On global regular solution branches and multiple solutions of the Boltzmann equation

0.23 MB·

by  Jörg Kampen

On global regular solution branches and multiple
solutions of the Boltzmann equation
J¨org Kampen
6
January 7, 2016
1
0
2
Abstract
n
a
ExistenceofglobalregularsolutionbranchesoftheBoltzmannCauchy
J
problem with continuously differentiable data in phase space dimension
6 2d≥6withpolynomialdecayatinfinityofordergreaterthan2disproved.

Therearedatainthisclassofinfiniterelativeentropywithrespecttothe
]
P Gaussian. Furthermore, there are weakly singular solution branches of
A the Boltzmann equation in spatial dimension d≥ 3, i.e., solutions of the
BoltzmannequationswhichareonlyLipschitzwithrespecttothevelocity
.
h variables at some point in phase space. This is in accordance with a.e.
t L1-uniquenessofrenormalizedsolutions(cf.[4])andmoreclassical results
a
m in function spaces of mixed regularity.
[

1 Introduction
1
v
3 The Boltzmann equation
4
2 ∂ F +v F =QS(F,F) (1)
t x
1 ∇
0
determines the dynamics of the number density
.
1
0 R R3 R3 (t,x,v) F(t,x,v) 0 (2)
6 × × ∋ → ≥
1 in phase space. The right side QS(F,F) is the Boltzmann collision integral,
:
v and, as the Boltzmann approachconsiders only binary collisions,it is naturally
i describedbyabilinearoperatorwithacharacteristicdependenceonthevelocity
X
v =(v ,v ,v ). For a hard sphere gas the collision integral is defined by
r 1 2 3
a
QS(F,F)= (F (v˜)F(v˜ ) F (v)F (v )) v v dv dσ (3)
Z(v∗,σ)∈R3×S2 ∗ − ∗ | − ∗| ∗
along with
v˜= 1(v+v )+ 1 v v σ
2 ∗ 2| − ∗|
(4)
v˜ = 1(v+v ) 1 v v σ.
∗ 2 ∗ − 2| − ∗|
1
Theresultsofthispapercanbe generalisedtovector-valuedmixtureequations,
but we consider this standard scalar equation for simplicity. Note that in the
definition of the sourceterm in (3) the dependence of the density function F of
space and time is suppressed as usual.
We compare the following global existence theory with the global existence
theory of Lion and Di Perna, which is a construction in weak function spaces.
In this global existence theory renormalized solutions are considered. These
are solutions which have a certain distance to the Gaussian-Maxwell normal
distribution. The required distance to this Gaussian is finite relative entropy.
We recall
Definition 1.1. Given measurable functions N,M :R3 R3 R with N 0
× → ≥
and M >0, the relative entropy of N relative to M is
N
H(N M)= Nln N +M dxdv >0. (5)
| ZR3×R3(cid:20) (cid:18)M(cid:19)− (cid:21)
Definition 1.2. A renormalized solution of the Boltzmann equation in (33) is
a function
0 F C R ,L1 R3 R3 , (6)
≤ ∈ + loc ×
such that (cid:0) (cid:0) (cid:1)(cid:1)
QS(F,F)
L1 R ,R3 R3 , (7)
1+F ∈ loc + ×
and (cid:0) (cid:1)
QS(F,F)
(∂ +v )ln(1+F)= (8)
t ·∇x 1+F
in the sense of distributions.
In the follwing we denote the parametrized Gaussian by
a x b2
G = exp | − | . (9)
(a,b,c) (2πc)3/2 − 2c
(cid:18) (cid:19)
A main result is (cf. [1])
Theorem 1.3. Given initial data F0 F(0,.) 0 a.e., where
≡ ≥
H F0 G < , (10)
(1,0,1)
| ∞
there exists a renormalized solu(cid:0)tion F of t(cid:1)he Boltzmann equation with initial
data F0. This renormalized solution satisfies
t
t 0 H F(t,.)G + R(F)(s,x)dxds H(F(0,.)G ) (11)
(1,0,1) (1,0,1)
∀ ≥ | Z0 ZR3 ≤ |
(cid:0) (cid:1)
Here
R(F)= QS(F,F)ln(F)dv 0 (12)
−ZR3 ≥
represents the entropy production rate.
