Fermi-Dirac-Fokker-Planck equation: well-posedness and long-time asymptotics

0.29 MB·English

by  José A. Carrillo

Fermi-Dirac-Fokker-Planck Equation: Well-posedness
& Long-time Asymptotics
8 Jos´e A. Carrillo∗ Philippe Laurenc¸ot † & Jesu´s Rosado‡
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0
February 2, 2008
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n
a
J

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Abstract
1

] A Fokker-Planck type equation for interacting particles with exclusion principle
P
is analysed. The nonlinear drift gives rise to mathematical difficulties in controlling
A
moments of the distribution function. Assuming enough initial moments are finite,
.
h
we can show the global existence of weak solutions for this problem. The natural
t
a associated entropy of the equation is the main tool to derive uniform in time a
m
priori estimates for the kinetic energy and entropy. As a consequence, long-time
[ asymptotics in L1 are characterized by the Fermi-Dirac equilibrium with the same

1 initial mass. This result is achieved without rate for any constructed global solution
v
and with exponential rate due to entropy/entropy-dissipation arguments for initial
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8 data controlled by Fermi-Dirac distributions. Finally, initial data below radial solu-
8 tions with suitable decay at infinity lead to solutions for which the relative entropy
2
towards the Fermi-Dirac equilibrium is shown to converge to zero without decay
.
1
rate.
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: 1 Introduction
v
i
X
Kinetic equations for interacting particles with exclusion principle, such as fermions, have
r
a been introduced in the physics literature in [9, 12, 13, 15, 14, 23] and the review [10].
∗ICREA(Instituci´oCatalanadeRecercaiEstudisAvanc¸ats)andDepartamentdeMatema`tiques,Uni-
versitat Aut`onoma de Barcelona, E–08193 Bellaterra, Spain. E-mail: [email protected], Internet:
http://kinetic.mat.uab.es/∼carrillo/
†Institut de Math´ematiques de Toulouse, CNRS (UMR 5219) & Universit´e de Toulouse, 118 route
de Narbonne, F–31062 Toulouse C´edex 9, France. E-mail: [email protected], Internet:
http://www.mip.ups-tlse.fr/∼laurenco/
‡Departament de Matema`tiques, Universitat Aut`onoma de Barcelona, E–08193Bellaterra, Spain. E-
mail: [email protected]
1
Spatially inhomogeneous equations appear from formal derivations of generalized Boltz-
mann equations and Uehling-Uhlenbeck kinetic equations both for fermionic and bosonic
particles. The most relevant questions related to these problems concern their long-time
asymptotics and the rate of convergence towards global equilibrium if any.
The spatially inhomogeneous situation has been recently studied in [22], where the
long time asympotics of these models in the torus is shown to be given by spatially
homogeneous equilibrium given by Fermi-Dirac distributions when the initial data is not
far from equilibrium in a suitable Sobolev space. This nice result is based on techniques
developed in previous works [20, 21]. Other related mathematical results for Boltzmann-
type models have appeared in [7, 19].
In this work, we focus on the global existence of solutions and the convergence of solu-
tions towards global equilibrium in the spatially homogeneous case without any smallness
assumption on the initial data. Preliminary results in the one-dimensional setting were
reported in [5]. More precisely, we analyse in detail the following Fokker-Planck equation
for fermions, see for instance [10],
∂f
= ∆ f +div [vf(1−f)], v ∈ RN,t > 0, (1.1)
v v
∂t
with initial condition f(0,v) = f (v) ∈ L1(RN), 0 ≤ f ≤ 1 and suitable moment condi-
0 0
tions to be specified below. Here, f = f(t,v) is the density of particles with velocity v at
time t ≥ 0.
