HYPOCOERCIVITY WITHOUT CONFINEMENT
EMERIC BOUIN, JEAN DOLBEAULT, STÉPHANE MISCHLER, CLÉMENT MOUHOT,
AND CHRISTIAN SCHMEISER
Abstract. Hypocoercivity methods are extended to linear kinetic equations
with mass conservation and without confinement, such that the long-time behavior has algebraic decay as in the case of the heat equation. Two alternative
approaches are developed: an analysis based on decoupled Fourier modes and
a direct approach where, instead of the Poincaré inequality for the Dirichlet
form, Nash’s inequality is employed. The former is also used to provide a proof
of exponential decay to equilibrium on the flat torus. Finally, the results are
extended to larger function spaces by a factorization method.
1. Introduction
We consider the Cauchy problem
∂t f + v · ∇x f = Lf ,
f (x, v, 0) = f0 (x, v) ,
(1)
(2)
for the distribution function f (x, v, t), with the position variable in the whole
space, x ∈ Rd , or in the flat d-dimensional torus, x ∈ Td (represented by the box
[0, 2π)d ), with the velocity variable v ∈ Rd , and with time t ≥ 0.
For the collision operator L two choices will be considered:
(a) Fokker-Planck:
(b) Scattering:
Lf = ∇v · (vf + ∇v f ) ,
ˆ
Lf =
σ(v, v 0 ) f (v 0 )M (v) − f (v)M (v 0 ) dv 0 ,
Rd
2
with the normalized Maxwellian M (v) = (2π)−d/2 e−|v| /2 . More general microscopic equilibria could be considered up to some technicalities, so we shall only
consider the Gaussian case for sake of simplicity. The scattering rate σ is assumed
to satisfy
(3)
1 ≤ σ(v, v 0 ) ≤ σ ,
v, v 0 ∈ Rd .
With σ = 1, case (b) includes the relaxation operator Lf = ρf M − f , with the
position density
ˆ
ρf (x, t) =
f (x, v, t) dv .
Rd
Date: June 1, 2017.
2
E. BOUIN, J. DOLBEAULT, S. MISCHLER, C. MOUHOT, AND C. SCHMEISER
Note that micro-reversibility, i.e. symmetry of σ, is not required. Both cases
share the properties that the null space of L is spanned by the Maxwellian M ,
and that L only acts on the velocity variable. Under the requirement
ˆ
σ(v, v 0 ) − σ(v 0 , v) M (v 0 ) dv 0 = 0 ,
v ∈ Rd ,
Rd
´
in case (b), both cases satisfy local mass conservation, i.e. Rd Lf dv = 0.
We introduce the Fourier representation with respect to x,
ˆ
f (x, v, t) =
fˆ(ξ, v, t) ei x·ξ dµ(ξ) ,
Rd
where dµ(ξ) = dξ is the Lesbesgue measure for the case x ∈ Rd , whereas dµ(ξ)
is discrete for x ∈ Td . The normalization of fˆ and of dµ(ξ) are chosen such that
the Parseval identity reads
f (·, v, t) 2
= fˆ(·, v, t)L2 (dµ(ξ)) .
L (dx)
In both cases, the problem for the Fourier transform is decoupled in the ξdirection:
(4)
∂t fˆ + i (v · ξ)fˆ = Lfˆ ,
(5)
fˆ(ξ, v, 0) = fˆ0 (ξ, v) .
We shall extend the approach of [2] to hypocoercivity theory (see [11]), which has
been inspired by [5], but is also close to the Kawashima compensating function,
see [8, 3].
In the following section we slightly strengthen the abstract hypocoercivity
result of [2] by allowing complex Hilbert spaces and by providing explicit formulas
for the coefficients in the decay rate. This result (Theorem 1) is applied in
Theorem 2 to the Fourier transformed problem (4), (5). In the case (b) of a
collision operator of scattering type the result can be extended to larger function
spaces by the factorization method of [4]. A simple, abstract version of this
method is formulated and proven in Section 3 (Theorem 3), and again applied
to (4), (5): see Theorem 4.
For the problem (1), (2) on the flat torus, exponential decay to equilibrium is
deduced from the Fourier transformed problem in Section 4, recovering a result
(Theorem 5) which can also be obtained directly with the method of [2]. Here
the factorization method is applied directly to (1), (2), allowing extension to a
larger space.
Sections 5 and 6 present two different approaches to the whole space problem.