2
Subsequent research is usually in this framework in the sense that exis-
tence andpropertiesofsolutionswithfinite relativeentropywithrespectto the
Gaussian are considered. There is an important progress concerning regularity
of solution,where the connectionto the Landauequationwas a mayorstep (cf.
[2]andreferencestherein). Theassumptionoffiniterelativeentropyofthedata
with respect to the Gaussian is in some sense a strong assumption and in some
sense a weak assumption. It is a weak assumption in the sense that the data
may be quite irregularlocally (essentially L1), but it is strongin the sense that
there are data which lie in strong function spaces but have no finite relative
entropy distance relative to the Gaussian. For example consider for 7 < s < 8
the function p:R3 R3 R with
× →
1
p(x,v):= .where r = x2+x2+x3+v2+v2+v2. (13)
1+rs 1 2 3 1 2 3
q
We have
p Hm R3 R3 forall positive integers m, (14)
∈ ×
but (cid:0) (cid:1)
H pG = pln p p+G dxdv
| (1,0,1) R3×R3 G(1,0,1) −
h (cid:16) (cid:17) i
(cid:0) (cid:1) R
= R3 R3 p ln p(2π)3/2exp |x2|2 −1 + (2π1)3/2 exp −|x2|2 dxdv
×
h (cid:16) (cid:16) (cid:16) (cid:17)(cid:17) (cid:17) (cid:16) (cid:17)i
R
= r>0(cid:18)ln(cid:18)1+1rs(2π)31/+2resxp(cid:18)|r2|2(cid:19)(cid:19)−1(cid:19) + (2π1)3/2 exp −|r2|2 r5drdΦ
(cid:16) (cid:17)
R R
 
c ln(cid:18)1+1rs exp(cid:18)|r2|2(cid:19)(cid:19) r5drdΦ ,as c , where s (6,8),
≥ 0  1+rs  ↑∞ ↑∞ ∈
R R
  (15)
and where Φ is the angle function of polar coordinates. The latter estimate
indicates thatthe finite relativeentropyconditionwith respectto the Gaussian
is a stronger condition with respect to the required order of decay at spatial
infinity than the multiplicative properties of the nonlinear part of the operator
andthelinearcoefficientsofthelineearpartoftheoperatorindicate(cf. below).
Thismotivatesadifferentperspectiveinthispaper: weusestrongerassumptions
with respect to the local regularity of the data, weaker conditions with respect
to the polynomial decay at infinity, and prove the existence of strong global
solutions. Naturalfunctionspacesfortheconstructionofglobalregularsolution
branches of the Boltzmann equation in 2d-dimensional phase space are
Ck Cs,d , k 1, s>2d. (16)
∩ pol,k ≥
The classical phase space is 2d=6, whered denotes the spatial dimension, and
we formulate our results in this case, although the generalisation to 2d 6
≥
is straightforward. In (16) Ck is the function space of k-times continuously
differentiable functions, and
c
Cs,2d = f :R2d R: c>0 z 1 0 γ m Dγf(z) ,
pol,m → ∃ ∀| |≥ ∀ ≤| |≤ x ≤ 1+ z s
n (cid:12) (cid:12) | |(o17)
(cid:12) (cid:12)
3
whered 1 is anarbitraryintegerdimensionnumber ofspace. Aclosedballof
’radius’ c≥in Cs,2d is denoted by
pol,m
c
Cs,2d,c = f :R2d R: x 1 0 γ m Dγf(z) . (18)
pol,m → ∀| |≥ ∀ ≤| |≤ x ≤ 1+ z s
n (cid:12) (cid:12) | | o
We construct classical solutions F of the Boltzma(cid:12)nn equa(cid:12)tion in (closed sub-
spaces of) the function space
C1 [0,T],C1 Cs,2d , for some s>2d. (19)
∩ pol,1
(cid:16) (cid:17)
In the case of spatial dimension d = 3 this means that global regular solution
