




















































CUBO, A Mathematical Journal

Vol. 24, no. 02, pp. 211–226, August 2022

DOI: 10.56754/0719-0646.2402.0211

Vlasov-Poisson equation in weighted Sobolev
space W m,p(w)

Cong He
1

Jingchun Chen
2, B

1Department of Mathematical Sciences,

University of Wisconsin-Milwaukee,

Milwaukee, WI 53201, USA.

conghe@uwm.edu

2Department of Mathematics and

Statistics, The University of Toledo,

Toledo, OH 43606, USA.

jingchunchen123@gmail.com B

ABSTRACT

In this paper, we are concerned about the well-posedness of

Vlasov-Poisson equation near vaccum in weighted Sobolev

space W m,p(w). The most difficult part comes from esti-

mates of the electronic term ∇xφ. To overcome this difficulty,

we establish the Lp-Lq estimates of the electronic term ∇xφ;

some weight is introduced as well to obtain the off-diagonal

estimate. The weight is also useful when it comes to control

the higher-order derivative term.

RESUMEN

En este art́ıculo, estamos interesados en que la ecuación de

Vlasov-Poisson está bien puesta cercana al vaćıo en el es-

pacio de Sobolev W m,p(w) con peso. La parte más dif́ıcil

proviene de estimaciones del término electrónico ∇xφ. Para

superar esta dificultad, establecemos las estimaciones Lp-Lq

del término electrónico ∇xφ; donde algún peso es también in-

troducido para obtener la estimación fuera de la diagonal. El

peso es también útil cuando se trata de controlar el término

de la derivada de alto orden.

Keywords and Phrases: Vlasov-Poisson, Lp-Sobolev, weighted estimates, Lp-Lq estimates.

2020 AMS Mathematics Subject Classification: 35Q83, 46E35, 35A01, 35A02.

Accepted: 24 March, 2022

Received: 29 June, 2021

c©2022 C. He et al. This open access article is licensed under a Creative Commons

Attribution-NonCommercial 4.0 International License.

http://cubo.ufro.cl/
https://doi.org/10.56754/0719-0646.2402.0211
https://orcid.org/0000-0001-9868-8728
mailto:jingchunchen123@gmail.com
https://orcid.org/0000-0002-8808-6077
mailto:conghe@uwm.edu
mailto:jingchunchen123@gmail.com


212 Cong He & Jingchun Chen CUBO
24, 2 (2022)

1 Introduction

Understanding the evolution of a distribution of particles over time is a major research area of

statistical physics. The Vlasov-Poisson equation is one of the key equations governing this evo-

lution. Specifically, it models particle behaviors with long range interactions in a non-relativistic

zero-magnetic field setting. Two principal types of long range interactions are Coulomb’s forces,

the electrostatic repulsion of similarly charged particles in a plasma, and Newtonian’s forces, the

gravitational attraction of stars in a galaxy. The general Cauchy’s problem for the Vlasov-Poisson

equation (VP equation) in n dimensional space is as follows:


















∂tf + v · ∇xf + ∇xφ · ∇vf = 0,

−∆xφ =

∫

Rn

f dv,

f(0, x, v) = f0(x, v),

(1.1)

where f(t, x, v) denotes the distribution function of particles, x ∈ Rn is the position, v ∈ Rn is the

velocity, and t > 0 is the time and n ≥ 3.

The Cauchy problem for the Vlasov-Poisson equation has been studied for several decades. The

first paper on global existence is due to Arsen’ev [3]. He showed the global existence of weak

solutions. Then in 1977 Batt [5] established the global existence for spherically symmetric data.

In 1981 Horst [9] extended the global classical solvability to cylindrically symmetric data. Next,

in 1985, Bardos and Degond [4] obtained the global existence for “small” data. Finally, in 1989

Pfaffelmoser [12] proved the global existence of a smooth solution with large data. Later, simpler

proofs of the same results were published by Schaeffer [13], Horst [10], and Lions and Pertharne

[11]. Nevertheless, most of them were concerned about solutions in L∞ or continuous function

spaces. Also, there are many papers studying Vlasov-Poisson-Boltzmman (Landau) equation in

L2 setting, see [2, 6, 7, 8] and the references therein. A natural question is whether we can obtain

the solutions in Lp context, for example, W m,p spaces. This becomes our main theme in this

paper.

