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2018
On existence of solutions for
Boltzmann Continuous Slowing Down transport equation
Tervo J
Elsevier BV
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http://dx.doi.org/10.1016/j.jmaa.2017.11.052
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On existence of solutions for Boltzmann Continuous Slowing Down transport equation
J. Tervo, P. Kokkonen, M. Frank, M. Herty
PII: S0022-247X(17)31053-3
DOI: https://doi.org/10.1016/j.jmaa.2017.11.052 Reference: YJMAA 21851
To appear in: Journal of Mathematical Analysis and Applications Received date: 9 March 2017
Please cite this article in press as: J. Tervo et al., On existence of solutions for Boltzmann Continuous Slowing Down transport equation,J. Math. Anal. Appl.(2018), https://doi.org/10.1016/j.jmaa.2017.11.052
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On existence of solutions for Boltzmann Continuous Slowing Down transport equation
J. Tervo∗ P. Kokkonen† M. Frank‡ M. Herty§ November 28, 2017
Abstract
The paper considers a linear Boltzmann transport equation (BTE), and its Con- tinuous Slowing Down Approximation (CSDA). These equations are used to model the transport of particles e.g. in dose calculation of radiation therapy. We prove the existence and uniqueness of weak solutions, under sufficient criteria and in appro- priateL2-based spaces, of a single (particle) CSDA-equation by using the theory of m-dissipative operators. Relevant a priori estimates are shown as well. In addition, we prove the corresponding results and estimates for a system of coupled transport equations. We also outline a related inverse problem.
Keywords: Linear Boltzmann transport equation, Continuous Slowing Down Approximation, Dose calculation, Inverse radiation therapy planning, Optimal control, Dissipative operators
1 Introduction
TheBoltzmann transport equation (BTE) models changes of the number density of par- ticles in phase space (position, velocity direction, energy). In this paper the species of particles include photons, electrons and positrons. The presented analysis of transport equations is mainly intended for dose calculation in radiation treatment planning but it applies for other similar kind of transport problems. Several journal issues have been devoted to the topic of modelling and numerical analysis of cancer therapy.
Dose calculation is of crucial importance in radiation therapy. Relevant dose calcu- lation models require (approximate) solution of a coupled system of (linear) transport equations for fluencies (number densities in the phase space) for all considered particles.
This is a difficult problem, at least from computational point of view, due to the different particle species and their dependence on a high–dimensional phase space. For that reason traditional dose calculation algorithms have applied some closed-form formulas which have their origins in analytical solutions (of BTEs), or Monte–Carlo derived solutions, of sim- plified problems. These methods, however, often contain empirically derived corrections to take more accurately into account the underlying particle physics. Commonly used models are based on the so-calledpencil beams, orpoint kernels, seeasadzadeh,borgers,mayles07
[1, 3, 20] andtillikainen08,ulmer
[30, 32]
for more details. A notable exception to these approximate (deterministic) methods is the
Acuros codevassiliev[33], which is based on a discretization of the BTE. We also refer to borgers99,FrankHenselKlar20
[4, 9, 16]
for more details on physical modeling and derived simplifications. A full review on the underlying equations and properties can be found e.g. in frank10,HenselIza-TeranSiedow2006aa
[11, 16].
∗University of Eastern Finland, Department of Applied Physics, P.O.Box 1627, FI-70211 Kuopio, Fin- land
†Varian Medical Systems Finland Oy, Paciuksenkatu 21, 00270 Helsinki, Finland
‡RWTH Aachen University, MATHCCES, Schinkelstrasse 2, 52062 Aachen, Germany
§RWTH Aachen University, IGPM, Templergraben 5, 52062 Aachen, Germany
The original exact BTE is a partial integro-differential operator. The related integral operator is called a collision operator. The kernels, so called differential cross-sections, of collision operators may have singularities or evenhyper-singularitiesand in these cases the integral appearing in the collision term must be interpreted as the Hadamard finite part integral. Due to the hyper-singularities the transport of new primary electrons (and positrons) isforward-peaked.