branches can be realised in case of polynomial decay of order s > 6 in phase
space of the initial density via compactness arguments. In this contextwe note
that sequences in is a closedball of the function space (such as in (17) for fixed
c > 0) are directly related to sequences classical Banach spaces on bounded
domains: for f C2m,d and x=(x , ,x ), y =(y , ,y ) the function
∈ pol,m 1 ··· d 1 ··· d
π π d
g :Ω:= , R, g(y)=f(x), y =arctan(x ), 1 j d (20)
j j
−2 2 → ≤ ≤
i h
lives in the Banach function space Cm π,π d defined below in (21). We
−2 2
recall (cid:16)(cid:3) (cid:2) (cid:17)
Proposition 1.4. For open and bounded Ω Rn and consider the function
⊂
space
Cm(Ω):= f :Ω R ∂αf exists for α m
→ | | |≤
n (21)
and ∂αf has an continuous extension to Ω
where α=(α , ,α ) denotes a multiindex and ∂α denotoe partial derivatives
1 n
···
with respect to this multiindex. Then the function space Cm Ω with the norm
f := f := ∂αf (cid:0) (cid:1) (22)
| |m | |Cm(Ω)
|αX|≤m(cid:12) (cid:12)
(cid:12) (cid:12)
is a Banach space. Here,
f := sup f(x). (23)
| |
x Ω
∈
(cid:12) (cid:12)
The construction is essentia(cid:12)lly(cid:12)in such spaces. We prove
Theorem 1.5. For initial data F0 F(0,.) 0 a.e., where
≡ ≥
F0 C1 Cs,6 , s>6, (24)
∈ ∩ pol,1
there exists a global classical regular solution F of the Boltzmann equation with
initial data F0, which satisfies
for all t≥0: F(t,.)∈C1∩Csp−ol1,1,6 (25)
Here, by ’classical solution’ we mean a solution of a differential equation which
satisfies the equation pointwise with classical derivatives in the sense of Weier-
strass.
4
WenotethataphysicalsolutionshouldsatisfyF 0a.e.,andthisrelationis
≥
satisfiedin case ofa physicalclassicalsolution,which is constructedhere. If we
use the term ’global regular solution branch’, then we refer to our expectation
thatglobalsolutionsoftheBoltzmannCauchyproblemarenotuniqueingeneral
and in case of dimension n 3. We have
≥
Theorem 1.6. In the situation of Theorem 1.5 and for d 3 thereexist weakly
≥
singular solution branches next to a global regular solution branch. Here the
term ’weakly singular solution’ means that there exists a Lipschitz continuous
distributional solution which is not classical everywhere.
Remark 1.7. A strong form of a weakly singular solution is a solution which is
classical on a domain except at one point of the phase space where it is only
Lipschitz. Such solution exist.
Global regular solution branches and weakly singular solutions are con-
structed via viscosity limits (ν 0) of local time solutions of the extended
↓
equations
∂ Fν +ν∆Fν +v Fν =QS(Fν,Fν), Fν(0,.)=F0, (26)
t x
∇
fortheapproximativeparticledensitiesFν(t,x,v), ν >0. Alocaltimesolution
has the representation
Fν =F0 g Γv +QS(Fν,Fν) gΓv, (27)
∗sp ν ∗ ν
where ∂ Γv +ν∆Γv +v Γv = 0. Here, the upper script g in g, and g
t ν ν ∇x ν ′ ′ ∗ ∗sp
indicates that these symbols denote generalizedconvolutedintegrals (not usual
convolutions as coefficients v are not constant) , i.e., we define
i
F0 g Γv(t,x,v):= F0(y)Γv(t,x,v;0,y)dy
∗sp ν R2d ν
R
QS(Fν,Fν) gΓv(t,x,v):= t QS(Fν,Fν)(s,y)Γv(t,x,v;s,y)dyds.