In this paper, our aim is to construct the solution to (1.1) in W m,p space. The difficulty lies in the

absence of Lp estimates of the electronic term ∇xφ. To handle this issue, we establish the L
p-Lq

off-diagonal estimates of ∇xφ which is highly important in estimating the higher order derivative

term. Also, it is necessary to introduce a weight w in order to obtain this off-diagonal estimate.

It is worthy to mention that this weight is crucial to deal with the higher-order derivative term.



CUBO
24, 2 (2022)

Vlasov-Poisson equation in weighted Sobolev space W m,p(w) 213

2 Preliminaries and main theorem

2.1 Notations and definitions

We first would like to introduce some notations.

• Given a locally integrable function f, the maximal function Mf is defined by

(Mf)(x) = sup
δ>0

1

|B(x, δ)|

∫

B(x,δ)

|f(y)|dy, (2.1)

where |B(x, δ)| is the volume of the ball of B(x, δ) with center x and radius δ.

• Weight w(v) = 〈v〉
γ
, γ · p

′

p
> n, 1

p
+ 1

p
′ = 1, n is the dimension.

• ‖f‖
p

L
p
x,v(w)

=:

∫

R2n

|f|pw dxdv.

• Define the higher-order energy norm as

E(f(t)) =: ‖f‖W m,p(w) =
∑

|α|+|β|≤m

‖∂αx ∂
β
v f(t)‖

p

L
p
x,v(w)

,

and

E(f0) =: E(f(0)) =
∑

|α|+|β|≤m

‖∂αx ∂
β
v f0‖

p

L
p
x,v(w)

,

where m ≥ 5 and n
3
< p < n

2
, n ≥ 3. Here α and β denote multi-indices with length |α|

and |β|, respectively. If each component of α1 is not greater than that of α, we denote the

condition by α1 ≤ α. We also define α1 < α if α1 ≤ α and |α1| < |α|. We also denote
(
α1
α

)

by Cα1α .

• A . B means there exists a constant c > 1 independent of the main parameters such that

A ≤ cB. A ∼ B means A . B and B . A.

Now we are ready to state our main theorem.

Theorem 2.1. For any sufficiently small M > 0, there exists T ∗(M) > 0 such that if

E(f0) =
∑

|α|+|β|≤m

‖∂αx ∂
β
v f0‖

p

L
p
x,v(w)

≤
M

2
,

then there is a unique solution f(t, x, v) to Vlasov-Poisson system (1.1) in [0, T ∗(M)) × Rn × Rn

such that sup
0≤t≤T ∗

E(f(t)) ≤ M, where m > n
p
+ 1 with n ≥ 3 and n

3
< p < n

2
.



214 Cong He & Jingchun Chen CUBO
24, 2 (2022)

Remark 2.2.

• One should pay attention to the differential index m in W m,p(Rn) which represents the weak

derivative, is not the classical derivative in C2(Rn). Indeed, for the space W 4,1.4(R6) in which

we could obtain solutions that could not be embedded into C(R6) (the continuous function

space) or L∞(R6), not to mention C2(R6) (the twice continuously differentiable function

space) due to the fact 4 · 1.4 < 6, i.e. W 4,1.4(R6) 6→֒ C2(R6) which implies that the classical

results in [3, 4] and [9]-[13] could not cover our results.

• In [4], C. Bardos and P. Degond also imposed the pointwise condition like

0 ≤ uα,0(x, v) ≤
ǫ

(1 + |x|)4 · (1 + |v|)4
.

However, the polynomial decay in the x variable is not needed at all in our proofs.

• Our working space W m,p(Rn) has more flexibility than C2(Rn) because of the triplet (m, n, p)

which implies that we can obtain the solutions in more spaces.