The exact BTE has various approximations which can be analysed by known methods and which enable to calculate numerical solutions. One of them is the so calledContinuous Slowing Down Approximation (CSDA). CSDA-approximation is based on the fact that hyper-singular part and non-singular part are separated in the analysis of the collision operator. The hyper-singular part produces a pseudo-differential-like term and the non- singular part produces a so called restricted collision operator, which is a partial Schur integral operator. Roughly speaking, the approximation of the pseudo-differential-like term by a partial differential operator leads to the CSDA. The approach is exposed in[28].tervo16-up We notice that traditionally, to overcome the computational complexity, one has applied to the evolution of electrons and positrons the CSDA-approximation. The reason for the use of CSDA-type equations forcharged particles is, however deeper: The underlying kinematic laws produce partial differential (more accurately pseudo-differential-like) terms with respect to energy (and angle). This can transparently be seen from the formula (89) given in[28], section 2.3.tervo16-up
This paper considers existence and uniqueness of solutions for a single (particle) CSDA–
equation. We note that this paper is a focused version on our preprint[28], and partiallytervo16-up relies on another preprint[29]. The CSDA–equation is given bytervo17-up
−∂(S0ψ)
∂E +ω· ∇xψ+ Σψ−Kψ=f inG×S×I (1) i-1 ψ|Γ− =g, ψ(·,·, Em) = 0 inG×S, (2) i-1-a inL2(G×S×I)-based spaces. HereG⊂R3is the spatial domain,Sis the unit sphere and
I is the energy interval. On the right in (i-11), the functionf represents an (internal) source and g in (2) is an (inflow) boundary source. The solutioni-1-a ψ of the problem describes the radiation flux of electrons (and positrons). It is a function of position (x), velocity direction (ω) and energy (E). Roughly speaking, the flux ψ = ψ(x, ω, E) is the flux of energy through a surface located at x and normal to the direction ω. The set Γ−, so called inflow boundary (see section fs2.1), consists of all elements (x, ω, E) ∈ ∂G×S×I such that ν(x)·ω <0, where ν(x) is the outward normal vector at point x. Besides the inflow boundary condition, one needs to impose on the solutionψan energy-boundary (an initial) condition. We make the reasonable assumption that at some (high enough, but finite) cut-off energy Em >0 the flux ψ vanishes, i.e. we take the initial condition to be ψ(x, ω, Em) = 0. The function Σ = Σr = Σr(x, ω, E) is the restricted total cross-section, S0=S0(x, E) is therestricted stopping powers. K=Kr is therestricted collision operator and it is assumed to be of the form
(Kψ)(x, ω, E) =
S×I
σ(x, ω, ω, E, E)ψ(x, ω, E)dωdE, (3) i-col where σ = σr = σr(x, ω, ω, E, E) is the restricted differential cross-section, which does
not include soft inelastic interactions (boman[2]). The operator is linear, because due to the high electron density in tissue, interactions of charged beam particles among themselves can be neglected in comparison to the interaction with the medium. We assume through- out that the restricted collision operator K is bounded, and Σ−K is dissipative. We, however, notify that the actual restricted collision operator is a sum of three different types of operators like (3) as mentioned below in sectioni-col coll2.2. The present analysis can
straightforwardly be extended for this sum of restricted collision operators. The obtained existence results are exposed in section3. They are based on thesingle-eq m-dissipativity analysis and partially on the theory of evolution operators.
In section cosyst4 analogous existence results are given for the coupled system
ω· ∇xψ1+ Σ1ψ1−K1ψ=f1 (4) intro5-1
−∂(Sjψj)
∂E +ω· ∇xψj+ Σjψj−Kjψ=fj, j= 2,3, (5) intro5-2 ψj|Γ−=gj, j = 1,2,3; ψj(·,·, Em) = 0 inG×S, j= 2,3, (6) intro5-3 where forj = 2,3
(Kjψ)(x, ω, E) = (Kj,rψ)(x, ω, E) =
S×Iσ1j(x, ω, ω, E, E)ψ1(x, ω, E)dωdE +
3 k=2
S×Iσkj,r(x, ω, ω, E, E)ψk(x, ω, E)dωdE. The differential cross sections related to the evolution of photons (such as Compton scat- tering) do not contain hyper-singularities with respect to energy. Hence the transport of photons can be modelled by (intro5-14). The system is coupled through the partial integral operatorsKj. In practical dose calculation the evolution of particles must be modelled by the coupled systemof the above form.
In radiation therapy thedose D(x) = (Dψ)(x) is calculated from the solution of the equations (4-intro5-16) byintro5-3
D(x) = 3 j=1
S×Iςj(x, E)ψj(x, ω, E)dωdE,
where ςj(x, E) are stopping powers, which in general can be different from the restricted stopping powers Sj,r. The dose calculation is a forward problem. The determination of the external particle flux g = (g1, g2, g3) and/or the distribution of internal source f = (f1, f2, f3) for the purpose of obtaining a certain dose profile is calledinverse radiation
treatment planning problem(IRTP) which is aninverse problem, see e.g. schepard,tervo17-up,webb,frank08,fran
[25, 29, 34, 10, 11].