In both cases algebraic decay to zero is shown with the same rate as for the heat
equation. In Section 5 the Fourier mode analysis is used, whereas in Section 6 we
develop a variant of the method of [2], where the so-called macroscopic coercivity
condition, typically taking the form of a Poincaré inequality, is replaced by an
application of Nash’s inequality, which can also be used for showing decay for the
HYPOCOERCIVITY WITHOUT CONFINEMENT
3
heat equation. The result by the Fourier mode analysis is stronger in the sense
that it also holds on larger spaces. On the other hand, the direct approach by
Nash’s inequality is also applicable to problems with non-constant coefficients.
2. Mode-by-mode hypocoercivity
We shall use the abstract approach of [2]. Although the extension of the
method to Hilbert spaces over the complex numbers is straightforward, we carry
it out here for completeness and also derive explicit information on the decay
rate.
Theorem 1. Let L and T be closed linear operators in the complex Hilbert space
(H, h·, ·i) with induced norm k · k. Let L be hermitian and T be anti-hermitian.
Let Π denote the orthogonal projection to the null space of L and define
−1
A := 1 + (TΠ)∗ TΠ (TΠ)∗ ,
where ∗ denotes the adjoint with respect to h·, ·i. Let positive constants λm , λM ,
and CM exist, such that, for any F ∈ H, the following properties are satisfied:
(6) microscopic coercivity:
− hLF, F i ≥ λm k(1 − Π)F k2 ,
(7) macroscopic coercivity: kTΠF k2 ≥ λM kΠF k2 ,
(8) parabolic macroscopic dynamics: ΠTΠF = 0 ,
(9) bounded auxiliary operators: kAT(1 − Π)F k + kALF k ≤ CM k(1 − Π)F k .
Then there exist positive constants C and λ, such that the semigroup generated
by L − T satisfies
ke(L− T)t k2 ≤ Ce−λt .
A possible choice for the constants is
λm λM
λM
min 1, λm ,
.
(10)
C = 3,
λ=
2
3 (1 + λM )
(1 + λM ) CM
Proof. The approach of [2] relies on the Lyapunov (or modified entropy) functional:
kF k2
+ δ RehAF, F i ,
H[F ] :=
2
with a small positive constant δ. Its time derivative along solutions of
dF
+ TF = LF
dt
is
(11)
d
H[F ] = hLF, F i − δ hATΠF, F i
dt
− δ RehAT(1 − Π)F, F i + δ RehTAF, F i + δ RehALF, F i
=: − D[F ] .
4
E. BOUIN, J. DOLBEAULT, S. MISCHLER, C. MOUHOT, AND C. SCHMEISER
Since the hermitian operator ATΠ can be interpreted as the application of the
z
map z 7→ 1+z
to (TΠ)∗ TΠ, the conditions (6) and (7) imply that the sum of the
first two terms in the entropy dissipation (11) is coercive:
δ λM
−hLF, F i + δ hATΠF, F i ≥ λm k(1 − Π)F k2 +
kΠF k2 .
1 + λM
For δ small enough, these two terms control the remaining three terms, if the
operators appearing there are bounded and act only on the microscopic part
(1 − Π)F of the distribution. Whereas the latter is straightforward for AT(1 − Π)
and for AL, for the operators A and TA it follows from (8). Whereas boundedness
of AT(1 − Π) and of AL is assumed in (9), it holds in general for A and TA by [2,
Lemma 1]:
1
(12)
kAF k ≤ k(1 − Π)F k ,
kTAF k ≤ k(1 − Π)F k .
2
The former shows the norm equivalence of H[F ] to kF k2 for δ < 1:
1+δ
1−δ
kF k2 ≤ H[F ] ≤
kF k2 .
(13)
2
2
In the following, we choose δ ≤ 1/2. With the abbreviations a := k(1 − Π)F k,
b := kΠF k, the dissipation term can be estimated by
δ λM 2
b − δ CM a b .
D[F ] ≥ (λm − δ) a2 +
1 + λM
In view of the first term we add the requirement δ ≤ λm /2. Finally, the choice
1
λm λM
δ = min 1, λm ,
2
2
(1 + λM ) CM
implies coercivity with
δ λM
1
2 δ λM
2 δ λM
λm 2
2
D[F ] ≥
a +
b ≥ min λm ,
H[F ] =
H[F ] ,
4
2 (1 + λM )
3
1 + λM
3 (1 + λM )
where, by δ ≤ 1/2, kF k2 ≥
4
3
H[F ] has been used.
This result will be applied to (4) for fixed ξ, with the choices
ˆ
ˆ
2
2
2
−1
H = L (dv/M ) ,
kF k =
|F | M dv ,
ΠF =
F dv M ,
Rd
Rd
and TF = i (v · ξ)F .