∗ ν 0 R2d ν
(28)
R R
For data F0 as in Theorem 1.5 or in Theorem 1.6 we shall use properties of
the adjoint Γv, of the fundamental solution Γv in order to observe that for all
ν∗ ν
t >0 and z =(x,v) R2d there exists a δ (0,1) and a finite constant C > 0
∈ ∈
(independent of ν) such that
ν∆Fν(t,z) = ν∆ F0 g Γv +QS(Fν,Fν) gΓv (t,z)
∗sp ν ∗ ν
(cid:12) (cid:12) (cid:12) (cid:0) (cid:1) (cid:12)
≤(cid:12) ν i F,0(cid:12)i∗gsp(cid:12)Γvν,,i∗+ QS(Fν,Fν) ,i∗gΓvν,.i∗ (t,z) (cid:12) (29)
(cid:12) P (cid:16) (cid:0) (cid:1) (cid:17) (cid:12)
ν(cid:12) C +ν C 0 as ν 0 and t>0. (cid:12)
≤ (νt)δ (ν)δ ↓ ↓
Here, C is a finite constant which is independent of ν. The properties of the
fundamental solution Γv and its adjoint are discussed in the appendix. The
ν
limit in (62) holds for data F0(.) C1 Cs,6 for s > 6, but it holds, e.g.,
∈ ∩ pol,1
also for data x r1+α0sin 1 1 , where r = d x2+ d v2, and
→ rα0 1+r8 i=1 i i=1 i
α0 ∈ (0,5) which are only Li(cid:0)psch(cid:1)itz at one point inqphPase space. PIt holds even
for some examples of data which are strongly H¨older continuous at one point
5
in space time. The multiplicative property the nonlinear term implies that the
functional increment
δFν :=Fν F0 g Γv :=QS(Fν,Fν) gΓv (30)
− ∗sp ν ∗ ν
canbeobtainedbylocaltimeiterativesolutionschemesFν,k =F0 g Γv+δFν,k
∗sp ν
which satisfy
δFν,k :=QS(Fν,k−1,Fν,k−1)∗gΓvν ∈C1∩C6pol,1, k ≥1, Fν,0 =F0∗gspΓvν. (31)
It follows that the sequence (δFν, ,k) with δFν, ,k(t,z) = δFν,k(t,tan(z)) has
∗ k ∗
a fixed point limit
δFν, (t,.) C1(Ω), (32)
∗
∈
where Ω is the bounded domain of (20) and t is in some short time interval
[t ,t +∆]whichimpliesthatδFν(t,.) C1 H1(Ω). ThefunctionFν :=F0 g
0 0 ∈ ∩ ∗sp
Γv+δFν satisfiestheCauchyproblemforallν >0andthecorrespondingfamily
ν
Fν, :=(F0 g Γv)(.,tan(.))+δFν(.,tan(.))turnsouttobeapointwisebounded
∗ ∗sp ν
andequicontinuousfamilyoffunctionswithaboundedsubsequenceFνk,∗,νk 0
↓
which converges along with Fνk,νk 0 to F∗ and F respectively with F
↓ ∈
C [t ,t +∆],C1 H1 . Moreover, the increment δFν inherits the order of
0 0
∩
polynomial spatial decay of order s > 6, where here and henceforth by ’spatial
(cid:0) (cid:1)
polynomial decay’ we mean decay with respect to the phase space variables as
theirmodulusgoestoinfinity. Since(62)holdsthefunctionF isaclassicallocal
time solution of the Boltzmann equation. Using the structure of the nonlinear
term we shall observe in the next section that this local time solution can be
extended to a global classical solution branch. As the local time argument can
be performedfor some mentioned data whichare only Lipschitz atone point in
phasespaceargumentitfollowsthatBoltzmann’sgastheoryisnotdeterministic
globally,orintrinsicallyincomplete. Thissituationhassomesimilaritieswiththe
caseoftheincompressibleEulerequation,sincebothequationsdonotloosewell-
posednessunderthetransformationt t(althoughwehaveacharacterisation
→−
of a time direction by the entropy in case of the Boltzmann equation). This is
inaccordancewithLionsresultthatthereisa.e. uniquenessofweaknormalized
solutions. In the next we provide the details of this proof. In the appendix we
add some information concerning the fundamental solution of Γv, its adjoint
ν
Γv, , and a priori estimates of these distributions and their spatial derivatives.