In section5 we outline a relevant inverse radiation treatment planning problem where theirtpre main results obtained in this paper become essential.
We notice that the single CSDA–equation is (symmetric) hyperbolic in nature and therefore the hyperbolic theory (e.g. rauch85[24] (section 4), nishitani98
[21] and their references) can be applied in the case where K = 0. For G= R3 the existence and regularity of solutions is, however, subtle because of the inflow boundary condition (the considered boundary problems are so–called multiple characteristic, and they have variable multiplicity). We also remark that the equation for photons, i.e. (4) is not hyperbolic.intro5-1
2 Preliminaries
pre
2.1 Notations, Assumptions and Introduction of Relevant Function Spaces fs
We assume that G is an open bounded convex set in R3 such that G is a C1-manifold with boundary (as a submanifold of R3; cf. [18]). In particular, it follows from thislee definition that Glies on one side of its boundary. The unit outward (with respect to G) pointing normal on∂Gis denoted byν. The Lebesgue measure onR3isdxand the surface measure (induced by the Lebesgue measure) on ∂G is written asdσ. We let S =S2 be the unit sphere in R3 equipped with the usual rotationally invariant surface measure dω.
Furthermore, let I= [0, Em] where 0< Em<∞. We could replaceI byI = [E0, Em] or I = [E0,∞[ where E0 ≥0 but we omit this generalization here. We shall denote by I◦ the interior ofI. The intervalI is equipped with the Lebesgue measuredE. All functions considered in this paper are real-valued, and all linear (Hilbert, Banach)spaces are real.
For (x, ω)∈G×S theescape time (in the direction ω)t(x, ω) =t−(x, ω) is defined by t(x, ω) = inf{s >0| x−sω∈G} (7)
= sup{T >0|x−sω∈Gfor all 0< s < T},
for (x, ω)∈G×S. The escape time function t(·,·) is known to be lower semi-continuous in general, and continuous if Gis convex, see e.g. tervo17-up[29]. Furthermore, we define
Γ := (∂G)×S, Γ := Γ×I, and
Γ0:={(y, ω)∈Γ |ω·ν(y) = 0}, Γ0= Γ0×I, Γ−:={(y, ω)∈Γ |ω·ν(y)<0}, Γ−:= Γ−×I, Γ+:={(y, ω)∈Γ |ω·ν(y)>0}, Γ+:= Γ+×I.
It follows that μΓ(Γ0) = 0 and Γ = Γ0∪Γ−∪Γ+.
Defineescape-time mappingsτ±(y, ω)from boundary to boundary in the directionωas follows
τ−(y, ω) := inf{s >0| y+sω∈G}, (y, ω)∈Γ−, τ+(y, ω) := inf{s >0| y−sω∈G}, (y, ω)∈Γ+.
Note that for (y, ω)∈Γ−, one has (y+, ω)∈Γ+, wherey+:=y+τ−(y, ω)ω, and moreover τ−(y, ω) =τ+(y+, ω). From[28], Proposition 2.6 it follows that we can (almost everywhere)tervo16-up uniquely set t(y, ω) = 0 for (y, ω)∈Γ−.
In the sequel we denote for k∈N0,
Ck(G×S×I) :={ψ∈Ck(G×S×I◦)|ψ=f|G×S×I◦, f ∈C0k(Rn×S×R)}, where for a Ck-manifold M without boundary, the set C0k(M) stands for the set of all Ck-functions onM with compact support. Define the (Sobolev) spaceW2(G×S×I) by
W2(G×S×I) ={ψ∈L2(G×S×I)| ω· ∇xψ∈L2(G×S×I)} and its subspaceW12(G×S×I) by
W12(G×S×I) =W2(G×S×I)∩H1(I, L2(G×S)).
Here ∇x is taken with respect to x-variable only, and ω· ∇xψ and ∂E∂ψ are understood in the distributional sense. Hk(I, X) is theX-valued L2-Sobolev space of order k. The spacesW2(G×S×I),W12(G×S×I) are equipped with the inner products, respectively,
ψ, vW2(G×S×I)=ψ, vL2(G×S×I)+ω· ∇xψ, ω· ∇xvL2(G×S×I)
and ψ, vW2
1(G×S×I)=ψ, vL2(G×S×I)+ω· ∇xψ, ω· ∇xvL2(G×S×I)+ ∂ψ
∂E, ∂v
∂E
L2(G×S×I)
. Both of these spaces are Hilbert spaces.
