On the Cauchy problem with large data for a space-dependent Boltzmann-Nordheim boson equation

0.19 MB·

by  L. Arkeryd

On the Cauchy problem with large data for a space-dependent
Boltzmann-Nordheim boson equation.
Leif ARKERYD and Anne NOURI
6
1 Mathematical Sciences, 41296 Go¨teborg, Sweden,
0
[email protected]
2
Aix-Marseille University, CNRS, Centrale Marseille, I2M UMR 7373, 13453 Marseille, France,
n
[email protected]
a
J

6
2
Abstract. This paper studies a Boltzmann Nordheim equation in a slab with two-dimensional

] velocity space and pseudo-Maxwellian forces. Strongsolutions are obtained for theCauchy problem
h
p withlargeinitialdatainanL1 L∞ setting. Themainresultsareexistence,uniquenessandstability
∩
- of solutions conserving mass, momentum and energy that explode in L if they are only local in
h ∞
t time. The solutions are obtained as limits of solutions to corresponding anyon equations.
a
m
[ 1 Introduction and main result.

1
v Inapreviouspaper[1],wehavestudiedtheCauchyproblemforaspace-dependentanyonBoltzmann
7
equation,
2
9
6 ∂ f(t,x,v)+v ∂ f(t,x,v) = Q (f)(t,x,v), f(0,x,v) = f (x,v), (t,x) R [0,1], v = (v ,v ) R2.
t 1 x α 0 + 1 2
0 ∈ × ∈
. (1.1)
1
0
6 The collision operator Qα in [1] depends on a parameter α ]0,1[ and is given by
∈
1
:
v Qα(f)(v) = B(v v ,n)[f′f′Fα(f)Fα(f ) ff Fα(f′)Fα(f′)]dv dn,
i ZIR2 S1 | − ∗| ∗ ∗ − ∗ ∗ ∗
X ×
r with the kernel B of Maxwellian type, f′, f′, f, f the values of f at v′, v′, v and v respectively,
a ∗ ∗ ∗ ∗
where
v = v (v v ,n)n, v = v +(v v ,n)n,
′ ′
− − ∗ ∗ ∗ − ∗
and the filling factor F
α
F (f)= (1 αf)α(1+(1 α)f)1 α.
α −
− −
Anyons are (quasi)particles that exist in one and two-dimensions besides fermions and bosons.
The exchange of two identical anyons may cause a phase shift different from π (fermions) and 2π
(bosons). In [1], also the limiting case α = 1 is discussed, a Boltzmann-Nordheim (BN) equation
[11] for fermions. In the present paper we shall consider the other limiting case, α = 0, which is a
12010 Mathematics Subject Classification. 82C10, 82C22, 82C40.
2Keywords;bosonicBoltzmann-Nordheimequation,lowtemperaturekinetictheory,quantumBoltzmannequation.
1
BN equation for bosons.
For the bosonic BN equation general existence results were first obtained by X. Lu in [7] in the
space-homogeneous isotropic boson large data case. It was followed by a number of interesting
studies in the same isotropic setting, by X. Lu [8, 9, 10], and by M. Escobedo and J.L. Vel´azquez
[5, 6]. Results with the isotropy assumption removed, were recently obtained by M. Briant and A.
Einav [3]. Finally a space-dependent case close to equilibrium has been studied by G. Royat in [12].
Thepapers[7,8,9,10]byLu,studytheisotropic, space-homogeneous BNequationbothforCauchy
data leading to mass and energy conservation, and for data leading to mass loss when time tends
to infinity. Escobedo and Vel´asquez in [5, 6], again in the isotropic space-homogeneous case, study
initial data leading to concentration phenomena and blow-up in finite time of the L -norm of the
∞
solutions. The paper [3] by Briant and Einav removes the isotropy restriction and obtain in poly-
nomially weighted spaces of L1 L type, existence and uniqueness on a time interval [0,T ). In
∞ 0
∩
[3] either T = , or for finite T the L -norm of the solution tends to infinity, when time tends
0 0 ∞
∞
to T . Finally the paper [12] considers the space-dependent problem, for a particular setting close
0
to equilibrium, and proves well-posedness and convergence to equilibrium.