ν∗
Most of this material is known from stochastic analysis.
2 Proof of Theorem 1.5
For the extended Cauchy problem in (58) in item i) below we prove local time
existenceofsolutionsFν inC1 [t ,t +∆],H1 C1 fort 0andsmall∆>
0 0 0
∩ ≥
0andfordatainFν,t0 =F(t0,.(cid:0))∈C1∩Csp,o6l,1. Then(cid:1)initemii)belowweobtain
a viscosity limit Fνk F for some zero sequence (νk)k, i.e. limk νk = 0,
→ ↑∞
whichisashorttimesolutionoftheoriginalBoltzmannequation. Furthermore,
we show that the increment δF =limk δFνk and first order derivatives with
respecttophasespacevariablespreserve↑∞spatialpolynomialdecayoforders>6.
Finally, in item iii) we use the special structure of the nonlinear term in order
toshowthattheincrementofthenonlineartermissmallcomparedtopotential
6
damping induced by a simple time transformation such that a global regular
upper bound can be constructed which is linear in time.
i) Local solutions of the Boltzmann Cauchy problem on a time interval
[t0,t0 + ∆] with data Ft0 ∈ C1 ∩ Csp,o6l,1 are constructed by a viscosity
limit (positive real parameter ν 0) of the extended system
↓
∂ Fν ν∆Fν +v Fν =QS(Fν,Fν), Fν(0,.)=F0, (33)
t x
− ∇
to be solved for the approximative particle density (t,x,v) Fν(t,x,v).
→
Here ∂ Fν = Fν denotes the partial derivative with respect to the time
t ,t
variable, and x,v Rd. If the coefficients v , 1 i d have a linear
i
∈ ≤ ≤
upper bound(which willbe verifiedby a localiterationscheme),then the
fundamental solution
∂ Γv +ν∆Γv +v Γv =0. (34)
t ν ν ∇x ν
exists, and a local solution of the extended Boltzmann Cauchy problem
ona time interval[t ,t +∆] has the representation(note that g, g are
0 0 ∗ ∗sp
not standard convolutions, cf. the definition in (28) above)
Fν =F0 g Γv +QS(Fν,Fν) gΓv, (cf. 28). (35)
∗sp ν ∗ ν
An alternative representation of the local solution is of the form
Fν =F0 G v Fν G +QS(Fν,Fν) G , (36)
sp ν x ν ν
∗ − ∇ ∗ ∗
where G is the fundamental solution of the equation ∂ G ν∆G =0.
ν t ν ν
−
Notethatinthelattercase and denoteindeedstandardconvolutions.