We have (cf. e.g. [14]. The proof can also be shown by the similar considerations asfriedrichs in[12], pp. 11-19)friedman
denseth Theorem 2.1 The spaceC1(G×S×I) is a dense subspace of bothW2(G×S×I) and W12(G×S×I).
On Γ− and Γ+ we define the spaces of L2-functions with respect to the measure
|ω·ν| dσdωdE (restricted onto Γ− and Γ+, respectively) which is denoted by T2(Γ±) that is,
T2(Γ±) :=L2(Γ±,|ω·ν| dσdωdE).
The spacesT2(Γ±) are Hilbert space with the respective inner products, h1, h2T2(Γ±)=
Γ±
h1(y, ω, E)h2(y, ω, E)|ω·ν|dσdωdE.
Similarly on Γ, we define Hilbert space
T2(Γ) :=L2(Γ,|ω·ν|dσdωdE), h1, h2T2(Γ)=
Γ
h1(y, ω, E)h2(y, ω, E)|ω·ν| dσdωdE.
Additionally we define
Tτ2±(Γ±) :=L2(Γ±, τ±(·,·)|ω·ν|dσdωdE) which are Hilbert spaces, equipped with the inner products
h1, h2Tτ2±(Γ±)=
Γ±
h1(y, ω, E)h2(y, ω, E)τ±(y, ω)|ω·ν| dσdωdE.
We formulate the following inflow trace theorem (see e.g.dautraylionsv6
[5, p. 252]) tth Theorem 2.2 The trace mappings
γ±:W2(G×S×I)→Tτ2±(Γ±); γ±(ψ) =ψ|Γ±
are (well-defined) bounded, surjective operators with bounded right inversesL±:Tτ2±(Γ±)→ W2(G×S×I) that is,γ±◦L±=I (the identity). The operators L± are calledlifts.
In fact, one can choose the above lift L− to be
(L−g)(x, ω, E) =g(x−t(x, ω)ω, ω, E), g ∈Tτ2−(Γ−), (8) lif-1 This lift L− is an isometric linear map and ω· ∇x(L−g) = 0, g ∈ Tτ2−(Γ−). Analogous
notes are valid for L+. We point out that the choice of lifts L± in Theorem 2.2 is nottth unique. In the sequel we denoteL:=L−. Especially from Theoremtth2.2 it follows that the trace mappingγ−:W2(G×S×I)→ L2loc(Γ−,|ω·ν|dσdωdE) is continuous and similarly for γ+.
By the Sobolev Embedding Theorem forψ∈C1(G×S×I) (cf. friedman[12], p. 22, or[31], p.treves 220)
ψ(·,·, E)L2(G×S)≤C
ψL2(G×S×I)+ ∂ψ
∂E L2
(G×S×I)
, ∀E∈I,
implying that the traces ψ(·,·,0), ψ(·,·, Em) ∈ L2(G×S) are well-defined for any ψ ∈ W12(G×S×I).
The traceγ(ψ) =ψ|Γ,ψ∈W2(G×S×I) is not necessarily in the spaceT2(Γ). Hence we define the space
W2(G×S×I) :={ψ∈W2(G×S×I) |γ(ψ)∈T2(Γ)}, which equipped with the inner product
ψ, vW2(G×S×I):=ψ, vW2(G×S×I)+γ(ψ), γ(v)T2(Γ), is a Hilbert space (cf. [29]).tervo17-up
The following Green’s formula is obtained by using Stokes’ Theorem first in the case where v, ψ∈C1(G×S×I), and then by applying a suitable limiting process.
Proposition 2.3 The Green’s formula
G×S×I
(ω· ∇xψ)v dxdωdE+
G×S×I
(ω· ∇xv)ψ dxdωdE =
∂G×S×I
(ω·ν)v ψ dσdωdE, holds for every v∈W2(G×S×I) andψ∈W2(G×S×I).
Also, we will work in the (energy independent)L2(G×S)-type spaces. The correspond- ing Hilbert spaces W2(G×S), T2(Γ±), T2(Γ) and W2(G×S) are defined in a similar manner as the above introduced spacesW2(G×S×I),T2(Γ±),T2(Γ) andW2(G×S×I), respectively. In addition, the traceγ(ψ) :=ψ|Γ ofψ∈W2(G×S) onto Γcan be defined in the same way asγ(ψ) =ψ|Γ.