This equation has been proposed in order to describe the dynamics of classical inter-
acting particles, obeying the exclusion-inclusion principle in [12]. In fact, equation (1.1)
is formally equivalent to
∂f f |v|2
= div f(1−f)∇ log +
v v
∂t 1−f 2
(cid:20) (cid:18) (cid:18) (cid:19) (cid:19)(cid:21)
from which it is easily seen that Fermi-Dirac distributions defined by
1
Fβ(v) :=
|v|2
1+βe
2
with β ≥ 0 are stationary solutions. Moreover, for each value of M ≥ 0, there exists a
unique β = β(M) ≥ 0 such that Fβ(M) has mass M, that is, kFβ(M)k = M. Throughout
1
the paper we shall denote Fβ(M) by F .
M
Another striking property of this equation is the existence of a formal Liapunov func-
tional, related to the standard entropy functional for linear and nonlinear Fokker-Planck
models [4, 2], given by
1
H(f) := |v|2f(v)dv+ [(1−f)log(1−f)+f log(f)]dv.
2
RN RN
Z Z
2
We will show that this functional plays the same role as the H-functional for the spatially
homogeneous Boltzmann equation, see for instance [24]. In particular it will be crucial to
characterize long-time asymptotics of (1.1). In fact, the entropy method will be the basis
of the main results in this work; more precisely by taking the formal time derivative of
H(f), we conclude that
2
d f
H(f) = − f(1−f) v +∇ log dv ≤ 0.
v
dt 1−f
ZRN (cid:12) (cid:18) (cid:19)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Therefore, to show the global equilibration of solutions to (1.1) we need to find the right
(cid:12) (cid:12)
functional setting to show the entropy dissipation. Furthermore, if we succeed in relating
functionally the entropy and the entropy dissipation, we will be able to give decay rates
towards equilibrium. These are the main objectives of this work. Let us finally mention
that these equations are of interest as typical examples of gradient flows with respect to
euclidean Wasserstein distance of entropy functionals with nonlinear mobility, see [1, 3]
for other examples and related problems.
In section 2, we will show the global existence of solutions for equation (1.1) based on
fixedpointarguments, estimates involving moment boundsandtheconservationofcertain
properties of the solutions. The suitable functional setting is reminiscent of the one used
inequations sharing a similar structure andtechnical difficulties asthose treated in[8,11].
The main technical obstacle for the Fermi-Dirac-Fokker-Planck equation (1.1) lies in the
control of moments. Next, in section 3, we show that the constructed solutions verify
that the entropy is decreasing, and from that, we prove the convergence towards global
equilibrium without rate. Again, here the uniform-in-time control of the second moment
is crucial. Finally, we obtain an exponential rate of convergence towards equilibrium if
the initial data is controlled by Fermi-Dirac distributions and the convergence to zero of
the relative entropy when controlled by radial solutions.
2 Global Existence of Solutions
In this section, we will show the global existence of solutions to the Cauchy problem to
(1.1). We start by proving local existence of solutions together with a characterization
of the time-span of these solutions. Later, we show further regularity properties of these
solutions with the help of estimates on derivatives. Based on these estimates we can
derive further properties of the solutions: conservation of mass, positivity, L∞ bounds,
comparison principle, moment estimates and entropy estimates. All of these uniform
estimates allow us to show that solutions can be extended and thus exist for all times.
3
2.1 Local Existence
We will prove the local existence and uniqueness of solutions using contraction-principle
arguments as in [1, 8, 11] for instance. As a first step, let us note that we can write (1.1)
as
∂f
= ∆ f +div (vf)−div (vf2) (2.1)
v v v
∂t
and, due to Duhamel’s formula, we are led to consider the corresponding integral equation
t
f(t,v) = F(t,v,w)f (w)dw− F(t−s,v,w)(div (wf(s,w)2))dwds (2.2)
0 w
ZRN Z0 ZRN
where F(t,v,w)is thefundamental solutionfor thehomogeneous Fokker-Planck equation:
∂f
= div (vf +∇ f)
v v
∂t
given by
F(t,v,w) := a(t)−N2 Mν(t)(a(t)−21v −w)
with
a(t) := e−2t , ν(t) := e2t −1 and Ml(ξ) := (2πl)−N2 e−|ξ2|l2
for any λ > 0. Let us define the operator F(t,v)[g] acting on functions g as:
F(t,v)[g(w)] = F(t,v,w)g(w)dw. (2.3)
RN
Z
Note that by integration by parts, the expression F(t,v)[div (wf2(w))] is equivalent
w
to:
eNt −|etv−w|2
e 2(e2t−1) div (wf(w)2)dw
N w
ZRN (2π(e2t −1))2 !
