HAL Id: hal-01147848 https://hal.archives-ouvertes.fr/hal-01147848v2 Submitted on 5 May 2016 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. On global existence and trend to the equilibrium for the Vlasov-Poisson-Fokker-Planck system with exterior confining potential Frédéric Hérau, Laurent Thomann To cite this version: Frédéric Hérau, Laurent Thomann. On global existence and trend to the equilibrium for the Vlasov- Poisson-Fokker-Planck system with exterior confining potential. Journal of Functional Analysis, El- sevier, 2016, 271 (5), pp.1301–1340. 10.1016/j.jfa.2016.04.030. hal-01147848v2
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HAL Id: hal-01147848https://hal.archives-ouvertes.fr/hal-01147848v2
Submitted on 5 May 2016
HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.
On global existence and trend to the equilibrium for theVlasov-Poisson-Fokker-Planck system with exterior
To cite this version:Frédéric Hérau, Laurent Thomann. On global existence and trend to the equilibrium for the Vlasov-Poisson-Fokker-Planck system with exterior confining potential. Journal of Functional Analysis, El-sevier, 2016, 271 (5), pp.1301–1340. �10.1016/j.jfa.2016.04.030�. �hal-01147848v2�
Abstract. — We prove a global existence result with initial data of low regularity, and prove thetrend to the equilibrium for the Vlasov-Poisson-Fokker-Planck system with small non linear termbut with a possibly large exterior confining potential in dimension d = 2 and d = 3. The proof relieson a fixed point argument using sharp estimates (at short and long time scales) of the semi-groupassociated to the Fokker-Planck operator, which were obtained by the first author.
1. Introduction and results
1.1. Presentation of the equation. — Let d = 2 or d = 3. We consider the Vlasov-Poisson-
Fokker-Planck system (VPFP for short) with external potential, which reads, for (t, x, v) ∈
where x 7→ Ve(x) is a given smooth confining potential (see Assumption 1 below). The constant
ε0 ∈ R is the total charge of the system and in the sequel we assume that either ε0 > 0 (repulsive
case) or ε0 < 0 (attractive case) in the case d = 3. The constant γ > 0 is the friction-diffusion
coefficient, and for simplicity we will take γ = 1.
The unknown f is the distribution function of the particles. We assume that f0 ≥ 0 and that∫f0(x, v)dxdv = 1, it is then easy to check that once a good existence theory is given, these
properties are preserved, namely that for all t ≥ 0
f ≥ 0 and
∫f(t, x, v)dxdv = 1,
and we refer to Section 3.1 for more details and other basic results.
2000 Mathematics Subject Classification. — 35Q83; 35Q84;35B40.Key words and phrases. — Vlasov-Poisson-Fokker-Planck equation; non self-adjoint operator; global solutions,return to equilibrium.
F.H. is supported by the grant ”NOSEVOL” ANR-2011-BS01019-01. L.T. is supported by the grant “ANAE”ANR-13-BS01-0010-03.The authors warmly thank Jean Dolbeault for enriching discussions and are grateful to Laurent Di Menza whowas at the origin of this collaboration.
2 FREDERIC HERAU & LAURENT THOMANN
This equation is a model for a plasma submitted to an external confining electric field (in the
repulsive case) and also a model for gravitational systems (in the attractive case). When there
is no external potential (Ve = 0), the equation has been exhaustively studied. First existence
results were obtained by Victory and O’Dwyer in 2d [23] and by Rein and Weckler [26] in 3d for
small data. Bouchut [2] showed that the equation is globally well-posed in 3 dimensions using
the explicit kernel. The long time behavior (without any rate) has been studied with or without
external potential by Bouchut and Dolbeault in [3], Carillo, Soler and Vazquez [4], and also by
Dolbeault in [10].
When there is a confining potential, arbitrary polynomial trend to the equilibrium was es-
tablished in [7] where a first notion of hypocoercivity [29] was developed and used later to
the full model [8]. The exponential trend to the equilibrium was shown in the linear case (the
Fokker-Planck equation) for a general external confining potential in [18] (see also [16]). So
far, in the non-linear case, there is no general result about exponential trend to the equilibrium.