2.2 On Collision Operators coll
The differential cross-sections may have singularities, or even hyper-singularities, which would lead to extra partial differential or pseudo-differential-like terms in the transport equation. Instead of explaining systematically the underlying theory, the following infor- mative description suffices for the purposes of this work.
The collision operator in (physical) literature is often given in the form (here we for clarity write I=I, S =S when the variable in integration isE, ω, respectively)
(Kψ)(x, ω, E) =
I
Sσ(x, ω, ω, E, E)ψ(x, ω, E)dωdE. (9) o-col In the case whereσ(x, ω, ω, E, E) has hyper-singularities (like in the case of Møller differ-
ential cross section given below) the integral
I
S occurring in the collision operator (9)o-col must be understood in the sense ofCauchy principal valuep.v.
I
S or more generally in the sense of Hadamard finite part integralp.f.
I
S hsiao
[17, Sec. 3.2]. Moreover, the (ω, ω)- dependence in differential cross-sections typically contain Dirac’s δ-distributions (on R).
More precisely, in σ(x, ωω, E, E) there may occur terms like δ(ω·ω−μ(E, E)) which require special treatment.
In [28] we verified that the differential cross sectiontervo16-up σ may be of the sum (this cross section is related to the so called Møller scattering)
σ(x, ω, ω, E, E) =χ(E, E) 1
(E−E)2σ2(x, ω, ω, E, E)
+ 1
E−Eσ1(x, ω, ω, E, E) +σ0(x, ω, ω, E, E) where χ(E, E) is a product of characteristic functions χR+(Em−E)χR+(E−E). Here each ofσj(x, ω, ω, E, E),j = 0,1,2 may contain the above mentionedδ-distributions, and
hence they are not necessarily measurable functions on G×S×S×I×I. The integral
S is originally interpreted as a distribution. In tervo16-up[28] we, however, showed that
S
σj(x, ω, ω, E, E)ψ(x, ω, E)dωdE= ˆσj(x, E, E) 2π
0
ψ(x, γ(E, E, ω)(s), E)ds, where γ(E, E, ω) : [0,2π] → S is a parametrization of the curve Γ(E, E, ω) = {ω ∈ S |ω·ω−μ(E, E) = 0}.So the terms ofK are of the form
p.f.
Iχ(E, E)ˆσj(x, E, E) 1 (E−E)j
2π 0
ψ(x, γ(E, E, ω)(s), E)ds
dE (10) kk-3 which can be ruled.
In [28] we gave a hyper-singular integral operator form and a pseudo-differential-liketervo16-up form of the resulting collision operator. It is well-known that certain hyper-singular in- tegral operators can be expressed as pseudo-differential operators, and conversely ([17],hsiao Chapter 7). The formulations therein reveal the nature of charged particles’ collisions:
Some interactions produce the first-order partial derivatives with respect to energy E combined with a Hadamard finite part operator ( a closer treatment shows that when unwrapping the terms, also partial derivatives with respect to ω appear). The problem- atic interactions are the primary electron-electron, primary positron-positron collisions and Bremsstrahlung. In[28] we additionally explored a CSDA-type approximation usingtervo16-up hyper-singular form of the transport equation but the error analysis remained open. These approaches require further study.
The restricted collision operator (that is, the collision operator whose kernel does not contain non-integrable singularities) related to Møller scattering is by (10)kk-3
(Kψ)(x, ω, E) =
I
2π 0
χ(E, E)ˆσ0(x, E, E)ψ(x, γ(E, E, ω)(s), E)dsdE. (11) coll3-aa Besides the collision operator of the form (11), at least two additional types of restrictedcoll3-aa
collision operators appear in applications, namely (Kψ)(x, ω, E) =
S×Iσ(x, ω, ω, E, E)ψ(x, ω, E)dωdE (12) coll1-aa and
(Kψ)(x, ω, E) =
Sσ(x, ω, ω, E)ψ(x, ω, E)dω. (13) coll2-aa The first (12) is related e.g. to the Bremstrahlung and the latter (coll1-aa coll2-aa13) to the elastic
scattering.