= − ∇ eNt e−|2e(tev2−t−w1|2) ·w f(w)2dw
w N
ZRN " (2π(e2t −1))2 ! #
= − e−t(∇ F(t,v,w)·w)f(w)2dw
v
RN
Z
=: −e−t∇ F(t,v)[wf(w)2]
v
so that (2.2) becomes
t
f(t,v) = F(t,v)[f (w)]+ e−(t−s)∇ F(t−s,v)[wf(s,w)2]ds. (2.4)
0 v
Z0
4
We will now define a space in which the functional induced by (2.4)
t
T [f](t,v) := F(t,v)[f (w)]+ e−(t−s)∇ F(t−s,v)[wf(s,w)2]ds (2.5)
0 v
Z0
has a fixed point. To this end, we define the spaces Υ := L∞(RN) ∩ L1(RN) ∩ Lp (RN)
1 m
and Υ := C([0,T];Υ) with norms
T
kf(t)kΥ := max{kf(t)k∞,kf(t)kL11,kf(t)kLpm} and kfkΥT := 0m≤ta≤xT kf(t)kΥ
for any T > 0, where we omit the N-dimensional euclidean space RN for notational
convenience and
1
p
kfkLpm := k(1+|v|m)fkp and kfkp := |f|pdv .
(cid:18)ZRN (cid:19)
In the following, we will see that for p > N, p ≥ 2, and m ≥ 1 we can choose q and r
satisfying
Np p mp p
< ≤ r ≤ < p and ≤ q ≤ p (2.6)
N +p 2 m+1 2
such that kT [f]k is bounded by kfk . Let us fix such parameters p, m, r, q and
ΥT ΥT
0 ≤ t ≤ T. Due to Proposition A.1 and q ≤ p ≤ 2q, we can compute
t eN(t−s)
kT[f](t)k ≤ CeNtkf k + C k|w|f2(s)k ds
∞ 0 ∞ Z0 ν(t−s)2Nq+21 q
t eN(t−s) 2−p p
≤ CeNtkf k + C kf(s)k qkf(s)kq ds
0 ∞ Z0 ν(t−s)2Nq+21 ∞ Lpm
t eN(t−s)
≤ CeNtkf k + C dskfk2
0 ∞ Z0 ν(t−s)2Nq+21 ΥT
≤ CeNtkf k +CI (t)kfk2 ,
0 ∞ 1 ΥT
where
1
I1(t) := χ−21(N−Nq−1)−1(1−χ)−12(Nq+1)dχ < ∞
Ze−2t
by the choice (2.6) of q. In the same way, since r satisfies (m+1)r ≤ mp and 2r ≥ p, we
5
get
kT [f](t)kLpm ≤ CeNp′tkf0kLpm +Z0tCν(t−esNp)′N(2t−(s1r)−p1)+12k|w|f2(s)kLrmds
Nt t eNp′(t−s) 2−p p
≤ Cep′ kf0kLpm +Z0 Cν(t−s)N2(1r−p1)+12kf(s)k∞ rkf(s)kLrpmds
≤ CeNp′tkf0kLpm +Z0tCν(t−esNp)′N(2t−(s1r)−p1)+12 dskfk2ΥT
≤ CeNp′tkf0kLpm +CI2(t)kfk2ΥT,
where
1
I2(t) := χ−21hpN′−(N(r1−p1)+1)i−1(1−χ)−N2(1r−p1)−12dχ < ∞
Ze−2t
by the choice (2.6) of r.