Thepresentpaperstudies aspace-dependent, large dataproblemfor theBNequation. Theanalysis
is based on the anyon results in [1], which are restricted to a slab set-up, since the proofs in [1] use
an estimate for the Bony functional only valid in one space dimension. Due to the filling factor
F (f), those proofs also in an essential way depend on the two-dimensional velocity frame. By a
α
limiting procedure relying on the anyon case when α 0, well-posedness and conservation laws are
→
obtained in the present paper for the BN problen.
With
v v
cos θ = n − ∗ ,
· v v
| − ∗|
the kernel B(v v ,n) will from now on be written B(v v ,θ) and assumed measurable with
| − ∗| | − ∗|
0 B B , (1.2)
0
≤ ≤
for some B > 0. It is also assumed for some γ,γ ,c > 0, that
0 ′ B
B(v v ,θ)= 0 for cos θ < γ , for 1 cos θ < γ , and for v v < γ, (1.3)
′ ′
| − ∗| | | −| | | − ∗|
and that
B(v v ,θ)dθ c > 0 for v v γ. (1.4)
B
Z | − ∗| ≥ | − ∗|≥
These strong cut-off conditions on B are made for mathematical reasons and assumed throughout
the paper. For a more general discussion of cut-offs in the collision kernel B, see [8]. Notice that
contrarytotheclassical BoltzmannoperatorwhererigorousderivationsofB fromvariouspotentials
have been made, little is known about collision kernels in quantum kinetic theory (cf [13]).
With v denotingthecomponentof v inthex-direction, theinitial valueproblemfor theBoltzmann
1
Nordheim equation in a periodic in space setting is
∂ f(t,x,v)+v ∂ f(t,x,v) = Q(f)(t,x,v), (1.5)
t 1 x
where
Q(f)(v) = B(v v ,θ)[f f F(f)F(f ) ff F(f )F(f )]dv dθ, (1.6)
′ ′ ′ ′
ZIR2 [0,π] | − ∗| ∗ ∗ − ∗ ∗ ∗
×
2
and
F(f)= 1+f. (1.7)
Denote by
f♯(t,x,v) = f(t,x+tv ,v) (t,x,v) R [0,1] R2. (1.8)
1 +
∈ × ×
Strong solutions to the Boltzmann Nordheim paper are considered in the following sense.
Definition 1.1 f is a strong solution to (1.5) on the time interval I if
f 1(I;L1([0,1] R2)),
∈ C ×
and
d
f♯ = Q(f) ♯, on I [0,1] R2. (1.9)
dt × ×
(cid:0) (cid:1)
The main result of this paper is the following.
Theorem 1.1 Assume (1.2)-(1.3)-(1.4). Let f L ([0,1] R2) and satisfy
0 ∞
∈ ×
(1+ v 2)f (x,v) L1([0,1] R2), sup f (x,v)dv = c < , inf f (x,v) > 0, a.a.v R2.
0 0 0 0
| | ∈ × Z x [0,1] ∞ x [0,1] ∈
∈ ∈
(1.10)
There exist a time T >0 and a strong solution f to (1.5) on [0,T ) with initial value f .
0
∞ ∞
For 0 < T < T , it holds
∞
f♯ 1([0,T );L1([0,1] R2)) L ([0,T] [0,1] R2). (1.11)
∞
∈ C ∞ × ∩ × ×
If T < + then
∞ ∞
lim f(t, , ) L ([0,1] R2)= + . (1.12)
t T k · · k ∞ × ∞
→ ∞
The solution is unique, depends continuously in L1 on the initial value f , and conserves mass,
0
momentum, and energy.
Remark.
A finite T may not correspond to a condensation. In the isotropic space-homogeneous case con-
∞
sidered in [5, 6], additional assumptions on the concentration of the initial value are considered in
order to obtain condensation.
The paper is organized as follows. In the following section, solutions f to the Cauchy prob-
α
lem for the anyon Boltzmann equation in the above setting are recalled, and their Bony functionals
are uniformly controlled with respect to α. In Section 3 the mass density of f is studied with
α
respect to uniform control in α. Theorem 1.1 is proven in Section 4 except for the conservations of
mass, momentum and energy that are proven in Section 5.