In the case of the torus (and V = 0), the strategy of Guo can be applied to many models (see
e.g. [12, 13, 14]). In the case when the potential is explicitly given by Ve(x) = C|x|2, a recent
result with small data is given in [20], following the micro-macro strategy of Guo.
In all previous cases (torus, Ve = 0 or polynomial of order 2), mention that one can compute
explicitly the Green function of the Fokker-Planck operator and also that exact computations
can be done thanks to vanishing commutators. Here instead we will rely on estimates (in short
and long time) of the linear solution of the Fokker-Planck operator obtained by the first author
in [17], and our approach allows us to deal with a large class of confining potentials Ve. Indeed,
in [17, Theorem 1.3 ] a first exponential trend to the equilibrium result for a VPFP type model
was given, but only for a mollified non-linearity. We will prove here a global existence result in
the full VPFP case, with trend to equilibrium assuming that the initial condition f0 is localised
and has some Sobolev regularity and under the assumption that the electric field is perturbative
in the sense that |ε0| ≪ 1.
Let us now precise our notations and hypotheses. We do not try to optimise the assumptions
on the confining potential Ve and first assume the following
Assumption 1. — The potential x 7→ Ve(x) satisfies
e−Ve ∈ S(Rd), with Ve ≥ 0 and V ′′e ∈ W∞,∞(Rd).
Observe that the assumption Ve ≥ 0 can be relaxed by assuming that Ve is bounded from
below and adding to it a sufficiently large constant.
We now introduce the Maxwellian of the equation (1.1)
(1.2) M∞(x, v) =e−(v2/2+Ve(x)+ε0U∞(x))
∫e−(v2/2+Ve(x)+ε0U∞(x))dxdv
,
where U∞ is a solution of the following Poisson-Emden type equation
(1.3) −∆U∞ =e−(Ve+ε0U∞)
∫e−(Ve(x)+ε0U∞(x))dx
.
Actually, one gets that under Assumption 1 and |ε0| small enough (assuming additionally that
ε0 > 0 in the case d = 2), the equation (1.3) has a unique (Green) solution U∞ which belongs
to W∞,∞(Rd) uniformly w.r.t |ε0| (see Propositions 3.5 and 3.6 following results from [9]). The
Maxwellian M∞ is then in S(Rdx × R
dv) and is the unique L1-normalised steady solution of
equation (1.1).
ON GLOBAL EXISTENCE FOR VLASOV-POISSON-FOKKER-PLANCK 3
In the case d = 2 and ε0 < 0, existence and uniqueness of solutions to (1.3) are unclear, that’s
why we do not consider this case.
For convenience, we now introduce the effective potential at infinity
(1.4) V∞def= Ve + ε0U∞ so that M∞(x, v) =
e−(v2/2+V∞(x))
∫e−(v2/2+V∞(x))dxdv
.
The second assumption on Ve is the following
Assumption 2. — The so-called Witten operator W = −∆x+|∂xVe|2/4−∆xVe/2 has a spectral
gap in L2(Rd). We denote by κ0 > 0 the minimum of this spectral gap and d/2.
Example 1.1. — As an example, we can check that if Ve satisfies Assumption 1 and is such
that
|∂xVe(x)| −→|x|−→∞
+∞
then it satisfies also Assumption 2 since it has a compact resolvent.
We introduce now the functional framework on which our analysis is done. We consider the
weighted space B built from the standard L2 space after conjugation with a half power of the
Maxwellian
(1.5) Bdef= M1/2
∞ L2 ={f ∈ S ′(R2d) s.t. f/M∞ ∈ L2(M∞dxdv)
}.
We define the natural scalar product
〈f, g〉 =
∫fgM−1
∞ dxdv,
and the corresponding norm
‖f‖2B = 〈f, f〉 =
∫f2M−1
∞ dxdv.
Next, consider the Fokker-Planck operator associated to the potential V∞ defined by
(1.6) K∞ = v.∂x − ∂xV∞(x).∂v − γ∂v. (∂v + v) .
The last object we need before writing our equation in a suitable way is the limit electric field
E∞(x) = ∂xU∞(x) = −1
|Sd−1|
x
|x|d⋆x
∫M∞(x, v)dv.