It is easily seen that (7)–(8) hold with
λM = |ξ|2 .
(14)
Microscopic coercivity (6) with λm = 1 holds as a consequence of the Poincaré
inequality [10]
ˆ
ˆ
ˆ
|∇u|2 M dv ≥
u2 M dv ,
if
u M dv = 0 ,
Rd
Rd
Rd
HYPOCOERCIVITY WITHOUT CONFINEMENT
5
with u = F/M in case (a). In case (b), by assumption (3), we have
− hLF, F i ≥ hF − ΠF, F i = k(1 − Π)F k2
as in [1, Proposition 2.2], and hence λm = 1. The operator A is defined by
ˆ
−iξ
v 0 F (v 0 ) dv 0 M (v)
(AF )(v) :=
·
1 + |ξ|2 Rd
and satisfies the estimate
ˆ
1
|v · ξ| |(1 − Π)F | dv
1 + |ξ|2 Rd
ˆ
1/2
1
|ξ|
2
k(1 − Π)F k
(v · ξ) M dv
k(1 − Π)F k .
=
2
1 + |ξ|
1 + |ξ|2
Rd
kAF k = kA(1 − Π)F k ≤
≤
For estimating AL we note for case (a) that
ˆ
ˆ
v LF dv = −
Rd
v F dv ,
Rd
and therefore AL = − A, whereas in case (b) the collision operator L is bounded:
ˆ
dv
2
2
(ρf M − f )2
= σ 2 k(1 − Π)F k2 .
kLF k ≤ σ
M
d
R
For both cases we therefore obtain
kALF k ≤
σ |ξ|
k(1 − Π)F k ,
1 + |ξ|2
where σ can be set to 1 in case (a). The operator
ˆ
M (v)
(ATF )(v) =
(v 0 · ξ)2 F (v 0 ) dv 0
1 + |ξ|2 Rd
can be estimated similarly:
√
kATF k ≤
3 |ξ|2
kF k ,
1 + |ξ|2
meaning that we have proven (9) with
(15)
CM
√
σ |ξ| + 3 |ξ|2
=
.
1 + |ξ|2
Thus, Theorem 1 can be applied, and with (14) and (15), the formula (10) for
the exponential decay rate gives
1 + |ξ|2
K |ξ|2
1
|ξ|2
√
(16)
min
1,
≥
=: λξ ,
K=
,
2
2
2
3 (1 + |ξ| )
1 + |ξ|
3 (3 + σ 2 )
(σ + 3 |ξ|)
which follows from
1 + t2
1
=
3 σ K = min
2
t>0 (1 + a t)
1 + a2
2
√
with a =
3
.
σ
6
E. BOUIN, J. DOLBEAULT, S. MISCHLER, C. MOUHOT, AND C. SCHMEISER
Theorem 2. Let ξ ∈ Rd , fˆ0 (ξ, ·) ∈ L2 (dv/M ), and fˆ be the solution of (4), (5).
Then, with λξ defined in (16),
2
2
ˆ
−λξ t ˆ
,
t ≥ 0.
≤
3
e
f
(ξ,
·)
f
(ξ,
·,
t)
0
2
2
L (dv/M )
L (dv/M )
The result holds for both the Fokker-Planck (with σ = 1) and the scattering
collision operators satisfying assumption (3).
3. Enlarging the space by factorization
Square integrability against the inverse of the Maxwellian is a rather restrictive
assumption on the initial data. In this section it will be relaxed with the help of
the abstract factorization method of [4]. Since this approach requires the splitting
of the collision operator in a dissipative loss term and a decay-generating gain
term, we only consider the case (b) of scattering models in this section.
We only need a special case of the factorization method which, for completeness, we state and prove below.
Theorem 3. Let B1 , B2 be Banach spaces and let B2 be continuously imbedded
in B1 , i.e. k · k1 ≤ c1 k · k2 . Let B and A + B be the generators of the strongly
continuous semigroups eBt and, respectively, e(A+B)t on B1 . Let there be positive
constants c2 , c3 , c4 , λ1 , and λ2 , such that, for all t ≥ 0,
Bt (A+B)t e ≤ c3 e−λ1 t ,
kAk1→2 ≤ c4 ,
≤ c2 e−λ2 t ,
e
1→1
2→2
where k·ki→j denotes the operator norm for linear mappings from Bi to Bj . Then
there exists a positive constant C = C(c1 , c2 , c3 , c4 ) such that, for all t ≥ 0,
C 1 + |λ1 − λ2 |−1 e− min{λ1 ,λ2 }t
for λ1 6= λ2 ,
(A+B)t ≤
e
C (1 + t) e−λ1 t
for λ1 = λ2 .