sp
∗ ∗
For positive ν > 0 we solve these fixed point equations by an iteration
scheme and then consider the viscosity limit. A localiterationscheme for
the Boltzmann Cauchy problem on the time interval [t ,t +∆] is given
0 0
by the list of functions (Fν) , where Fν :=F0 g Γv and for k 1 the
approximationFν is deterkmkin≥e0d recursiv0ely by ∗sp ν ≥
k
Fν =F0 g Γv +QS(Fν ,Fν ) gΓv. (37)
k ∗sp ν k−1 k−1 ∗ ν
For the first functional increment of this series we have
δFν :=Fν F0 g Γv =QS(Fν,Fν) gΓv, (38)
1 1 − ∗sp ν 0 0 ∗ ν
and for k 1 we have
≥
δFν :=Fν Fν =δQS(Fν,Fν) gΓv, (39)
k+1 k+1− k k k ∗ ν
where
δQS(Fν,Fν) gΓv = QS(Fν,Fν) QS(Fν ,Fν ) gΓv. (40)
k k ∗ ν k k − k−1 k−1 ∗ ν
We write (cid:0) (cid:1)
δQS(Fν,Fν)=QS(Fν,Fν) QS(Fν ,Fν)
k k k k − k−1 k (41)
+QS(Fν ,Fν) QS(Fν ,Fν )
k−1 k − k−1 k−1
7
For functions H :[t ,t +∆] Rd Rd R we define
0 0
× × →
H ts0u,p∆ :=sup(t,x,v)∈[t0,t0+∆]×Rd×Rd H(t,x,v) , (42)
and (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
H t0,∆ :=sup H(t,x,v) + H(t,x,v) ,
sup,1 (t,x,v)∈[t0,t0+∆]×Rd×Rd ∇(x,v)
(43)
(cid:12) (cid:12) (cid:0)(cid:12) (cid:12) (cid:12) (cid:12)(cid:1)
w(cid:12)her(cid:12)e denotesthe gradientwithr(cid:12)espectto(cid:12)the(cid:12)spatialphasesp(cid:12)aces
(x,v)
∇
variables(x,v) R2d. We also write
∈
H t0,∆,s := (1+rs)H t0,∆, (44)
sup sup
and (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
H t0,∆,s := (1+rs)H t0,∆ , (45)
sup,1 sup,1
(cid:12) (cid:12) (cid:12) (cid:12)
where r = d x2+(cid:12) (cid:12) d v2 is(cid:12)the radial(cid:12)variable of the phase space.
i=1 i i=1 i
Assume indquctively that for some finite constant C >0 we have
P P
Fν t0,∆,s−1 C, (46)
k−1 sup,1 ≤
whereweaccountforthelin(cid:12)eargr(cid:12)owthofthefirstorderterm. Wecontinue
(cid:12) (cid:12)
to suppress the dependence on (t,x) for brevity. For the first difference
on the right side of (41) we have
QS(Fν,Fν)(v) QS(Fν ,Fν)(v) =
k k − k−1 k
(cid:12) (cid:12)
(cid:12) (δFν(v˜)Fν(v˜ ) δF(cid:12)ν(v)Fν(v )) v v dv dσ
(v∗,σ)∈R3×S2 k k ∗ − k k ∗ | − ∗| ∗ (47)
(cid:12)R (cid:12)
(cid:12) c δFν t0,∆ C˜ δFν t0,∆ (cid:12)
k sup C rrd 1dr k sup
≤ r>0 (cid:12) 1+r(cid:12)s 1+rs−1! − ≤ 1+(cid:12)r2s−(cid:12)d−2
(cid:12)(cid:12)R (cid:12) (cid:12) (cid:12)(cid:12) (cid:12) (cid:12)
for so(cid:12)me finite constant c which depend(cid:12)s only on phase space dimension
2d. Similarly, for the second difference on the right side of (41) we have
QS(Fν,Fν)(v) QS(Fν ,Fν)(v) =
k k − k−1 k
(cid:12) (cid:12)
(cid:12) (Fν(v˜)δFν(v˜ ) F(cid:12)ν(v)δFν(v )) v v dv dσ
(v∗,σ)∈R3×S2 k k ∗ − k k ∗ | − ∗| ∗ (48)
(cid:12)R (cid:12)
(cid:12) c δFν t0,∆ C˜ δFν t0,∆ (cid:12)
k sup C rrd 1dr k sup .
