2.3.4 and its proof as well as [8].Modified Maxwell conditions have also been studied in detail.Here [4] should be mentioned.
Bounds on solutions Lower bounds on solutions to various classes of Boltzmann equations are of interest for decades.For the spatially homogeneous case, there are a number of results.For example, a Maxwellian lower bound not depending on time has been established in [10].
Eternal solutions Eternal solutions to Boltzmann type equations are non-stationary, nonnegative solutions on t ∈ R. For example, the infinite energy solution introduced in [1] is eternal.It has been analyzed numerically in [12], Sections 4.4 and B.7.
Stationary solutions Stationary solutions are of course eternal.In the paper [2] a particular Boltzmann equation with diffusive boundary conditions but unspecified redistribution densities has been examined.One of the goals of [2] is to establish a unique stationary solution.It has been demonstrated that, in case of a velocity whose modulus is positively bounded from below and suitable initial configurations, the unique stationary solution can be represented as a certain limit of the distribution of an N -particle system.
Boltzmann's H-theorem This is a major theorem in order to prove irreversibility of solutions to Boltzmann type equations.However, several forms of boundary conditions do not permit to adapt known proofs of the H-theorem.In particular, the proof of Theorem 1 in Section 1.1.2 of [11] cannot be adjusted to cope with general diffusive boundary conditions, since those boundary conditions may not conserve energy.
Entropy Denoting by p(r, v, t) the solution to the Boltzmann equation (r is the position, v the velocity, t the time), the modified ) dv dr is used in order to describe a modified entropy term in case of diffusive boundary conditions.Here β w denotes the inverse temperature.As demonstrated in [3], (4.1), (4.2) in Chapter 9, this H decreases in time in case of certain isothermal diffusive boundary conditions.Furthermore, relative entropy has been introduced and analyzed in [13].

Preliminaries Let
where the initial value is S(0 Here n(r) denotes the outer normal at r ∈ ∂ (1) Ω, where ∂ (1) is obtained from ∂Ω by taking away all vertices and all edges if d = 3.The symbol "•" stands for the scalar product in R d .

The boundary conditions
There is a specular reflective component and a diffusive component ω ∈ (0, 1).Here, v denotes the velocity and Ω × V models its specular reflection at r ∈ ∂ (1) Ω.
• Furthermore, • The function M is defined on on {(r, v) : It quantifies the diffusive part of the boundary conditions.It is positive, continuous, and uniformly bounded from below and above.• The relation between the impact of the reflective and diffusive part of the boundary condition is displayed by ω ∈ (0, 1).
Remark The particle densities from r ∈ ∂Ω directed to the outside of Ω at time t, i.e. v •n(r) ≥ 0, contribute to the integral J(r, t)(S(•)p 0 ) = v •n(r) S(t)p 0 (r, v) dv which ranges over all v with v • n(r) ≥ 0. The term J refers to the diffusive part.Moreover, S(t)p 0 (r, v) also goes into the reflective part.
For the particle densities from r ∈ ∂Ω directed to the inside of Ω at time t, i.e. v • n(r) ≤ 0, there are two sources.On the the one hand, there is specular reflection with probability ω.On the other hand, the particle densities arise from J(r, t)(S(•)p 0 ) with probability 1−ω.The latter are distributed over {v : v • n(r) ≤ 0} with respect to the redistribution density M (r, v) relative to the measure |v • n(r)| dv.

The boundary shape
Let us assume ∂Ω = n ∂ i=1 Γ i with all Γ i being manifolds of dimension (d − 1).The Γ i are supposed to be open sets in the topology of ∂Ω, i ∈ {1, . . ., n ∂ }, which are smooth up to the boundary.For an intersection Γ i ∩ Γ j , i = j, suppose that it is either the empty set or, in case of d = 2, just a single point in R 2 or, in case of d = 3, a closed smooth curve in R 3 .
Consider neighboring Γ i and Γ j and let x ∈ Γ i ∩ Γ j .Denote by R i and R j the rays from x tangential to Γ i as well as Γ j being orthogonal to Γ i ∩ Γ j if d = 3, and let ξ(i, j; x) be the angle between R i and R j .Suppose that there exists ξ ∈ (0, π) independent of such i as well as j and x such that ξ < ξ(i, j; x) < 2π − ξ .
Let us assume that all conditions of Section 2 are satisfied throughout the rest of the paper.