3
2 Preliminaries on anyons and the Bony functional.
The Cauchy problem for a space-dependent anyon Boltzmann equation in a slab was studied in [1].
That paper will be the starting point for the proof of Theorem 1.1, so we recall the main results
from [1].
Theorem 2.1
Assume (1.2)-(1.3)-(1.4). Let the initial value f be a measurable function on [0,1] R2 with values
0
×
in ]0, 1], and satisfying (1.10). For every α ]0,1[, there exists a strong solution f of (1.1) with
α ∈ α
1
f♯ 1([0, [;L1([0,1] R2)), 0 < f (t, , ) < for t >0,
α ∈ C ∞ × α · · α
and
sup f♯(s,x,v)dv c (t), (2.1)
α α
Z ≤
(s,x) [0,t] [0,1]
∈ ×
for some function c (t) > 0 only depending on mass and energy. There is t > 0 such that for any
α m
T > t , there is η > 0 so that
m T
1
f (t, , ) η , t [t ,T].
α T m
· · ≤ α − ∈
The solution is unique and depends continuously in ([0,T];L1([0,1] R2)) on the initial L1-datum.
C ×
It conserves mass, momentum and energy.
The conditions f L ([0,1] R2) and (1.10) are assumed throughout the paper.
0 ∞
∈ ×
To obtain Theorem 1.1 for the boson BN equation from the anyon results, we start from a fixed
initial value f bounded by 2L with L N. We shall prove that there is a time T > 0 independent
0
∈
of 0 < α < 2 L 1, so that the solutions are bounded by 2L+1 on [0,T]. For that, some lemmas
− −
from the anyon paper are sharpened to obtain control in terms of only mass, energy and L. We
then prove that the limit f of the solutions f when α 0 solves the corresponding bosonic BN
α
→
problem. Iterating the result from T on, it follows that f exists up to the first time T when
∞
limt T fα(t, , ) L ([0,1] R2)= .
→ ∞ k · · k ∞ × ∞
We observe that
Lemma 2.2
Given f 2L and satisfying (1.10), there is for each α ]0,2 L 1[ a time T > 0 so that the
0 − − α
≤ ∈
solution f to (1.1) is bounded by 2L+1 on [0,T ].
α α
Proof of Lemma 2.2.
Split the Boltzmann anyon operator Q into Q = Q+ Q , where the gain (resp. loss) term Q+
α α α −α α
−
(resp. Q ) is defined by
−α
Q+(f)(v) = Bf f F (f)F (f )dv dθ (resp. Q (f)(v) = Bff F (f )F (f )dv dθ). (2.2)
α ′ ′ α α −α α ′ α ′
Z ∗ ∗ ∗ Z ∗ ∗ ∗
4
The solution f to (1.1) satisfies
α
t t
f♯(t,x,v) = f (x,v)+ Q (f )(s,x+sv ,v)ds f (x,v)+ Q+(f )(s,x+sv ,v)ds.
α 0 α α 1 0 α α 1
Z ≤ Z
0 0
Hence
t
supf♯(s,x,v) f (x,v)+ Q+(f )(s,x+sv ,v)ds (2.3)
α 0 α α 1
≤ Z
s t 0
≤
t
= f (x,v)+ Bf (s,x+sv ,v )f (s,x+sv ,v )F (f )(s,x+sv ,v)F (f )(s,x+sv ,v )dv dθds
0 α 1 ′ α 1 ′ α α 1 α α 1
Z0 Z ∗ ∗ ∗
B 1 2(1 2α) t
2L + 0 1 − f (s,x+sv ,v )dv dθds,
α 1 ′
≤ α (cid:16)α − (cid:17) Z0 Z ∗
since the maximum of F on [0, 1] is (1 1)1 2α for α ]0, 1[. With the angular cut-off (2.2),
α α α − − ∈ 2
v v is a change of variables. Using it and (2.1) for t 1 leads to
′
∗ → ≤
B c (1) 1 2(1 2α)
sup f♯(s,x,v) 2L +c 0 α 1 − t
α ≤ α α −
s t,x (cid:16) (cid:17)
≤
2Lα3 4α(1 α)2(2α 1)
2L+1 for t min 1, − − − .
≤ ≤ { cB c (1) }
0 α
The lemma follows.