Finally we can estimate
t C
kT[f](t)k ≤ Ckf k + k|w|f2(s)k ds
L11 0 L11 ν(t−s)1 L11
Z0 2
where by interpolation, we get as p ≥ 2 and m ≥ 1
k|w|f2k = (1+|w|)|w|f2dw ≤ (1+|w|)2f2dw
L1
1 RN RN
Z Z
p−2 1
p−1 p−1
≤ (1+|w|)f dw (1+|w|)pfpdw
(cid:18)ZRN (cid:19) (cid:18)ZRN (cid:19)
p−2 p
≤ kfkp−1kfkp−1. (2.7)
L11 Lpm
Consequently
1
kT[f](t)kL11 ≤ Ckf0kL11 +C χ−23(1−χ)−21dχkfk2ΥT.
Ze−2t
We next check the existence of a fixed point of (2.5) in Υ . To this end, we define a
T
sequence (f ) by f = T[f ] for n ≥ 0. Collecting all the above estimates, we can
n n≥1 n+1 n
write
kf (t)k ≤ C (N,t)kf k +C (N,p,q,r,t)kf k2
n+1 Υ 1 0 Υ 2 n ΥT
for any 0 ≤ t ≤ T and any T > 0, with
C (N,t) := CeNt
1
1
C2(N,p,q,r,t) := Cmax I1(t),I2(t), χ−23(1−χ)−21dχ
( Ze−2t )
6
which are clearly increasing with t and C (t) tends to 0 as t does. Thus, for any T > 0
2
kf k ≤ C (T)kf k +C (T)kf k2
n+1 ΥT 1 0 Υ 2 n ΥT
with C (T) = C (N,T) and C (T) = C (N,p,q,r,T), both being increasing functions of
1 1 2 2
T. We may also assume that C (T) ≥ 1 without loss of generality.
1
From now on, we will follow the arguments in [18]. We will first show that if T is small
enough, the functional T is bounded in Υ , which will in turn imply the convergence. Let
T
us take T > 0 and δ > 0 which verify
1
kf k < δ and 0 < δ < .
0 Υ
4C (T)C (T)
1 2
Then, let us prove by induction that kf k < 2C (T)δ for all n. By the choice of T and
n ΥT 1
δ we have kf k < C (T)δ < 2C (T)δ. If we suppose that kf k < 2C (T)δ, we have
0 Υ 1 1 n ΥT 1
kf k < C (T)δ +4C2(T)C (T)δ2 < 2C (T)δ,
n+1 ΥT 1 1 2 1
hence the claim. Now, computing the difference between two consecutive iterations of the
functional and proceeding with the same estimates as above, we can see for any 0 ≤ t ≤ T
that
t
kf −f k = e−(t−s)∇ F(t−s,v) w f2 −f2 ds
n+1 n ΥT v n n−1
(cid:13)(cid:13)Z0 (cid:2) (cid:2) (cid:3)(cid:3) (cid:13)(cid:13)ΥT
(cid:13) (cid:13)
≤ (cid:13)(cid:13)C2(T)sup fn +fn−1 ∞ fn −fn−1 ΥT (cid:13)(cid:13)
[0,T]
(cid:13) (cid:13) (cid:13) (cid:13)
≤ C (T) f(cid:13) + f(cid:13) (cid:13) f −(cid:13)f
2 n ΥT n−1 ΥT n n−1 ΥT
≤ 4C (T)(cid:16)C(cid:13) (T(cid:13))δ f (cid:13)−f (cid:13) (cid:17)(cid:13)≤ (4C (T)(cid:13)C (T)δ)n f −f .
1 (cid:13)2 (cid:13) n(cid:13) n−(cid:13)1 ΥT (cid:13) 1 (cid:13)2 1 0 ΥT
(cid:13) (cid:13) (cid:13) (cid:13)
Since 4C (T)C (T)δ < 1 we can conclude that there exists a function f in Υ which is
1 2 (cid:13) (cid:13) (cid:13)∗ T(cid:13)
a fixed point for T , and hence a solution to the integral equation (2.2). It is not difficult
to check that the solution f ∈ Υ to the integral equation is a solution of (1.1) in the
T
sense of distributions defining our concept of solution. We summarize the results of this
subsection in the following result.