Let us illustrate our strategies for proving Theorem 2.1. As is known, the routine to prove the

existence of solution is to get a uniform-in-k estimate for the energy norm E(fk+1(t)). In this paper,

we adopt the Lp version energy method, i.e. to do the dual with |∂αx ∂
β
v f

k+1|p−2(∂αx ∂
β
v f

k+1)w (see

(4.5)). We expect all the estimates Ji in Section 4 can be controlled by

E(f(t)) =:
∑

|α|+|β|≤m

‖∂αx ∂
β
v f(t)‖

p

L
p
x,v(w)

.

To achieve our goal, some estimates related to the electronic term ∇xφ are needed. The L
p-Lq

estimate is established to deal with the higher-order derivative. For instance, when |α| = m, the

Lp-Lq estimate comes in to handle the highest order derivative term ∂αx ∇xφ
k :

〈

∂αx ∇xφ
k · ∇vf

k+1, |∂αx f
k+1|p−2 · ∂αx f

k+1 · w
〉

. ‖∂αx ∇xφ
k‖Lqx‖∇vf

k+1‖Ln
x,v

(w)‖∂
α
x f

k+1‖
p−1
L

p
x,v(w)

.

(2.2)

In turn, in order to get this Lp-Lq estimate involving ∇xφ, we introduce weight w; surprisingly, this

weight w also plays another crucial role to deal with the higher order derivative. More precisely,

we do this trick when |α| + |β| = m, w could “absorb” the extra derivative in ∇v as follows:

〈

∇xφ
k · ∇v∂

α
x ∂

β
v f

k+1
, |∂αx ∂

β
v f

k+1|p−2(∂αx ∂
β
v f

k+1)w
〉

∼
〈

∇xφ
k · ∇v|∂

α
x ∂

β
v f

k+1|p, w
〉

∼ −
〈

∇xφ
k · |∂αx ∂

β
v f

k+1|p, ∇vw
〉

.

(2.3)

Before we give the proof of the main theorem, we would like to establish the following Lp-Lq

estimates.



CUBO
24, 2 (2022)

Vlasov-Poisson equation in weighted Sobolev space W m,p(w) 215

3 Lp-Lq estimates

In this section, we are going to prove the Lp-Lq estimate which plays an essentially important role

in our proofs.

Lemma 3.1. Suppose 1 < p <
n

2
and

1

q
=

1

p
−

1

n
. If −∆φ =

∫

Rn

fdv =: g, then it holds that

‖∇xφ‖Lq(Rn) . ‖g‖Lp(Rn), (3.1)

Proof. Note that ∇xφ = ∇x(I2 ∗ g), with I2(x) =
1

(n − 2)ωn−1
·

1

|x|n−2
, for more details, see the

last section Appendix. Therefore there holds

‖∇xφ‖Lq(Rn) = ‖∇x(I2 ∗ g)‖Lq(Rn)

. ‖(Mg)
1
2 · (I2 ∗ |g|)

1
2 ‖Lq(Rn)

. ‖(Mg)
1
2 ‖Lq1 (Rn) · ‖(I2 ∗ |g|)

1
2 ‖Lq2 (Rn)

. ‖Mg‖
1
2

L
q1
2 (Rn)

· ‖I2 ∗ |g|‖
1
2

L
q2
2 (Rn)

,

where we applied (5.3) in the second line, and Hölder’s inequality with

1

q1
+

1

q2
=

1

q
, qi > 1,

in the third line separately.

On the one hand, the boundedness of Hardy-Littlewood operator M as defined by identity (2.1)

yields that

‖Mg‖
L

q1
2 (Rn)

. ‖g‖
L

q1
2 (Rn)

= ‖g‖Lp(Rn), (3.2)

since we require that
q1

2
= p, i.e.

2

q1
=

1

p
. (3.3)

On the other hand, by Lemma 5.3, we have

‖I2 ∗ |g|‖
L

q2
2 (Rn)

. ‖g‖Lp(Rn), (3.4)

where
2

q2
=

1

p
−

2

n
. (3.5)

Consequently, ‖∇xφ‖Lq(Rn) . ‖g‖
1
2

Lp(Rn)
· ‖g‖

1
2

Lp(Rn)
= ‖g‖Lp(Rn).