1→1
Proof. Integrating the identity
d (A+B)s B(t−s) e
e
= e(A+B)s A eB(t−s)
ds
with respect to s ∈ [0, t] gives
ˆ t
(A+B)t
Bt
e
=e +
e(A+B)s A eB(t−s) ds ,
0
and therefore
(A+B)t e
1→1
ˆ t
(A+B)s B(t−s) ≤ c3 e
+ c1
Ae
ds
e
1→2
0
ˆ t
≤ c3 e−λ1 t + c1 c2 c3 c4 e−λ1 t
e(λ1 −λ2 )s ds .
−λ1 t
0
The proof is completed by a straightforward computation.
HYPOCOERCIVITY WITHOUT CONFINEMENT
7
Theorem 3 can be applied to solutions of (4), (5) with a collision operator of
scattering type (b) and
ˆ
AF (v) = M (v)
σ(v, v 0 )F (v 0 ) dv 0 ,
d
R ˆ
0
0
0
B2 = L2 (dv/M ) .
σ(v, v )M (v ) dv F (v) ,
BF (v) = − iv · ξ +
Rd
For the larger Banach space we consider
(17)
B1 = L2 (wk dv) ,
with wk (v) = (1 + |v|2 )k/2 ,
k > d.
Then the assumptions of Theorem 3 are satisfied with λ2 = λξ /2 ≤
defined in (16)), with λ1 = 1 because of
ˆ
σ(v, v 0 )M (v 0 ) dv 0 ≥ 1 .
1 |ξ|2
24 1+|ξ|2
(as
Rd
The boundedness of A : B1 → B2 follows from (3) and
ˆ
1/2
−1
(18)
kAF kL2 (dv/M ) ≤ σ kF kL1 (dv) ≤ σ
wk dv
kF kL2 (wk dv) .
Rd
Theorem 4. Let ξ ∈ Rd , let fˆ0 (ξ, ·) ∈ L2 (wk dv) with wk as in (17), and let fˆ be
the solution of (4), (5) with a collision operator of scattering type (b) satisfying
assumption (3). Then, there exists a constant C = C(k, d, σ) such that, with λξ
defined in (16),
2
2
ˆ
−λξ t ˆ
f
(ξ,
·,
t)
≤
C
e
f
(ξ,
·)
,
t ≥ 0.
2
0
2
L (wk dv)
L (wk dv)
4. Exponential decay to equilibrium in the periodic case
The unique global equilibrium in the case x ∈ Td is given by
¨
1
f0 dx dv .
(19)
f∞ (x, v) = ρ∞ M (v) ,
with ρ∞ = d
|T | Td ×Rd
If we represent the flat torus Td by the box [0, 2π)d with periodic boundary
conditions, the Fourier variable satisfies ξ ∈ Zd . For ξ = 0, the microscopic
coercivity with λm = 1 (see Section 2) implies
kfˆ(0, ·, t) − fˆ∞ (0, ·)kL2 (dv/M ) ≤ e−t kfˆ0 (0, ·) − fˆ∞ (0, ·) kL2 (dv/M ) .
For all other modes fˆ∞ (ξ, v) = 0, ξ 6= 0, and we can use Theorem 2 with
λξ ≥ K/2, withe notations of (16). An application of the Parseval identity
then proves
√
f (·, ·, t) − f∞ 2
≤ 3 e−tK/4 f0 − f∞ L2 (dx dv/M ) .
L (dx dv/M )
8
E. BOUIN, J. DOLBEAULT, S. MISCHLER, C. MOUHOT, AND C. SCHMEISER
This result can also be derived by directly applying Theorem 1 to (1), (2), as
in [2]. For collision operators of scattering type (b) it can be improved by the
factorization method applied to (1) with
ˆ
σ(v, v 0 )f (v 0 ) dv 0 ,
Af (v) = M (v)
d
R
ˆ
σ(v, v 0 )M (v 0 ) dv 0 ,
Bf (v) = − v · ∇x f (v) − f (v)
Rd
¨
2
1
B1 = f ∈ L (dx; L (dv)) :
f dx dv = 0 ,
Td ×Rd
¨
2
f dx dv = 0 .
B2 = f ∈ L (dx dv/M ) :
Td ×Rd
1
With the notation of the preceding section, we have λ2 = 48
, and λ1 = 1 follows
from the explicit representation
ˆ
Bt
0
0
0
(e f0 )(x, v) = exp −t
σ(v, v )M (v ) dv f0 (x − vt, v)
Rd
from which we deduce
keBt f0 k1 ≤ e−t kf0 k1 .