≤ r>0 (cid:12) 1+r(cid:12)s 1+rs−1! − ≤ 1+(cid:12)r2s−(cid:12)d−2
(cid:12)(cid:12)R (cid:12) (cid:12) (cid:12)(cid:12) (cid:12) (cid:12)
Henc(cid:12)e, using a priori estimates for Γv a(cid:12)nd first order phase space deriva-
ν
tives of Γv for ∆>0 small enough we have
ν
δFν t0,∆,s = δQS(Fν,Fν) gΓv t0,∆,s 1 δFν t0,∆,s, (49)
k+1 sup,1 k k ∗ ν sup,1 ≤ 2 k sup,1
such th(cid:12)at we(cid:12)have the(cid:12)desired contraction(cid:12). We rema(cid:12)rk th(cid:12)at for first order
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(α = 1) derivatives Dα with respect to phase space variables z = (x,v)
| | z
we may use the adjoint in order to obtain
DαδFν = DαδQS(Fν,Fν) gΓv, , (50)
z k+1 z k k ∗ ν∗
(cid:0) (cid:1)
8
where DαδQS(Fν,Fν) can be represented in terms of DβδFν for 0
z k k z k ≤
β 1. Hence, the estimates are naturally indpendent of ν as is the
| | ≤
contraction. For the solution one order of decay at phase space infinity is
lost due to the fact that the linear first order coefficient implies a linear
factor for the Gaussian upper bound of the fundamental solution Γv (cf.
ν
appendix).
ii) Next we consider viscositylimits for the localtime solution of item i), i.e.
on a time interval where we have contraction. For each ν > 0 we have
obtained fixed point solutions Fν of the extended Boltzmann equation,
whereforallmultiindicesα=(α , ,α )wit0 α 1andz =(x,v)
1 2d
··· ≤| |≤
we have
DαFν =F0 g DαΓv +QS(Fν,Fν) gDαΓv, , t [t ,t +∆], (51)
z ∗sp z ν ∗ z ν∗ ∈ 0 0
or
DαFν =F0 DαG v Fν DαG +QS(Fν,Fν) DαzG , (52)
z ∗sp z ν − ∇x ∗ z ν ∗ ν
also for t [t ,t +∆]. Since the contraction argument in item i) holds
0 0
∈
independently of the viscosity ν >0 we have
Fν(t,.) t0,∆,s−1 C, (53)
sup,1 ≤
(cid:12) (cid:12)
where C > 0 is a finite c(cid:12)(cid:12)onstant(cid:12)(cid:12) which is independent of ν > 0. Note
the upper script s 1 due to the linear coefficient of the linear term
−
of the Boltzmann equation. Moreover and accordingly, the fundamental
solutions and their spatial derivatives up to first order involved in (51)
have the a prior upper bound
(1+ v ) z y 2
(cid:12)DzαΓvν,∗(t,z,s,y)(cid:12)≤cν22d(t−s|)2|d+2|α| exp(cid:18)−λν|(t−−s|)(cid:19) (54)
(cid:12) (cid:12)
forsom(cid:12)e finite constant(cid:12)sλ,c>0, andwhere 0 α 1. A similarupper
≤| |≤
bound without factor 1+ v holds for the Gaussian G , of course. The
ν
| |
estimates in (60) imply that we have local estimates
C˜(1+ v )
DzαΓvν,∗(t,z,s,y) ≤ νδ(t s)δ z |y|2d+α 2δ, z =(x,v), (55)
(cid:12) (cid:12) − | − | | |−
(cid:12) (cid:12)
and w(cid:12)here C˜ is a finit(cid:12)e constant. For 0 α 1 and z = (x,v) the
≤ | | ≤
families (DαFν) are equicontinuous and pointwise bounded (by the
z ν>0
constant C). Since Fν(.) C1 Cs,6 for some s > 6 the corresponding
∈ ∩ pol,1
families (DαFν(.,tan(.))) for 0 α 1 and z =(x,v) are uniformly
z ν>0 ≤| |≤
pointwiseboundedandequicontinuousfamiliesinC1([t ,t +∆],Ω))and
0 0
we get a zero sequence (ν ) and limits
k k
DαF(.,tan(.)):= limDαFνk(.,tan(.)), 0 α 1. (56)
z k z ≤| |≤
↑∞
where we have a corresponding limit in the whole space
DαF := limDαFνk, 0 α 1. (57)
z k z ≤| |≤
↑∞
9
Note that for the latter we have DαF(t,.) Cs,6 for all t [t ,t +∆].