A “derivative version” is immediate:



216 Cong He & Jingchun Chen CUBO
24, 2 (2022)

Corollary 3.2. With the same assumptions as in Lemma 3.1, we have

‖∂αx ∇xφ‖Lq(Rn) . ‖∂
α
x g‖Lp(Rn).

Proof. One only needs to observe that

∇x∂
α
x φ = ∂

α
x ∇xφ = ∂

α
x ∇x(I2 ∗ g) = ∇x(I2 ∗ ∂

α
x g).

Applying Lemma 3.1 with φ and g replaced by ∂αφ and ∂αg respectively, the desired result is

immediate.

Now we adapt Corollary 3.2 to the “kinetic version”. To achieve this goal, we need to introduce a

weight w.

Corollary 3.3. Take g =

∫

Rn

fdv in Corollary 3.2, then we have

‖∂αx ∇xφ‖Lqx(Rn) . ‖∂
α
x f‖Lpx,v(w).

Proof. Hölder’s inequality leads to

∣
∣
∣
∣

∫

Rn

∂
α
x fdv

∣
∣
∣
∣
.

(∫

Rn

|∂αx f|
p
wdv

) 1
p
(∫

Rn

w
−

p
′

p dv

) 1
p
′

.

Note that w = 〈v〉
γ
and γ · p

′

p
> n, which implies that

(∫

Rn

w
−

p
′

p dv

) 1
p
′

≤ c.

Thus we end the proof of Corollary 3.3.

An L∞ estimate is also needed in the proof of the main Theorem 2.1.

Lemma 3.4. Suppose −∆φ =

∫

Rn

fdv. If 0 ≤ |α| ≤ m − 2, m ≥ 3, then

‖∂αx ∇xφ‖L∞x .
∑

|i|≤2

‖∂i+αx f‖Lpx,v(w). (3.6)

Proof. Choose a q such that q > n
2
and p ≤ q, then W 2,q →֒ L∞. Thus we have

‖∂αx ∇xφ‖L∞x . ‖∂
α
x ∇xφ‖W 2,q(Rnx ).

Combining Corollary 3.2 and Corollary 3.3 leads to

‖∂αx ∇xφ‖W 2,q =
∑

|i|≤2

‖∂ix∂
α
x ∇xφ‖Lqx .

∑

|i|≤2

‖∂i+αx f‖Lpx,v(w),

i.e.

‖∂αx ∇xφ‖L∞x .
∑

|i|≤2

‖∂i+αx f‖Lpx,v(w).



CUBO
24, 2 (2022)

Vlasov-Poisson equation in weighted Sobolev space W m,p(w) 217

4 Proof of main theorem

Now we are in the position to prove Theorem 2.1. We split the proof into two parts which are

existence and uniqueness.

Part I: Proof of existence. To prove the existence of the solution to (1.1), we adopt the Lp-

version energy method and iteration method. In this process, we will apply the Lp-Lq

estimate of electronic term ∇xφ proved in Lemma 3.1 to estimate J3.

Proof. We consider the following iterating sequence for solving the Vlasov-Poisson system

(1.1),



















∂tf
k+1 + v · ∇xf

k+1 + ∇xφ
k · ∇vf

k+1 = 0,

− ∆φk =

∫

Rn

fkdv,

fk+1(0, x, v) = f0(x, v).

(4.1)

(4.2)

(4.3)

Step 1. Applying ∂αx ∂
β
v to (4.1) with β 6= 0, |α|+|β| ≤ m, starting with f

0(t, x, v) = f0(x, v),

we have

(∂t + v · ∇x + ∇xφ
k · ∇v)∂

α
x ∂

β
v f

k+1 +
∑

β1<β

C
β1
β
∂β−β1v v · ∂

β1
v ∇x∂

α
x f

k+1

= −
∑

06=α1≤α

Cα1α ∂
α1
x ∇xφ

k · ∂α−α1x ∂
β
v ∇vf

k+1.