From (18) we also have
kAf k2 ≤ σ kf k1 ,
and
ˆ
ˆ
¨
2
f2
dv dx = kf k22 .
M
d
d
d
d
T
R
T ×R
Therefore the assumptions of Theorem 3 are satisfied.
kf k21 =
|f | dv
dx ≤
Theorem 5. For collision operators of scattering type (b) satisfying assumption (3) there exists a constant C such that for the solution f of (1), (2) in the
case x ∈ Td with f0 ∈ L2 (dx; L1 (dv)),
f (·, ·, t) − f∞ 2
≤ C e−λt f0 − f∞ L2 (dx;L1 (dv))
L (dx;L1 (dv))
holds with f∞ defined in (19) and λ =
1
.
12 (3+σ 2 )
5. Whole space – algebraic decay by Fourier analysis
Since the macroscopic limit of (1) is the heat equation, algebraic decay to zero
can be expected in the case x ∈ Rd .
Theorem 6. There exists a constant c > 0 such that for the case x ∈ Rd with
f0 ∈ L2 (dv/M ; L1 (dx)) ∩ L2 (dx dv/M ), the solution f of (1), (2) satisfies
c
2
2
kf
k
+
kf
k
kf (·, ·, t)k2L2 (dx dv/M ) ≤
2
2
1
0
0
L (dx dv/M )
L (dv/M ;L (dx)) ,
(1 + t)d/2
HYPOCOERCIVITY WITHOUT CONFINEMENT
where
ˆ
ˆ
9
2
dv
.
M
Rd
Rd
The result holds for Fokker-Planck and for scattering collision operators satisfying
assumption (3). In the latter case the weight M −1 can also be replaced by wk ,
given in (17).
kf0 k2L2 (dv/M ;L1 (dx))
:=
|f0 | dx
Proof. We start with the result of Theorem 2 and integrate with respect to ξ.
Then the Parseval identity implies
¨
dv
−1
2
e−λξ t |fˆ0 |2 dξ
kf (·, ·, t)kL2 (dx dv/M ) ≤ C
M
Rd ×Rd
for some C > 0. The same holds with the weight wk , if we start from Theorem 4.
With the splitting
ˆ
ˆ
ˆ
e−λξ t |fˆ0 |2 dξ =
e−λξ t |fˆ0 |2 dξ +
e−λξ t |fˆ0 |2 dξ = I1 + I2 ,
Rd
|ξ|≤1
we get the estimate
I1 ≤ C
|ξ|>1
ˆ
kf0 (·, v)k2L1 (dx)
e−|ξ|
|ξ|≤1
2 tK/2
dξ ≤ C
2π
Kt
d
2
kf0 (·, v)k2L1 (dx) ,
where the decay as t → ∞ is shown by the coordinate transformation η = ξ
t ≥ 1. The second part decays exponentially:
√
t,
I2 ≤ C e−tK/2 kf0 (·, v)k2L2 (dx) ,
where again the Parseval identity has been used. An integration with respect
to v completes the proof.
6. Whole space – application of Nash’s inequality
A direct application of the hypocoercivity approach of [2] to the whole space
problem fails by lack of a Poincaré inequality or, in the terminology of [2], by
lack of macroscopic coercivity. In this section, the approach will be adapted by
relaxing macroscopic coercivity using Nash’s inequality.
We use the abstract setting of Section 2, applied to (1) with x ∈ Rd and the
notation
Tf = v · ∇x f ,
with the scalar product h·, ·i on L2 (dx dv/M ) and the induced norm k · k. We use
the modified entropy
−1
kf k2
+ δ hAf, f i ,
with A := 1 + (TΠ)∗ TΠ (TΠ)∗ ,
H[f ] :=
2
and its time derivative
d
(20)
H[f ] = hLf, f i−δ hATΠf, f i−δ hAT(1−Π)f, f i+δ hTAf, f i+δ hALf, f i .
dt
10
E. BOUIN, J. DOLBEAULT, S. MISCHLER, C. MOUHOT, AND C. SCHMEISER
As in the previous sections, the first term on the right hand side satisfies the
microscopic coercivity condition
− hLf, f i ≥ λm k(1 − Π)f k2 .
Although the second term is not coercive, we shall proceed as in the original
approach and show that the last three terms can be dominated by the first two
for small enough δ > 0. By (12) we have
|hTAf, f i| = |hTA(1 − Π)f, (1 − Π)f i| ≤ k(1 − Π)f k2 .