z ∈ pol,1 ∈ 0 0
We remark that continuous spaces of functions on Euclideanspace of any
dimensionwhich vanishat spatialinfinity are indeedclosed, but this may
be less well-known. Here we note also that continuous differentiability
with respectto time followsfrom continuousdifferentiability with respect
to variables of phase space. In order to show that the functions in (56)
and (57) are solutions of the Boltzmann Cauchy problem (in transformed
coordinates in the former case of course), we first remark that all the
members Fνk, k of the series above are classical local time solutions
↑∞
of the extended equations
∂tFνk+νk∆Fνk+v xFνk =QS(Fνk,Fνk), Fνk(0,.)=F0, νk >0. (58)
∇
We have to show that
limνk↓0 νk∆Fνk(t,z) =0 (59)
(cid:12)(1+ v ) (cid:12) z y 2
(cid:12)DzαΓvν,∗(t,z,s,y)(cid:12)≤cν22d(cid:12)(t−s|)2|d+2|α| e(cid:12)xp(cid:18)−λν|(t−−s|)(cid:19) (60)
(cid:12) (cid:12)
forsom(cid:12)e finite constant(cid:12)sλ,c>0, andwhere 0 α 1. A similarupper
≤| |≤
bound without factor 1+ v holds for the Gaussian G , of course. The
ν
| |
estimates in (60) imply that we have local estimates
C˜(1+ v )
DzαΓvν,∗(t,z,s,y) ≤ νδ(t s)δ z |y|2d+α 2δ, z =(x,v), (61)
(cid:12) (cid:12) − | − | | |−
(cid:12) (cid:12)
We us(cid:12)e the adjoint of t(cid:12)he fundamental solution and a priori estimates in
(60) such that for some finite constants C,C >0 and for t>0 we have
0
ν∆Fν(t,z) = ν∆ F0 g Γv +QS(Fν,Fν) gΓv (t,z)
∗sp ν ∗ ν
(cid:12) (cid:12) (cid:12) (cid:0) (cid:1) (cid:12)
≤(cid:12) ν 2i=d1 (cid:12)F,0i∗(cid:12)gspΓvν,,i∗+ QS(Fν,Fν) ,i∗gΓvν,.i∗ (t,z) (cid:12)
(cid:12) P (cid:16) (cid:0) (cid:1) (cid:17) (cid:12)
≤(cid:12)R2d2d ν1+|Cz|2dν22d(1t+)2|dv+|2)|α| exp −λ|z−νty|2 dz+ (cid:12)
R (cid:12)(cid:12) (cid:16) (cid:17)(cid:12)(cid:12)
tt00+∆ R2(cid:12)d2d ν1+C|z|2dν22d((t1+s|)v2|d)+2|α| exp −λν|(cid:12)z(−t−ys|2) dzds (62)
R R (cid:12)(cid:12) − (cid:16) (cid:17)(cid:12)(cid:12)
≤ B12d(z)2d ν(cid:12)1+|Cz|2dνδ(t−s)Cδ˜|(z1−+y|v|2|d)+|α|−2δ dz+νC (cid:12)
(cid:12) (cid:12)
R (cid:12) (cid:12)
+ tt00+∆ B12(cid:12)d(z)2d ν1+|Cz|2dνδ(t−s)Cδ˜|(z1−+y|v|2|d)+(cid:12)|α|−2δ dzds
(cid:12) (cid:12)
R R (cid:12) (cid:12)
ν C0 +ν C0 (cid:12)0 as ν 0. (cid:12)
≤ (νt)δ (ν)δ ↓ ↓
Hence the viscosity limit ν 0 is indeed a local time solution of the
k
↓
Boltzmann equation.
iii) The difference structure of the nonlinear source term QS(F,F) encodes
some spatial effects of the operator and can be exploited in order to ob-
tainanupperboundofthesolutionincrementδF overalocaltimeinterval
10