The estimate of the Bony functional
1
B¯ (t) := v v 2Bf f F (f )F (f )dvdv dθdx, t 0,
α α α α α′ α α′
Z0 Z | − ∗| ∗ ∗ ∗ ≥
from the proof of Theorem 2.1 for f 2L+1 , can be sharpened.
α
≤
Lemma 2.3
For α 2 L 1 and T > 0 such that f (t) 2L+1 for 0 t T, it holds
− − α
≤ ≤ ≤ ≤
T
B¯ (t)dt c (1+T),
Z α ≤ ′0
0
with c independent of T and α, and only depending on f (x,v)dxdv, v 2f (x,v)dxdv and L.
′0 0 | | 0
R R
Proof of Lemma 2.3.
Denote f by f for simplicity. The proof is an extension of the classical one (cf [2], [4]), together
α
with the control of the filling factor F when v R2, as follows.
α
∈
The integral over time of the momentum v f(t,0,v)dv (resp. the momentum flux
1
v2f(t,0,v)dv ) is first controlled. Let Rβ C1([0,1]) be such that β(0) = 1 and β(1) = 1.
1 ∈ −
MR ultiply (1.1) by β(x) (resp. v β(x) ) and integrate over [0,t] [0,1] R2. It gives
1
× ×
t 1
v f(τ,0,v)dvdτ = β(x)f (x,v)dxdv β(x)f(t,x,v)dxdv
1 0
Z Z 2 Z −Z
0 (cid:0)
t
+ β (x)v f(τ,x,v)dxdvdτ ,
′ 1
Z Z
0 (cid:1)
5
resp.
(cid:16)
t 1
v2f(τ,0,v)dvdτ = β(x)v f (x,v)dxdv β(x)v f(t,x,v)dxdv
Z Z 1 2 Z 1 0 −Z 1
0 (cid:0)
t
+ β (x)v2f(τ,x,v)dxdvdτ .
′ 1
Z0 Z (cid:1)(cid:17)
Consequently, using the conservation of mass and energy of f,
t t
v f(τ,0,v)dvdτ + v2f(τ,0,v)dvdτ c(1+t). (2.4)
|Z Z 1 | Z Z 1 ≤
0 0
Here c is of magnitude of mass plus energy uniformly in α. Let
(t)= (v v )f(t,x,v)f(t,y,v )dxdydvdv .
1 1
I Zx<y − ∗ ∗ ∗
It results from
(t) = (v v )2f(t,x,v)f(t,x,v )dxdvdv +2 v (v v )f(t,0,v )f(t,x,v)dxdvdv ,
′ 1 1 1 1 1
I −Z − ∗ ∗ ∗ Z ∗ ∗ − ∗ ∗
and the conservations of the mass, momentum and energy of f that
t 1
(v v )2f(s,x,v)f(s,x,v )dvdv dxds
1 1
Z0 Z0 Z − ∗ ∗ ∗
2 f (x,v)dxdv v f (x,v)dv+2 f(t,x,v)dxdv v f(t,x,v)dxdv
0 1 0 1
≤ Z Z | | Z Z | |
t
+2 v (v v )f(τ,0,v )f(τ,x,v)dxdvdv dτ
1 1 1
Z0 Z ∗ ∗ − ∗ ∗
2 f (x,v)dxdv (1+ v 2)f (x,v)dv+2 f(t,x,v)dxdv (1+ v 2)f(t,x,v)dxdv
0 0
≤ Z Z | | Z Z | |
t t
+2 ( v2 f(τ,0,v )dv )dτ f (x,v)dxdv 2 ( v f(τ,0,v )dv )dτ v f (x,v)dxdv
Z0 Z ∗1 ∗ ∗ Z 0 − Z0 Z ∗1 ∗ ∗ Z 1 0
t t
c 1+ v2f(τ,0,v)dvdτ + v f(τ,0,v)dvdτ .