With all the previous notations, the VPFP equation (1.1) can be rewritten
(1.7)
∂tf +K∞f = ε0(E − E∞)∂vf,
E(t, x) = −1
|Sd−1|
x
|x|d⋆x ρ(t, x), where ρ(t, x) =
∫f(t, x, v)dv,
f(0, x, v) = f0(x, v).
We define the operator
Λ2x = −∂x.
(∂x + ∂xV∞
)+ 1
which is up to a conjugation with M1/2∞ the Witten operator introduced in Assumption 2 but
defined on B, and
Λ2v = −∂v.(∂v + v) + 1,
which is again up to a conjugation the harmonic oscillator in velocity. They both are non-
negative selfadjoint unbounded operators in B. We also introduce
Λ2 = −∂x.(∂x + ∂xV∞
)− ∂v.(∂v + v) + 1 = Λ2
x +Λ2v − 1.
4 FREDERIC HERAU & LAURENT THOMANN
It is clear that
1 ≤ Λ2x, Λ
2v ≤ Λ2.
As we mentioned previously, if Ve satisfies Assumptions 1 and 2, then V∞ = Ve + ε0U∞ also
does, and we check in Subsection 3.3 that the operator
−∂x.(∂x + ∂xV∞
)− ∂v.(∂v + v) = Λ2 − 1
has 0 as single eigenvalue and a spectral gap bounded in B which is, uniformly w.r.t |ε0| small,
bounded from below by κ0/2.
In the sequel, we will need the anisotropic chain of Sobolev spaces: for α, β ≥ 0
(1.8) Bα,β = Bα,βx,v (R
2d) ={f ∈ B : Λα
xf ∈ B and Λβvf ∈ B
},
and we endow this space by the norm
‖f‖Bα,β = ‖Λαxf‖B + ‖Λβ
v f‖B .
In the case α = β we simply define
Bα = Bα,αx,v (R
2d) ={f ∈ B : Λαf ∈ B
},
with the norm
‖f‖Bα = ‖Λαf‖B ∼ ‖f‖Bα,α .
We observe that M∞ ∈ Bα,β for all α, β ≥ 0, since we have M∞ ∈ S(R2d).
1.2. Main results. — We are now able to state our global well-posedness results.
Theorem 1.2. — Let d = 2 and let f0 ∈ B(R4). Assume moreover that Assumptions 1 and 2
are satisfied. Then if ε0 > 0 is small enough, there exists a unique global mild solution f to (1.1)
in the class
f ∈ C([0,+∞[ ; B(R4)
).
Moreover, the following convergence to equilibrium holds true
‖f(t)−M∞‖B ≤ C0e−κ0t/c, ∀t ≥ 1,
and
‖E(t)− E∞‖L∞(R2) ≤ C1e−κ0t/c, ∀t ≥ 1.
By mild, we mean f and E which satisfy the integral formulation of (1.7), namely
(1.9)
f(t) = e−K∞f0 + ε0
∫ t
0e−(t−s)K∞(E(s) − E∞)∂vf(s)ds,
E(t) = −1
|Sd−1|
x
|x|d⋆x
∫f(t)dv.
In the case d = 3, we need to assume more regularity on the initial condition, but the
known results about the uniqueness of the Poisson-Emden equation (see Subsection 3.2) allow
to consider also the case ε0 < 0.
Denote by
(1.10) U0 =1
4π|x|⋆x
∫f0dv,
which is such that ∆U0 =
∫f0dv. Then
ON GLOBAL EXISTENCE FOR VLASOV-POISSON-FOKKER-PLANCK 5
Theorem 1.3. — Let d = 3 and 1/2 < a < 2/3. Assume that f0 ∈ Ba,a(R6) ∩ L∞(R6) is such
that U0 ∈ W 2,∞(R3). Assume moreover that Assumptions 1 and 2 are satisfied. Then if |ε0| is
small enough, there exists a unique global mild solution f to (1.1) in the class
f ∈ C([0,+∞[ ; Ba,a(R6)
)∩ L∞
loc
([0,+∞[ ; L∞(R6)
).