Theorem 2.1 (Local Existence) Let m ≥ 1, p > N, p ≥ 2, and f ∈ Υ. Then there
0
exists T > 0 depending only on the norm of the initial condition f in Υ, such that (1.1)
0
has a unique solution f in C([0,T];Υ) with f(0) = f .
0
Remark 2.2 The previous theorem is also valid for f ∈ (L∞ ∩ Lp ∩ L1)(RN), with a
0 m
solution defined in C([0,T];(L∞∩Lp ∩L1)(RN)) but we will need to have the first moment
m
of the solution bounded in order to be able to extend it to a global in time solution. We
thus include here this additional condition.
7
Remark 2.3 With the same arguments used to prove Theorem 2.1 we can prove an equi-
valent result for the Bose-Einstein-Fokker-Planck equation
∂f
= ∆ f +div [vf(1+f)], v ∈ RN,t > 0.
v v
∂t
2.2 Estimates on Derivatives
Let us now work on estimates on the derivatives. By taking the gradient in the integral
equation, we obtain
t
∇ f(t,v) = ∇ F(t,v)[f(w)]− ∇ F(t−s,v)[div (wf2(s,w))]ds. (2.8)
v v v w
Z0
where ∇ F(t,v)[g] is defined as the vector:
v
∇ F(t,v)[g] := ∇ F(t,v,w)g(w)dw
v v
RN
Z
for the real-valued function g. Here, we will consider a space X with suitable weighted
T
norms for the derivatives
1 1
kfkXT = max kfkΥT, sup ν(t)2k∇vf(t)kLpm, sup ν(t)2k∇vf(t)kL11
(cid:26) 0<t<T 0<t<T (cid:27)
wherefornotationalsimplicity werefertok|∇vf|kLpm ask∇vfkLpm. LetusestimatetheLpm-
and L1-norms of ∇ f using again the results in Proposition A.1 as follows: for r ∈ [1,p)
v
satisfying (2.6)
e“pN′+1”t t
k∇vf(t)kLpm ≤ C ν(t)1 kf0kLpm + k∇vF[2f(w·∇wf)]+Nf2kLpmds
2 Z0
e“pN′+1”t t e“pN′+1”(t−s)
≤ C ν(t)1 kf0kLpm +C ν(t−s)1 kf(s)kLpmkf(s)k∞ds
2 Z0 2
t e“pN′+1”(t−s)
+C kf(w·∇ f)k ds
Z0 ν(t−s)N2(1r−p1)+21 w Lrm
≤ Ce“νpN(′t+)112”tkf0kLpm +Ckfk2ΥT Ze−12t χ−N2+p2′p′(1−χ)−12ds
+C sup ν(s)1/2kf(s)(w·∇ f(s))k I(t)
w Lr
m
0<s<T
(cid:8) (cid:9)
8
where
I(t) ≤ e−t 1 et“N+r′2r′”(1−χ)−(N2(1r−p1)+21)(χ−e−2t)−12dχ
2
Ze−2t
≤ 1et(Nr+′r′) 1+e2−2t 1−e−2t −N2(1r−p1)−12 (χ−e−2t)−21dχ
2 2
"Ze−2t (cid:18) (cid:19)
+ 1 (χ−e−2t)−(N2(1r−p1)−12 1−e−2t −21 dχ
1+e−2t 2 #
Z 2 (cid:18) (cid:19)
≤ CetNr+′r′(1−e−2t)−N2(r1−p1)
≤ Cet(N−Nr+1+Nr−Np)ν(t)−12ν(t)12−N2(r1−p1)
≤ Ch(t)ν(t)−12
with h(t) := et(Np+′p′)ν(t)12−N2(1r−p1) which is an increasing function of time with h(0) = 0
since p > r > Np/(N +p). It remains to estimate kf(w·∇ f)k :
w Lr
m
1
r
kf(w·∇ f)k ≤ C fr|∇ f|rdw + |w|(m+1)rfr|∇ f|rdw
w Lr w w
m
(cid:18)ZRN ZRN (cid:19)
Now, we can bound these integrals by using H¨older’s inequality to obtain
p−r r
pr p p
fr|∇wf|rdw ≤ fp−rdw |∇wf|pdw
ZRN (cid:18)ZRN (cid:19) (cid:18)ZRN (cid:19)
and
p−r r
pr pr p p
|w|(m+1)rfr|∇wf|rdw ≤ |w|p−rfp−rdw |w|mp|∇wf|pdw .