≤ (cid:16) Z0 Z 1 |Z0 Z 1 |(cid:17)
And so, by (2.4),
t 1
(v v )2f(s,x,v)f(s,x,v )dvdv dxds c(1+t). (2.5)
1 1
Z0 Z0 Z − ∗ ∗ ∗ ≤
Denote by u1 = R v1ffddvv. Recalling (1.2) it holds
R
t 1
(v u )2Bχ ff F (f )F (f )(s,x,v,v ,θ)dvdv dθdxds
1 1 j j ′ j ′
Z0 Z0 Z − ∗ ∗ ∗ ∗
t 1
c (v u )2ff (s,x,v,v )dvdv dxds
1 1
≤ Z0 Z0 Z − ∗ ∗ ∗
c t 1
= (v v )2ff (s,x,v,v )dvdv dxds
1 1
2 Z0 Z0 Z − ∗ ∗ ∗ ∗
c(1+t). (2.6)
≤
6
Here c also contains supF (f )F (f ) which is of magnitude bounded by 22L. So c is of magnitude
α ′ α ′
22L(mass+energy) and uniformly in∗α. Multiply equation (1.1) for f by v2, integrate and use that
1
v2Q (f)dv = (v u )2Q (f)dv and (2.6). It results
1 α 1− 1 α
R R
t
(v u )2Bf f F (f)F (f )dvdv dθdxds
1 1 ′ ′ α α
Z0 Z − ∗ ∗ ∗
t
= v2f(t,x,v)dxdv v2f (x,v)dxdv + (v u )2Bff F (f )F (f )dxdvdv dθds
Z 1 −Z 1 0 Z0 Z 1− 1 ∗ α ′ α ∗′ ∗
< c (1+t),
0
where c is a constant of magnitude 22L(mass+energy).
0
After a change of variables the left hand side can be written
t
(v u )2Bff F (f )F (f )dvdv dθdxds
Z0 Z 1′ − 1 ∗ α ′ α ∗′ ∗
t
= (c n [(v v ) n])2Bff F (f )F (f )dvdv dθdxds,
1 1 α ′ α ′
Z0 Z − − ∗ · ∗ ∗ ∗
where c = v u . And so,
1 1 1
−
t
n2[(v v ) n])2Bff F (f )F (f )dvdv dθdxds
Z0 Z 1 − ∗ · ∗ α ′ α ∗′ ∗
t
c (1+t)+2 c n [(v v ) n]Bff F (f )F (f )dvdv dθdxds.
0 1 1 α ′ α ′
≤ Z0 Z − ∗ · ∗ ∗ ∗
The term containing n2[(v v ) n]2 is estimated from below. When n is replaced by an orthogonal
1 − ∗ ·
(direct) unit vector n , v and v are shifted and the product ff F (f )F (f ) is unchanged. In
′ ′ α ′ α ′
R2 the ratio between⊥the sum of∗the integrand factors n2[(v v )∗ n]2 +n2 ∗[(v v ) n ]2 and
v v 2, is, outside of the angular cut-off (1.3), uniformly1bou−nde∗d ·from belo⊥w1 by−γ2.∗ In·de⊥ed, if θ
′
(|re−sp.∗|θ ) denotes the angle between v v and n (resp. the angle between e and n, where e is a
1 v−v∗ 1 1
unit vector in the x-direction), | − ∗|
v v v v
n21[ v−v∗ ·n]2+n2⊥1[ v−v∗ ·n⊥]2 = cos2θ1 cos2θ+sin2θ1 sin2θ
| − ∗| | − ∗|
γ2cos2θ +γ (2 γ )sin2θ
′ 1 ′ ′ 1
≥ −
γ2, γ < cosθ < 1 γ , θ [0,2π].
′ ′ ′ 1
≥ | | − ∈
This is where the condition v R2 is used.
∈
That leads to the lower bound
t
n2[(v v ) n]2Bff F (f )F (f )dvdv dθdxds
Z0 Z 1 − ∗ · ∗ α ′ α ∗′ ∗
γ2 t
′ v v 2Bff F (f )F (f )dvdv dθdxds.