Some Properties of the Knudsen Type Semigroup
Lemma 1 (a) The Knudsen type semigroup S(t), t ≥ 0, is strongly continuous in L 1 (Ω × V ).

Towards the Knudsen Type Group
Recall the following classical Theorem, see e. g. [9], Theorem I.6.5.
Theorem Let T (t) be a strongly continuous semigroup of bounded operators acting in some Banach space.If 0 belongs to the resolvent set ρ (T (t 0 )) for some t 0 > 0 then 0 ∈ ρ(T (t)) for all t > 0 and T (t) can be embedded in a strongly continuous group.

Procedure:
Step (1) Introduce the space L 1 u of all equivalence classes of measurable functions on {(r, w) ∈ ∂ (1) Let U be a map on L 1 u defined by T Ω (r,w) 0 e −βµ g(r − βw, w) dβ .
Let (A, D(A)) be the infinitesimal generator of the Knudsen type semigroup S(t), t ≥ 0.
Lemma 2 Let µ ∈ C and g ∈ L 1 (Ω × V ).There exists a unique f ∈ D(A) with µf − Af = g if and only if there exists a unique f ∈ L 1 u satisfying Step (2) For (r, w) ∈ ∂ (1) Ω × V such that w • n(r) ≥ 0 define as well as Note the difference between the the infinitesimal generator A in L 1 (Ω × V ) and the operator A(µ) in L 1 u .According to the definitions of a(µ) and U we can decompose a(µ)f (r, w) = A(µ)f (r, w) + B(µ)f (r, w) .

Now we can take advantage of the relation
to analyze and solve the equation Step (3) Here it is the aim to demonstrate bijectivity of id − A(µ).For this, we recall the ergodic theory of mathematical billiards in dimensions d = 2, 3.In particular, for the times l k ≡ l k (r, v) between the kth and the (k + 1)st specular reflection there exists the limit lim As explained in [6] Then the operator id − A(µ) is bijective in L 1 u .
Step (4) Here we are concerned with bijectivity of id − (id (a) Assume ω ∈ (0, 1) and µ ∈ M m .Then, for all f ∈ L 1 u , the sum In this way, a bounded linear operator X µ in L 1 u is defined.
Then there exists −∞ < m 1 ≤ m in a way that for Here the infinite sum converges in L 1 u .In addition, the operator id − (id − A(µ)) −1 B(µ) : u is bijective.

Knudsen Group
Step (6) Recall the objective formulated in the beginning of Section 4. Assume the contrary, i.e. 0 does not belong to ρ(S(t), for any t > 0.
It follows from Theorem I.6.5 of [9], see Section 4 above, and Corollary 5(c) that λ = 0 is for any t > 0 an isolated point in the spectrum of S(t).Leading this to a contradiction we obtain the following.
denote the velocity space and Ω ⊂ R d the physical space.For a class of physical spaces which contains certain convex polygons in dimension d = 2 and certain convex polyhedrons if d = 3, let us consider the Knudsen type semigroup S(t), t ≥ 0. It is given by

Fig. 1 :
Fig.1: Assume that the horizontal edges of the rectangle Ω are longer than the vertical edges which are of length s > 0.Here the crucial condition is satisfied for k 0 = 2 and σ min = s.Fig.2:Let Ω be a disk.For a fixed starting point (y, v) ∈ ∂(1) Ω × V where v • n(y) ≤ 0, the distances |y (j) − y (j+1) | are preserved with respect to j.However, depending on v • n(y), these distances can be arbitrarily small.Therefore the crucial condition fails.