The restricted collision operators are typically partial Schur integral operators ([15],halmos p. 20, cf. also dautraylionsv6
[5], pp. 227-228) with respect to the pertinent measures. Hence they are bounded operatorsL2(G×S×I)→L2(G×S×I). We give a precise statement for the boundedness ofK given by (12).coll1-aa
Theorem 2.4 Suppose thatσ:G×S2×I2→ Ris non-negative measurable function, and that there are constants M1, M2>0 such that for a.e. (x, ω, E)∈G×S×I,
S×Iσ(x, ω, ω, E, E)dωdE≤M1<∞,
S×I
σ(x, ω, ω, E, E)dωdE≤M2<∞. (14) bound-K1 ThenK :L2(G×S×I)→L2(G×S×I) is bounded andK ≤√
M1M2.
Another typical feature of restricted collision operators is that the bounded operator Σ−K : L2(G×S×I) → L2(G×S×I) possesses accretivity (coercitivity) properties.
We formulate for the operator Σ−K where K is given by (12).coll1-aa
accre-K Theorem 2.5 Suppose that Σ ∈ L∞(G×S ×I), and that σ : G×S2×I2 → R is non-negative measurable function such that a.e. (x, ω, E)∈G×S×I
Σ(x, ω, E)−
S×Iσ(x, ω, ω, E, E)dωdE ≥c, Σ(x, ω, E)−
S×Iσ(x, ω, ω, E, E)dωdE ≥c, (15) accre-K1 for some constantc≥0. Then
(Σ−K)ψ, ψL2(G×S×I)≥cψ2L2(G×S×I), ∀ψ∈L2(G×S×I). (16) coer-K1 This result can be verified by utilizing the Cauchy-Schwarz inequality as in dautraylionsv6
[5], p. 241.
Note that (16) implies that Σcoer-K1 −K −cI is accretive. We remark that the operator Σ−K−cI:L2(G×S×I)→L2(G×S×I) is accretive if and only if−(Σ−K−cI) =
−Σ +K+cI is dissipative.
The restricted collision operators (coll2-aa13) and (11) satisfy analogous properties as (coll3-aa coll1-aa12).
The proofs for (coll2-aa13) can again be verified by using the Cauchy-Schwarz inequality as indautraylionsv6
[5], p. 241, but the proofs for (11) require certain additional (differential geometric) treatmentscoll3-aa concerning for change of variables on sphere in the relevant integrals.
In this paper we apply the CSDA-approximation which among others approximates the above hyper-singular terms (or equivalently pseudo-differential-like terms) to partial differential terms (cf. tervo16-up[28], the end of section 2.3). The pertinent restricted collision operator K =Kr is the sum of operators like (coll3-aa11), (coll1-aa12), (13). For simplicity we assumecoll2-aa that the restricted collision operator is everywhere of the form (12). The given analysiscoll1-aa is, however, valid for case where the real restricted collision operators have been included.
This is only a technicality. Finally, we remark that the lower limit E0 of energy must be positive since the differential cross sections may have singularities at E = 0. This modification is a technicality as well.
3 Single Continuous Slowing Down Equation
single-eq
3.1 Some Preliminaries and Change of Variables At first we consider a single CSDA transport equationgiven by
−∂(S0ψ)
∂E +ω· ∇xψ+ Σψ−Kψ=f onG×S×I, (17) se1 where the solution satisfies inflow boundary and initial value conditions
ψ|Γ− =g on Γ−, (18) se2
ψ(·,·, Em) = 0 onG×S. (19) se3
We assume that
Σ∈L∞(G×S×I), Σ≥0 a.e. on G×S×I. (20) ass1 Furthermore, the restricted collision operator is given forψ∈L2(G×S×I) by
(Kψ)(x, ω, E) =
S×Iσ(x, ω, ω, E, E)ψ(x, ω, E)dωdE,
where σ:G×S2×I2 →Ris a non-negative,σ≥0, measurable function such that (bound-K114) holds (for some constants M1, M2>0), and that there exists c≥0 such that
Σ(x, ω, E)−
S×Iσ(x, ω, ω, E, E)eC(E−E)dωdE≥c, Σ(x, ω, E)−
S×Iσ(x, ω, ω, E, E)eC(E−E)dωdE≥c, (21) ass3 for a.e. (x, ω, E) ∈ G×S×I, and where the constant C ≥ 0 is specified below after exponential shift (see (eq:def_C26)). For some special cases this assumption has been relaxed in
schlottbom2014
[7] or[6] (cf. alsoegger2014 dautraylionsv6
[5], Remark 15, pp. 241-242).