For the following steps, the abstract operators need
´ to be evaluated for the present
situation. A straightforward computation using Rd v ⊗ v M dv = id gives
(TΠ)∗ TΠf = − ΠT2 Πf = − M ∆x ρf .
Therefore
ATΠf = −M ∆x wf ,
implying
with
ˆ
ˆ
hATΠf, f i = −
ρf ∆x wf dx = −
Rd
= k∆x wf k2L2 (dx) +
(21)
wf − ∆x wf = ρf ,
(wf −
Rd
k∇x wf k2L2 (dx) .
∆x wf )∆x wf dx
For the third term on the right hand side of (20) we use adjoint operators as
in [2]:
hAT(1 − Π)f, f i = − h(1 − Π)f, TA∗ f i .
Thus we need
A∗ f
−1
−1
= TΠ 1 + (TΠ)∗ TΠ f = T 1 + (TΠ)∗ TΠ Πf
= T(wf M ) = v · ∇x wf M
and
TA∗ f = (v · ∇x )2 wf M .
Therefore
kTA∗ f k2 ≤ 3 k∇2x wf k2L2 (dx)
and
|hAT(1 − Π)f, f i| ≤
√
√
3 k(1 − Π)f k k∇2x wf kL2 (dx) = 3 k(1 − Π)f k k∆x wf kL2 (dx) .
Considering the representation (21), we have for any γ > 0:
√
γ√
3
|hAT(1 − Π)f, f i| ≤
3 hATΠf, f i +
k(1 − Π)f k2 .
2
2γ
It remains to estimate the last term on the right hand side of (20). As in Section 2
we have either AL = − A (case (a)) or kLf k ≤ σ kf k (case (b)). Therefore in both
cases
|hALf, f i| ≤ σ k(1 − Π)f k kA∗ f k
HYPOCOERCIVITY WITHOUT CONFINEMENT
11
with the convention that σ = 1 in case (a). Again (21) implies
|hALf, f i| ≤ σ k(1 − Π)f k k∇x wf kL2 (dx) ≤
γσ
σ
hATΠf, f i +
k(1 − Π)f k2 .
2
2γ
Using these results in (20) gives
√
√
d
γ ( 3+σ)
2
λm − δ − δ ( 23+σ)
k(1
−
Π)f
k
+
δ
1
−
hATΠf, f i
− H[f ] ≥
γ
2
dt
λ
√m
=
k(1 − Π)f k2 + hATΠf, f i ,
3 + ( 3 + σ)2
with
γ=
√1
3+σ
,
δ=
2√λm
3+( 3+σ)2
.
We still need to show that hATΠf, f i controls the macroscopic contribution,
i.e. ρf . We start by representing the norm of the macroscopic contribution in
terms of wf :
kΠf k2 = kρf k2L2 (dx) = kwf k2L2 (dx) + 2 k∇x wf k2L2 (dx) + k∆x wf k2L2 (dx)
≤ kwf k2L2 (dx) + 2 hATΠf, f i .
At this point we use Nash’s inequality [10],
4
2d/(2+d)
kwf k2L2 (dx) ≤ CNash kwf kLd+2
1 (dx) k∇x wf kL2 (dx)
,
and observe that
kwf (·, t)kL1 (dx) ≤ kρf (·, t)kL1 (dx) ≤ kf0 kL1 (dv dx) =: m ,
t ≥ 0.
Consequentially,
d
4
kΠf k2 ≤ 2 hATΠf, f i + CNash m d+2 hATΠf, f i d+2 =: Φ−1 hATΠf, f i ,
The function Φ : [0, ∞) → [0, ∞) satisfies 0 < Φ0 < 21 , which implies
1
2
k(1 − Π)f k2 + Φ(kΠf k2 ) ≥ Φ(kf k2 ) ≥ Φ 1+δ
H[f ] ,
2
with the last inequality again a consequence of (13). On the other hand, for
0 < z ≤ z0 ,
2
4
d
4
d
z
d+2
−1
d+2
d+2
d+2
Φ (z) = 2 z0
+ CNash m
z
≤ 2 z0 + CNash m
z d+2 .
z0
H[f0 ]
2
2
0]
With the choice z0 := H[f
1+δ , we have Φ 1+δ H[f ] ≤ Φ 1+δ H[f0 ] ≤ 1+δ = z0 ,
and therefore
k(1 − Π)f k2 + hATΠf, f i ≥
Φ
2
1+δ
c H[f ]
H[f ] ≥ H[f0 ]
2
d+2
d+2
d
+m
4
d+2
d+2 ≥
d
c H[f ]
d+2
d
2
2 (H[f0 ] + m2 ) d
,
12
E. BOUIN, J. DOLBEAULT, S. MISCHLER, C. MOUHOT, AND C. SCHMEISER
where from now on c is an explicit constant possibly changing from line to line.