α ′ α ′
≥ 2 Z0 Z | − ∗| ∗ ∗ ∗
7
And so,
t
γ2 v v 2Bff F (f )F (f )dvdv dθdxds
′ α ′ α ′
Z0 Z | − ∗| ∗ ∗ ∗
t
2c (1+t)+4 (v u )n [(v v ) n]Bff F (f )F (f )dvdv dθdxds
0 1 1 1 α ′ α ′
≤ Z0 Z − − ∗ · ∗ ∗ ∗
t
2c (1+t)+4 v (v v )n n Bff F (f )F (f )dvdv dθdxds,
0 1 2 2 1 2 α ′ α ′
≤ Z0 Z (cid:16) − ∗ (cid:17) ∗ ∗ ∗
since
u (v v )n2Bff F (f )F (f )dvdv dθdx
Z 1 1− ∗1 1 ∗ α ′ α ∗′ ∗
= u (v v )n n Bff F (f )F (f )dvdv dθdx = 0,
1 2 2 1 2 α ′ α ′
Z − ∗ ∗ ∗ ∗
by an exchange of the variables v and v . Moreover, exchanging first the variables v and v ,
∗ ∗
t
2 v (v v )n n Bff F (f )F (f )dvdv dθdxds
1 2 2 1 2 α ′ α ′
Z0 Z − ∗ ∗ ∗ ∗
t
= (v v )(v v )n n Bff F (f )F (f )dvdv dθdxds
1 1 2 2 1 2 α ′ α ′
Z0 Z − ∗ − ∗ ∗ ∗ ∗
8 t
(v v )2n2Bff F (f )F (f )dvdv dθdxds
≤γ′2 Z0 Z 1 − ∗1 1 ∗ α ′ α ∗′ ∗
γ2 t
+ ′ (v v )2n2Bff F (f )F (f )dvdv dθdxds
8 Z0 Z 2− ∗2 2 ∗ α ′ α ∗′ ∗
8πc γ2 t
0(1+t)+ ′ (v v )2n2Bff F (f )F (f )dvdv dθdxds.
≤ γ′2 8 Z0 Z 2− ∗2 2 ∗ α ′ α ∗′ ∗
It follows that
t
v v 2Bff F (f )F (f )dvdv dθdxds c (1+t),
Z0 Z | − ∗| ∗ α ′ α ∗′ ∗ ≤ ′0
withc uniformlywithrespecttoα, of thesamemagnitudeas c , only dependingon f (x,v)dxdv,
′0 0 0
v 2f (x,v)dxdv and L. This completes the proof of the lemma. R
0
| |
R
3 Control of phase space density.
This section is devoted to obtaining a time T > 0, such that
sup f♯(t,x,v) 2L+1,
α
≤
t [0,T],x [0,1]
∈ ∈
uniformly with respect to α ]0,2 L 1[ . We start from the case of a fixed α 2 L 1. Up to
− − − −
∈ ≤
Lemma 3.3 the time interval when the solution does not exceed 2L+1, may beα-dependent. Lemma
8
3.4 implies that this time interval can be chosen independent of α.
Lemma 3.1
Given T > 0 such that f (t) 2L+1 for 0 t T, the solution f of (1.1) satisfies
α α
≤ ≤ ≤
sup f♯(t,x,v)dxdv < c +c T, α ]0,2 L 1[,
Z α ′1 ′2 ∈ − −
t [0,T]
∈
where c and c are independent of T and α, and only depend on f (x,v)dxdv, v 2f (x,v)dxdv
′1 ′2 0 | | 0
and L. R R
Proof of Lemma 3.1.
Denote f by f for simplicity. By (2.3),
α
T
sup f♯(t,x,v) f (x,v)+ Q+(f)(t,x+tv ,v)dt.
0 α 1
≤ Z
t [0,T] 0
∈
Integrating the previous inequality with respect to (x,v) and using Lemma 2.3, gives
T
sup f♯(t,x,v)dxdv f (x,v)dxdv + B
0
Z ≤ Z Z Z
0 t T 0
≤ ≤
f(t,x+tv ,v )f(t,x+tv ,v )F (f)(t,x+tv ,v)F (f)(t,x+tv ,v )dvdv dθdxdt
1 ′ 1 ′ α 1 α 1
∗ ∗ ∗
1 T
f (x,v)dxdv + B v v 2
≤ Z 0 γ2 Z0 Z | − ∗|
f(t,x,v )f(t,x,v )F (f)(t,x,v)F (f)(t,x,v )dvdv dθdxdt
′ ′ α α
∗ ∗ ∗
c (1+T) C +C T
f (x,v)dxdv + ′0 := 1 2 .