(4.4)

Multiplying |∂αx ∂
β
v f

k+1|p−2(∂αx ∂
β
v f

k+1)w on both sides of (4.4), and then integrating

over Rnx × R
n
v yields that

1

p

d

dt
‖∂αx ∂

β
v f

k+1‖
p

L
p
x,v(w)

+
∑

β1<β

C
β1
β

〈

∂β−β1v v · ∂
β1
v ∇x∂

α
x f

k+1, |∂αx ∂
β
v f

k+1|p−2(∂αx ∂
β
v f

k+1)w
〉

︸ ︷︷ ︸

J1

=
〈

∇xφ
k · |∂αx ∂

β
v f

k+1|p, ∇vw
〉

︸ ︷︷ ︸

J2

−
∑

06=α1≤α

C
α1
α

〈

∂
α1
x ∇xφ

k · ∂α−α1x ∂
β
v ∇vf

k+1
, |∂αx ∂

β
v f

k+1|p−2(∂αx ∂
β
v f

k+1)w
〉

︸ ︷︷ ︸

J3

.

(4.5)

We now estimate (4.5) term by term.



218 Cong He & Jingchun Chen CUBO
24, 2 (2022)

For J1, note that |∂
β−β1
v v| ≤ c, β1 < β. Thus,

J1 .
∑

β1<β

∫

R2n

|∂β1v ∇x∂
α
x f

k+1|w
1
p · |∂αx ∂

β
v f

k+1|p−1w
1

p
′
dxdv

.
∑

β1<β

(∫

R2n

|∂β1v ∇x∂
α
x f

k+1|pw dxdv

) 1
p
(∫

R2n

|∂αx ∂
β
v f

k+1|(p−1)p
′

w dxdv

) 1
p
′

.
∑

β1<β

‖∂β1v ∇x∂
α
x f

k+1‖Lpx,v(w) · ‖∂
α
x ∂

β
v f

k+1‖
p−1
L

p
x,v(w)

,

where 1
p
+ 1

p
′ = 1, i.e. (p − 1)p

′

= p, p
p
′ = p − 1.

For J2, note |∇vw| ≤ w, by Lemma 3.4, we have

J2 . ‖∇xφ
k‖L∞

x
‖∂αx ∂

β
v f

k+1‖
p

L
p
x,v(w)

.
∑

|i|≤2

‖∂ixf
k‖Lpx,v(w)‖∂

α
x ∂

β
v f

k+1‖
p

L
p
x,v(w)

.

For J3, we consider two cases individually.

Case 1: Recall |α| ≤ m − 1, if 0 < |α1| ≤ m − 2, m ≥ 3, Lemma 3.4 leads to

‖∂α1x ∇xφ
k‖L∞

x
.

∑

|i|≤2

‖∂i+α1x f
k‖Lpx,v(w).

Note |i| + |α1| ≤ m − 2 + 2 = m, the order of the derivatives does not exceed m, then

we obtain,

J3 .
∑

0<|α1|≤m−2

∫

Rn

‖∂α1x ∇xφ
k‖L∞

x
‖∂α−α1x ∂

β
v ∇vf

k+1‖Lpx

∥
∥
∥|∂

α
x ∂

β
v f

k+1|p−1
∥
∥
∥
L

p
′

x

w dv

.
∑

0<|α1|≤m−2

∑

|i|≤2

‖∂i+α1x f
k‖Lpx,v(w)‖∂

α−α1
x ∂

β
v ∇vf

k+1‖Lpx,v(w)‖∂
α
x ∂

β
v f

k+1‖
p−1
L

p
x,v(w)

,

where |α − α1| + |β| + 1 ≤ |α| + |β| ≤ m.