In the second inequality we have used (a + b)p ≤ 2p−1 (ap + bp ), a, b ≥ 0, p ≥ 1,
with p = 2+d
2 . Finally, we obtain the entropy decay inequality
d
− H[f ] ≥
dt
(22)
c√λm
3+( 3+σ)2
which proves algebraic decay of H[f ],
(
H[f ] ≤
1+
c√λm
3+( 3+σ)2
H[f ]
d+2
d
2
(H[f0 ] + m2 ) d
,
) !− d
2
2
H[f0 ] d
2
(H[f0 ] + m2 ) d
t
H[f0 ] ,
implying decay of kf k.
Theorem 7. Let f0 ∈ L2 (dx dv/M ) with x ∈ Rd . Then there exists two constants
c1 > 0 and c2 > 0 such that the solution f of (1), (2) satisfies
c1
kf0 k2L2 (dx dv/M ) ,
(23)
kf (·, ·, t)k2L2 (dx dv/M ) ≤
(1 + γ t)d/2
−2/d
with γ = c2 1 + kf0 k2L1 (dv dx) /kf0 k2L2 (dx dv/M )
. The result holds for both
Fokker-Planck operators and scattering collision operators satisfying assumption (3).
Remark 8. In contrast to the Fourier mode analysis, the approach of this section
does not require translation invariance of the problem. In particular, the results
also hold for x-dependent scattering rates σ, satisfying the bounds (3), or more
general Fokker-Planck operators of the form Lf = ∇v · (D(x)(vf + ∇v f )). Note
also that for large t the right hand side of (23) is asymptotically proportional to
d
kf0 k2L1 (dx dv) + kf0 k2L2 (dx dv/M ) t− 2 ,
which is similar to the result of Theorem 6, because k · kL1 (dv) ≤ k · kL2 (dv/M ) .
Remark 9. If we consider the scaled equation
dF
1
ε
+ TF = LF
dt
ε
(which formally corresponds to a parabolic rescaling t 7→ ε2 t, x 7→ εx), it is
straightforward to check that in the estimate (10) for λ, the gap constant λm
has to be replaced by λm /ε while, with the notations of Theorem 1, CM can be
replaced by CM /ε for ε < 1. In the asymptotic regime as ε → 0+ , we obtain that
d
λM
λm λM ε
ε H[F ] ≤ −D[F ] ≤ −
2 D[F ]
dt
3 (1 + λM ) (1 + λM ) CM
which proves that the estimate of Theorem 1 becomes
λ≥
λm λ2M
2
3 (1 + λM )2 CM
HYPOCOERCIVITY WITHOUT CONFINEMENT
13
for ε > 0, small enough. We observe that this rate is independent of ε.
In the context of Theorem 7, σ has to be replaced by σ/ε and in the limit as
ε → 0+ , (22) becomes
d+2
H[f ] d
d
c λm
− H[f ] ≥ 2
dt
σ (H[f0 ] + m2 ) d2
which is again independent of ε.
7. Whole space – An explicit computation for the kinetic
Fokker-Planck equation
Let us consider the kinetic Fokker-Planck equation of case (a)
(24)
∂t f + v · ∇x f = ∇v · (v f + ∇v f )
on (0, +∞)×Rd ×Rd 3 (t, x, v). The characteristics associated with the equations
dx
dv
= v,
= −v
dt
dt
suggest to change variables and consider the distribution function g such that
f (t, x, v) = ed t g t, x + 1 − et v, et v
∀ (t, x, v) ∈ (0, +∞) × Rd × Rd .
The kinetic Fokker-Planck equation is changed into a heat equation in both
variables x and v with t dependent coefficients, which can be written as
(25)
∂t g = ∇ · Ḋ ∇g
where ∇g = (∇v g, ∇x g) and Ḋ is the t-derivative of the bloc-matrix
a Id b Id
1
D=2
with a = e2t −1 , b = 2 et −1−e2t , c = e2t −4 et +2 t+3 .
b Id c Id
Here Id is the identity matrix on Rd . We observe that Ḋ is degenerate: it is
nonnegative but its lowest eigenvalue is 0. However, the change of variables
allows the computation of a Green function.
Lemma 10. With the above notations, the Green function associated with (25)
is given by
G(t, x, v) =
a |x|
1
−
e
(2π (a c − b2 ))d
2 −2 b x·v+c |v|2
2 (a c−b2 )
∀ (t, x, v) ∈ (0, +∞) × Rd × Rd .