ZRN (cid:18)ZRN (cid:19) (cid:18)ZRN (cid:19)
Since p < pr/(p−r) ≤ mp or equivalently (m+1)r/m ≤ p < 2r by (2.6), we have for
any 0 < t ≤ T
kfk2r
fr|∇ f|rdw ≤ kfk2r−pkfkp−rk∇ fkr ≤ XT
w ∞ p w p r
ν(t)
RN 2
Z
and
kfk2r
|w|(m+1)rfr|∇ f|rdw ≤ kfk2r−pkfkp−rk∇ fkr ≤ XT .
RN w ∞ Lpm w Lpm ν(t)2r
Z
Putting together the above estimates we have shown that,
ν(t)1/2kf(t)(w·∇ f(t))k ≤ Ckfk2
w Lrm XT
9
and
ν(t)12k∇vf(t)kLpm ≤ C11(T,N,p)kf0kLpm +C21(T,N,p,r)kfk2XT (2.9)
with C1 and C1 increasing functions of T and for any 0 < t ≤ T. Analogously, we reckon
1 2
et t et−s
k∇ f(t)k ≤C kf k +C kf(s)k kf(s)k ds
v L11 ν(t)1 0 L11 ν(t−s)1 ∞ L11
2 Z0 2
t e(t−s)
+C kf(w·∇ f)(s)k ds
ν(t−s)1 w L11
Z0 2
where by taking p ≥ 2 and by interpolation as in (2.7), we have
1 1
kf(w·∇wf)kL1 ≤ k|w|2fk2k|w|2|∇wf|k2
1
p−2 p p−2 p
≤ kfk2(p−1)kfk2(p−1)k∇ fk2(p−1)k∇ fk2(p−1)
L11 Lpm w L11 w Lpm
kfk2
≤ XT .
ν(t)1/2
Putting together the last estimates, we deduce
ν(t)12k∇vf(t)kL11 ≤ C13(T,N,p)kf0kL11 +C23(T,N,p,r)kfk2XT (2.10)
with C3 and C3 increasing functions of T, for any 0 < t ≤ T. From (2.9) and (2.10) and
1 2
all the estimates of the previous section, we finally get
kfk ≤ C (T,N,p)kf k +C (T,N,p,r)kfk2
XT 1 0 Υ 2 XT
for any T > 0. From these estimates and proceeding as at the end of the previous section,
it is easy to show that we have uniform estimates in X of the iteration sequence and the
T
convergence of the iteration sequence in the space X . From the uniqueness obtained in
T
the previous section, we conclude that the solution obtained in this new procedure is the
same as before and lies in X . Summarizing, we have shown:
T
Theorem 2.4 Let m ≥ 1, p > N, p ≥ 2, and f ∈ Υ. Then there exists T > 0 depending
0
only on the norm of the initial condition f in Υ such that (1.1) has a unique solution
0
1
in C([0,T];Υ) with f(0) = f and velocity gradients verifying that t 7→ ν(t) |∇ f(t)| ∈
0 2 v
BC((0,T),(Lp ∩L1)(RN)).
m
2.3 Properties of the solutions
As (1.1) belongs to the general class of convection-diffusion equation, it enjoys several
classicalpropertieswhichwegatherinthissection. Theproofoftheseresultsusesclassical
approximation arguments, see [8, 25] for instance. Since these arguments are somehow
standard we will only give the detailed proof of the L1-contraction property below.
10