Case 2: |α1| = m − 1, we have

J3 .
∑

|α1|=m−1

∫

Rn

w(v)‖∂α1x ∇xφ
k‖Lqx‖∂

α−α1
x ∂

β
v ∇vf

k+1‖Ln
x

∥
∥
∥|∂

α
x ∂

β
v f

k+1|p−1
∥
∥
∥
L

p
′

x

dv

.
∑

|α1|=m−1

‖∂α1x ∇xφ
k‖Lqx

∫

Rn

w(v)
∑

|i|≤m−2

‖∂ix∂
α−α1
x ∂

β
v ∇vf

k+1‖Lpx

∥
∥
∥|∂

α
x ∂

β
v f

k+1|p−1
∥
∥
∥
L

p
′

x

dv

.
∑

|α1|=m−1

‖∂α1x f
k‖Lpx,v(w)

∑

|i|≤m−2

‖∂ix∂
α−α1
x ∂

β
v ∇vf

k+1‖Lpx,v(w)‖∂
α
x ∂

β
v f

k+1‖
p−1
L

p
x,v(w)

,

where in the first inequality, we applied Hölder’s inequality with respect to x with

1

q
+

1

n
+

1

p
′ = 1,

1

p
+

1

p
′ = 1.

And in the second inequality, we used the embedding W m−2,p →֒ Ln, with

m >
n

p
+ 1, p ≤ n. (4.6)



CUBO
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Vlasov-Poisson equation in weighted Sobolev space W m,p(w) 219

In the third inequality, we applied Corollary 3.3 and Hölder’s inequality in v.

Finally, plugging all the estimates of J1, J2, and J3 into (4.5) yields that

d

dt
‖∂αx ∂

β
v f

k+1‖
p

L
p
x,v(w)

.
∑

β1<β

‖∂β1v ∇x∂
α
x f

k+1‖Lpx,v(w) · ‖∂
α
x ∂

β
v f

k+1‖
p−1
L

p
x,v(w)

+
∑

|i|≤2

‖∂ixf
k‖Lpx,v(w)‖∂

α
x ∂

β
v f

k+1‖
p

L
p
x,v(w)

+
∑

0<|α1|≤m−2

∑

|i|≤2

‖∂i+α1x f
k‖Lpx,v(w)‖∂

α−α1
x ∂

β
v ∇vf

k+1‖Lpx,v(w)‖∂
α
x ∂

β
v f

k+1‖
p−1
L

p
x,v(w)

+
∑

|α1|=m−1

∑

|i|≤m−2

‖∂α1x f
k‖Lpx,v(w)‖∂

i
x∂

α−α1
x ∂

β
v ∇vf

k+1‖Lpx,v(w)‖∂
α
x ∂

β
v f

k+1‖
p−1
L

p
x,v(w)

.

(4.7)

Step 2. β = 0, |α| ≤ m, applying ∂αx to (4.1) on both sides, we have

(∂t + v · ∇x + ∇xφ
k · ∇v)∂

α
x f

k+1 = −
∑

06=α1≤α

Cα1α ∂
α1
x ∇xφ

k · ∂α−α1x ∇vf
k+1. (4.8)

We could completely repeat the process of step 1, the only difference is that we do not

need to estimate J1, thus we give the estimates as below but omit the process of proof

in details.

d

dt
‖∂αx f

k+1‖
p

L
p
x,v(w)

.
∑

|i|≤2

‖∂ixf
k‖Lpx,v(w)‖∂

α
x f

k+1‖
p

L
p
x,v(w)

+
∑

0<|α1|≤m−2

∑

|i|≤2

‖∂i+α1x f
k‖Lpx,v(w)‖∂

α−α1
x ∇vf

k+1‖Lpx,v(w)‖∂
α
x f

k+1‖
p−1
L

p
x,v(w)

+
∑

m−1≤|α1|≤m

∑

|i|≤m−2

‖∂α1x f
k‖Lpx,v(w)‖∂

i
x∂

α−α1
x ∇vf

k+1‖Lpx,v(w)‖∂
α
x f

k+1‖
p−1
L

p
x,v(w)

.

(4.9)

Collecting the estimates of J1, J2 and J3 and integrating over [0, t] of (4.5), summing

over |α| + |β| ≤ m, we deduce from the definition of E(f(t)) that

E(fk+1(t)) ≤ E(f0) + Ct sup
0≤s≤t

E(fk+1(s)) + Ct sup
0≤s≤t

(E(fk(s)))
1
p · sup

0≤s≤t
E(fk+1(s)).