Moreover, for all a ≤ α < 2/3 and a ≤ β < 1 such that 3α− 1 < β < 1
(1.11) f ∈ C(]0,+∞[ ; Bα,β(R6)
),
and the following convergence to equilibrium holds true
‖f(t)−M∞‖Bα,β ≤ C0e−κ0t/c, ∀t ≥ 1,
and
‖E(t)− E∞‖L∞(R3) ≤ C1e−κ0t/c, ∀t ≥ 1.
In the previous lines, the constants c, C1, C2 > 0 only depend on ‖V∞‖W 2,∞ where V∞ was
defined in (1.4), on ‖U0‖W 2,∞ and on f0.
Notice that in Theorem 1.3, the parameters (α, β) can be chosen independently from a. It
is likely that the assumption a < 2/3 is technical, but our proof needs that β < 1 (see e.g.
Corollary 2.6). Since in this work we focus on low regularity issues, we did not try to relax this
hypothesis.
It is likely that the assumption made on U0 is technical. It is needed here in order to guarantee
that the linearised equation near t = 0 enjoys reasonable spectral estimates. Observe (see
Remark 3.17 for more details), that the assumption f0 ∈ Ba,a(R6)∩L∞(R6) alone ensures that
U0 ∈ W 2,p(R3) for any 2 ≤ p < +∞.
An analogue of the regularizing estimate (1.11) can also be obtained in Theorem 1.2. This
can be proven by getting estimates in some spaces Bα,βx,v as in the proof of Theorem 1.3 (see
Section 5). We did not include it here in order to simplify the argument.
The proof uses estimates of e−tK∞ in the space B , obtained in [17] by the first author.
Theorem 1.3 extends [17, Theorem 1.3] where he considered a regularised version of the electric
field E in (1.1), which was so that E(t) ∈ L∞(R3) for any f ∈ B. Here we tackle this difficulty
by using the Sobolev regularity of f and a gain given by the integration in time. The proof
relies on a fixed point argument in a space based on Bα,β in the (x, v) variables, and allowing
an exponential decay in time.
As a consequence of Theorems 1.2 and 1.3, we directly obtain the exponential decay of the
relative entropy. Let us define
H(f(t),M∞) =
∫∫f(t) ln
( f(t)
M∞
)dxdv,
then
Corollary 1.4. — Let d = 2 or d = 3. Then under the assumptions of Theorem 1.2 or Theo-
rem 1.3, the corresponding solution f of (1.1) satisfies
0 ≤ H(f(t),M∞) ≤ Ce−κt/c,
where C, c > 0 only depend on second order derivatives of Ve + ε0U∞ and on f0.
We refer to [17, Corollary 1.4] for the proof of this result.
6 FREDERIC HERAU & LAURENT THOMANN
1.3. Notations and plan of the paper. —
Notations. — In this paper c, C > 0 denote constants the value of which may change from line
to line. These constants will always be universal, or uniformly bounded with respect to the other
parameters.
The rest of the paper is organised as follows. In Section 2 we prove some linear estimates on
e−tK (whereK is a generic linear Fokker-Planck operator). In Section 3 we gather some estimates
on solutions of (1.1). Finally, Sections 4 and 5 are devoted to the proofs of Theorems 1.2 and 1.3
with fixed points arguments.
2. Semi-group estimates
In this section, we denote by V a generic potential satisfying Assumptions 1 and 2. We also
denote by K the associated generic linear Fokker-Planck operator
As a conclusion, if ε0 > 0 is small enough, Φ has a unique fixed point in Γ1 ⊂ Z. This shows
the existence of a unique h ∈ C(]0,+∞[ ; Bα,β(R6)
)such that f = M∞+e−tKg0+h solves (1.1).
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Frederic Herau, Laboratoire de Mathematiques J. Leray, UMR 6629 du CNRS, Universite de Nantes, 2, ruede la Houssiniere, 44322 Nantes Cedex 03, France • E-mail : [email protected]
Laurent Thomann, Laurent Thomann, Institut Elie Cartan, Universite de Lorraine, B.P. 70239, F-54506Vandoeuvre-les-Nancy Cedex, France • E-mail : [email protected]