The method is standard and goes back to [9] (also see, for instance, [6, 7]). It
has been repeatedly used in the theory of the kinetic Fokker-Planck equation.
Proof. By a Fourier transformation in x and v, with associated variables ξ and η,
we find that
log C − log Ĝ(t, ξ, η) = (η, ξ) · D(η, ξ) = 12 a |η|2 + 2 b η · ξ + c |ξ|2
2
2
= 12 a η + ba ξ + 12 A |ξ|2 , A = c − ba
14
E. BOUIN, J. DOLBEAULT, S. MISCHLER, C. MOUHOT, AND C. SCHMEISER
for some constant C > 0 which is determined by the mass normalization condition
kG(t, · ·)kL1 (Rd ×Rd ) = 1. We take the inverse Fourier transform with respect to η ,
ˆ
b
1
|v|2
C
− 2 a −i a v·ξ − 2 A |ξ|2
e
eiv·η Ĝ(t, ξ, η) dη =
(2π)−d
e
(2π a)d
Rd
2
2
1 b b
2
|v|2
C
− 2 a − 2 A ξ+i a A v − 2 a2 A |v|
e
e
,
(2π a)d
and then the inverse Fourier transform with respect to ξ, so that we obtain
|v|2
2
2
b
C
− 1+ ab A 2 a − |x|
x·v
2A eaA
e
G(t, x, v) =
e
d
d
(2π a) (2π A)
=
2
2
b C
− 21A x− a v − |v|
2a .
=
e
e
(4π 2 a A)d
It is easy to check that C = (2π)d .
Let us consider a solution g of (25) with initial datum g0 ∈ L1 (Rd × Rd ). From
the representation
g(t, ·, · = G(t, ·, ·) ∗x,v g0
we obtain the estimate
kg(t, ·, ·)kL∞ (Rd ×Rd ) ≤ kG(t, ·, ·)kL∞ (Rd ×Rd ) kg0 kL1 (Rd ×Rd )
−d
= 4π et − 1 (t − 2) et + t + 2
∼ t−d e−d t
as t → +∞. As a consequence, we obtain that the solution of (24) with a
nonnegative initial datum f0 satisfies
kf0 kL1 (Rd ×Rd )
(1 + o(1)) as t → +∞ .
kf (t, ·, ·)kL∞ (Rd ×Rd ) =
(4π t)d
Using the simple Hölder interpolation inequality
1/p
1−1/p
kf kLp (Rd ×Rd ) ≤ kf kL1 (Rd ×Rd ) kf kL∞ (Rd ×Rd ) ,
we obtain the following decay result.
Corollary 11. If f is a solution of (24) with a nonnegative initial datum f0 ∈
L1 (Rd × Rd ), then we have the decay estimate
kf (t, ·, ·)kLp (Rd ×Rd ) =
kf0 kL1 (Rd ×Rd )
(4π t)d (1−1/p)
(1 + o(1))
as
t → +∞
for any p ∈ (1, +∞].
By taking f0 (x, v) = G(1, x, v), it is moreover straightforward to check that this
estimate is optimal. With p = 2, this also proves that the decay rates obtained in
Theorems 6 and 7 for the Fokker-Planck operator, i.e., case (a), are the optimal
ones.
HYPOCOERCIVITY WITHOUT CONFINEMENT
15
Acknowledgments. This work has been partially supported by the Projects STAB (J.D., ANR12-BS01-0019) and Kibord (E.B., J.D., ANR-13-BS01-0004) of the French National Research Agency
(ANR). Support by the Austrian Science Foundation (grants no. F65 and W1245) is acknowledged by
c 2017 by the authors. This
C.S., who is also grateful for the hospitality at Université Paris Dauphine. paper may be reproduced, in its entirety, for non-commercial purposes.
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(E. Bouin) CEREMADE (CNRS UMR n◦ 7534), PSL research university, Université
Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris 16, France
E-mail address: [email protected]
(J. Dolbeault) CEREMADE (CNRS UMR n◦ 7534), PSL research university, Université Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris 16, France
E-mail address: [email protected]
(S. Mischler) CEREMADE (CNRS UMR n◦ 7534), PSL research university, Université
Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris 16, France
E-mail address: [email protected]
(C. Mouhot) DPMMS, Center for Mathematical Sciences, University of Cambridge,
Wilberforce Road, Cambridge CB3 0WA, UK
E-mail address: [email protected]
(C. Schmeiser) Fakultät für Mathematik, Universität Wien, Oskar-MorgensternPlatz 1, 1090 Wien, Austria
E-mail address: [email protected]