≤ Z 0 γ2 γ2
Lemma 3.2
Given T > 0 such that f(t) 2L+1 for 0 t T, and δ > 0, there exist δ > 0 and t > 0
1 2 0
≤ ≤ ≤
independent of T and α and only depending on f (x,v)dxdv, v 2f (x,v)dxdv and L, such that
0 0
| |
R R
sup sup f♯(s,x,v)dxdv < δ , α ]0,2 L 1[, t [0,T].
α 1 − −
Z ∈ ∈
x0 [0,1] x x0<δ2 t s t+t0
∈ | − | ≤ ≤
Proof of Lemma 3.2.
Denote f by f for simplicity. For s [t,t+t ] it holds,
α 0
∈
t+t0
f♯(s,x,v) =f♯(t+t ,x,v) Q (f)(τ,x+τv ,v)dτ
0 α 1
−Z
s
t+t0
f♯(t+t ,x,v)+ Q (f)(τ,x+τv ,v)dτ.
0 −α 1
≤ Z
s
9
And so
t+t0
sup f♯(s,x,v) f♯(t+t ,x,v)+ Q (f)(s,x+sv ,v)ds.
0 −α 1
≤ Z
t s t+t0 t
≤ ≤
Integrating with respect to (x,v), using Lemma 2.3 and the bound 2L+1 from above for f, gives
sup f♯(s,x,v)dxdv
Z
x x0<δ2t s t+t0
| − | ≤ ≤
f♯(t+t ,x,v)dxdv
0
≤ Z
x x0<δ2
| − |
t+t0
+ Bf♯(s,x,v)f(s,x+sv ,v )F (f)(s,x+sv ,v )F (f)(s,x+sv ,v )dvdv dθdxds
1 α 1 ′ α 1 ′
Zt Z ∗ ∗ ∗
1 t+t0
f♯(t+t ,x,v)dxdv + B v v 2f♯(s,x,v)f(s,x+sv ,v )
≤ Zx x0<δ2 0 λ2 Zt Zv v λ | − ∗| 1 ∗
| − | | − ∗|≥
F (f)(s,x+sv ,v )F (f)(s,x+sv ,v )dvdv dθdxds
α 1 ′ α 1 ′
∗ ∗
t+t0
+c22L Bf♯(s,x,v)f(s,x+sv ,v )dvdv dθdxds
1
Zt Zv v <λ ∗ ∗
| − ∗|
c (1+t )
f♯(t+t ,x,v)dxdv + ′0 0 +c23Lt λ2 f (x,v)dxdv
≤ Z 0 λ2 0 Z 0
x x0<δ2
| − |
1 c (1+t )
v2f dxdv+cδ 2LΛ2+ ′0 0 +c23Lt λ2 f (x,v)dxdv.
≤ Λ2 Z 0 2 λ2 0 Z 0
Depending on δ , suitably choosing Λ and then δ , λ and then t , the lemma follows.
1 2 0
The previous lemmas imply for fixed α 2 L 1 a bound for the v-integral of f# only depend-
− − α
≤
ing on f (x,v)dxdv, v 2f (x,v)dxdv and L.
0 0
| |
R R
Lemma 3.3
With T defined as the maximum time for which f (t) 2L+1, t [0,T ], take T = min 1,T .
α′ α α′ α α′
≤ ∈ { }
The solution f of (1.1) satisfies
α
sup f♯(t,x,v)dv c , (3.1)
α 1
Z ≤
(t,x) [0,Tα[ [0,1]
∈ ×
where c is independent of α 2 L 1 and only depends on f (x,v)dxdv, v 2f (x,v)dxdv and
1 − − 0 0
≤ | |
L. R R
Proof of Lemma 3.3.
Denote by E(x) the integer part of x R, E(x) x < E(x)+1.
∈ ≤
By (2.3),
t
supf♯(s,x,v) f (x,v)+ Q+(f)(s,x+sv ,v)ds
0 α 1
≤ Z
s t 0
≤
t
= f (x,v)+ Bf(s,x+sv ,v )f(s,x+sv ,v )F (f)(s,x+sv ,v)F (f)(s,x+sv ,v )dv dθds
0 1 ′ 1 ′ α 1 α 1
Z0 Z ∗ ∗ ∗
f (x,v)+c22LA, (3.2)
0
≤
10