In what follows, we assume that the stopping power S0 : G×I → R satisfies the following assumptions:
S0∈L∞(G×I),
∂S0
∂E ∈L∞(G×I), (22) d-s0
κ:= inf
(x,E)∈G×IS0(x, E)>0, (23) csda9a
∇xS0∈L∞(G×I). (24) csda9b
csdale0 Lemma 3.1 For allψ∈C1(G×S×I), ∂(S0ψ)
∂E , ψ
L2(G×S×I)
≤qψ2L2(G×S×I)
+1 2
G×S
S0(x, Em)ψ2(x, ω, Em)−S0(x,0)ψ2(x, ω,0) dxdω, where
q:= 1
2 ess sup
(x,E)∈G×I
∂S0
∂E(x, E). (25) eq:def_q
Proof. Integrating by parts, we have ∂(S0ψ)
∂E , ψ
L2(G×S×I)
= ∂S0
∂Eψ, ψ
L2(G×S×I)
+ ∂ψ
∂E, S0ψ
L2(G×S×I)
= ∂S0
∂Eψ, ψ
L2(G×S×I)
−
ψ,∂(S0ψ)
∂E
L2(G×S×I)
+
G×S
S0(x, Em)ψ(x, ω, Em)2−S0(x,0)ψ(x, ω,0)2 dxdω, and therefore
2
∂(S0ψ)
∂E , ψ
L2(G×S×I)
= ∂S0
∂Eψ, ψ
L2(G×S×I)
+
G×S
S0(x, Em)ψ(x, ω, Em)2−S0(x,0)ψ(x, ω,0)2 dxdω
≤2qψ2L2(G×S×I)+
G×S
S0(x, Em)ψ(x, ω, Em)2−S0(x,0)ψ(x, ω,0)2 dxdω.
This finishes the proof.
Let
C:= max{q,0}
κ , (26) eq:def_C
where q and κare defined in (eq:def_q25) and (23). Note that ifcsda9a E →S0(x, E) is decreasing for everyx∈Gthenq≤0 and thereforeC vanishes. In the case whereC= 0 no exponential shift is needed.
We make a change of the unknown function (an exponential shift) by replacing ψby
φ(x, ω, E) :=eCEψ(x, ω, E). (27) eq:exp_trick This substitution changes the equation (17) tose1
−∂(S0φ)
∂E +ω· ∇xφ+CS0φ+ Σφ−KCφ=eCEf, where KC is given by
(KCφ)(x, ω, E) =
S×Iσ(x, ω, ω, E, E)eC(E−E)φ(x, ω, E)dωdE. Furthermore we define
fC:=eCEf, gC:=eCEg.
The inflow boundary and the initial conditions are φ|Γ− =gC, φ(x, ω, Em) = 0,
the latter (initial) condition holding for a.e. (x, ω)∈G×S.
From Theorem accre-K2.5 it follows that under the the conditions (14), (bound-K120) and (ass1 ass321) the operator
Σ−KC:L2(G×S×I)→L2(G×S×I) is a bounded operator and it satisfies the following accretivity condition
(Σ−KC)φ, φL2(G×S×I)≥cφ2L2(G×S×I), ∀φ∈L2(G×S×I).
3.2 On Weak and Strong Solutions m-d
In this section we apply results of dissipative operators to retrieve existence results for the problem (17), (se1 se218), (se319). Let
PC(x, ω, E, D)φ:=−∂(S0φ)
∂E +ω· ∇xφ+CS0φ,
where S0∈C1(I, L∞(G×S)) andC is a constant. WhenC = 0, we writePC(x, ω, E, D) simply asP(x, ω, E, D), that is
P(x, ω, E, D)φ=−∂(S0φ)
∂E +ω· ∇xφ, Recall that theformal transpose (adjoint) ofPC(x, ω, E, D) is
PC(x, ω, E, D)v:=S0∂v
∂E −ω· ∇xv+CS0v. (28) eq:P_C_prime
Define linear operators PC, PC : L2(G×S ×I) → L2(G×S ×I) with domains of definition D(PC),D(PC) by setting
D(PC) :=C1(G×S×I), PCφ:=PC(x, ω, E, D)φ, D(PC) :=C0∞(G×S×I◦), PCv:=PC(x, ω, E, D)v.
It is clear that bothPC andPC are densely defined.
Let PC∗ : L2(G×S ×I) → L2(G×S ×I) be the adjoint operator of PC. Then φ∈L2(G×S×I) is said to be aweak solution of
PCφ=f, f∈L2(G×S×I) (29) d1 if and only if
φ∈D(PC∗) and PC∗φ=f.
Since the adjointPC∗ is a closed operator the space HPC(G×S×I◦) :={φ∈L2(G×S×I)|
PC(x, ω, E, D)φ∈L2(G×S×I) in the weak sense} is a Hilbert space when equipped with the inner product
φ, vHPC(G×S×I◦):=φ, vL2(G×S×I)+PC(x, ω, E, D)φ, PC(x, ω, E, D)vL2(G×S×I). In particular, whenC= 0 this Hilbert space is written as HP. Notice that
HPC(G×S×I◦) =D(PC∗), (30) eq:H_P_is_D-P- whenD(PC∗) is equipped with the graph norm of PC∗.
We say thatφ∈L2(G×S×I) is astrong solution of (d129) (without boundary conditions) if there exists a sequence {φn} ⊂C1(G×S×I) (=D(PC)) such that
φ−φnL2(G×S×I)+PCφn−fL2(G×S×I)−→0 whenn→ ∞.
Let ˜PC : L2(G×S ×I) → L2(G×S×I) be the smallest closed extension (closure) of PC. Then one sees thatφ is a strong solution (without boundary conditions) if and only ifφ∈D( ˜PC) and ˜PCφ=f.
One immediately sees that every strong solution of (d129) is its weak solution. When the boundary ∂Gis smooth enough the converse is also true, the result which goes back toFriedrich friedrich44
[13] (the main theorem on page 135 together with Theorem 4.2, p. 144); see also[24].rauch85
Theorem 3.2 LetG⊂Rn be an open bounded subset, lying on one side of its boundary, and with boundary of class C1. Then every weak solution of the equation (29) is itsd1 strong solution (without boundary conditions), or equivalentlyC1(G×S×I) is dense in HPC(G×S×I◦).
The claim of the above theorem can also be equivalently stated as P˜C =PC∗.
We formulate below an existence result of strong solutions concerning homogeneous initial inflow boundary value problems. To this end, the following modified definition of
the strong solution is needed. One says that φ ∈ L2(G×S ×I) is a strong solution of (d129) with homogeneous initial inflow boundary conditions if there exists a sequence {φn} ⊂W2(G×S×I)∩H1(I, L2(G×S)) such that
φ−φnL2(G×S×I)+PC(x, ω, E, D)φn−fL2(G×S×I)−→0, whenn→ ∞, and
φn|Γ− = 0, φn(·,·, Em) = 0.
Define a linear operator PC,0:L2(G×S×I)→L2(G×S×I) by
D(PC,0) :={φ∈W2(G×S×I)∩H1(I, L2(G×S))|φ|Γ− = 0, φ(·,·, Em) = 0} PC,0φ:=PC(x, ω, E, D)φ.
Let ˜PC,0:L2(G×S×I)→L2(G×S×I) be the smallest closed extension (closure) ofPC,0. Then one sees that φ is a strong solution with the homogeneous initial inflow boundary conditions if and only if φ ∈ D( ˜PC,0) and ˜PC,0φ = f. Moreover, one sees that every strong solution with homogeneous initial inflow boundary conditions is a weak solution of PC(x, ω, E, D)φ=f, that is ˜PC,0⊂PC∗.
Since ˜PC,0 is a closed operator the space
HPC,0(G×S×I◦) :=D( ˜PC,0) (31) eq:H_tilde-P_0 is a Hilbert space when equipped with the inner product
φ, vHPC,0(G×S×I◦):=φ, vL2(G×S×I)+
P˜C,0φ,P˜C,0v
L2(G×S×I).
Remark 3.3 When φ∈ HPC,0(G×S ×I◦) we say that the (homogeneous) initial and boundary conditions
φ|Γ− = 0, φ(·,·, Em) = 0 are valid in the strong sense.
3.3 An Existence Result for a Problem Without Collisions ex-non-au
In the first place we show an existence and uniqueness result where the collision operator is zero that is, K = 0. The proof is based on the theory of non-autonomous evolution operators. The below result is more generally valid for collision operators satisfying the property (Kψ)(E) =K(E)ψ(E) whereψ(E)(x, ω) =ψ(x, ω, E)(see[28], Theorem 3.36).tervo16-up This property is valid for example, for the collision operators related to elastic scattering.
Here we utilize the result of this section only as an aid for the proof of Theorem3.7 andmd-evoth the caseK = 0 is sufficient.
Consider the problem (17), (se1 se218), (19) withse3 K = 0. We make the following change of variables and of the unknown function (note that in this section the constant C is not related to (26))eq:def_C
ψ(x, ω, E) :=˜ ψ(x, ω, Em−E),
φ:=e−CEψ,˜ (32) ecsd2