Inductively, assume sup
0≤s≤T ∗(M)

E(fk(s)) ≤ M, T ∗(M) and M are sufficiently small; note

that f0(t, x, v) ≡ f0(x, v), E(f0) ≤
M
2
, we have

E(fk+1(t)) ≤
M

2
+ Ct sup

0≤s≤t
E(fk+1(s)) + CM

1
p · t sup

0≤s≤t
E(fk+1(s)),

i.e.

(1 − CT ∗ − CM
1
p T ∗(M)) sup

0≤s≤T ∗(M)

E(fk+1(s)) ≤
M

2
.

Thus sup
k

sup
0≤s≤T ∗(M)

E(fk(s)) ≤ M, i.e. we get a uniform-in-k estimate.

As a routine, let k → ∞, we obtain the solution and complete the proof of existence.



220 Cong He & Jingchun Chen CUBO
24, 2 (2022)

Remark 4.1. We summarize the indices as follows:

























































2

q1
=

1

p
, 1 < p <

n

2
, n ≥ 3,

2

q2
=

1

p
−

2

n
,

1

q1
+

1

q2
=

1

q
, q > 1, qi > 1, i = 1, 2,

q >
n

2
,

m >
n

p
+ 1, m, n ∈ N,

1

q
+

1

p1
=

1

p
,

γ > n(p − 1).

(4.10)

(4.11)

(4.12)

(4.13)

(4.14)

(4.15)

(4.16)

In fact, for any given (m, n, p) satisfying














m > n
p
+ 1, m ∈ N,

n ≥ 3,

n
3
< p < n

2
,

(4.17)

we could designate












q1 = 2p,

q2 =
2np
n−2p

,

q =
np
n−1

.

(4.18)

Let us move on to proving the uniqueness.

Part II: Proof of uniqueness. The proof of the uniqueness is analogous to the existence part.

However, we use a different energy norm E1(f(t)) =:
∑

|α|+|β|≤m−1

‖∂αx ∂
β
v f(t)‖

p

L
p
x,v(w)

because

of a difficult term J̃4. In J̃4, there is a term

〈

∇x(φf − φg) · ∂
α
x ∂

β
v ∇vg, |∂

α
x ∂

β
v (f − g)|

p−2 · ∂αx ∂
β
v (f − g)w

〉

.

If we still work with E(f(t)) =
∑

|α|+|β|≤m

‖∂αx ∂
β
v f(t)‖

p

L
p
x,v(w)

as in the existence part, the order

of derivative of ∂αx ∂
β
v ∇vg will be m + 1 which exceeds m when |α| + |β| = m. This is the

main reason we choose E1(f(t)) instead of E(f(t)).



CUBO
24, 2 (2022)

Vlasov-Poisson equation in weighted Sobolev space W m,p(w) 221

Proof. Assume another solution g exists such that sup
0≤s≤T ∗

E(g(s)) ≤ M, taking the difference,

we have 
















(∂t + v · ∇x + ∇xφf · ∇v)(f − g) + (∇xφf − ∇xφg) · ∇vg = 0,

−∆x(φf − φg) =

∫

Rn

(f − g) dv,

f(0, x, v) = g(0, x, v).

(4.19)

Step 1. Applying ∂αx ∂
β
v on both sides of (4.19)1 with β 6= 0, |α| + |β| ≤ m − 1, we have

(∂t + v · ∇x + ∇xφf · ∇v) · ∂
α
x ∂

β
v (f − g) +

∑

β1<β

C
β1
β
∂β−β1v v · ∂

β1
v ∇x∂

α
x (f − g)

= −
∑

06=α1≤α

Cα1α ∂
α1
x ∇xφf · ∂

α−α1
x ∂

β
v ∇v(f − g)

−
∑

0≤α1≤α

C
α1
α ∂

α1
x (∇xφf − ∇xφg) · ∂

α−α1
x ∂

β
v ∇vg.

(4.20)

Multiplying |∂αx ∂
β
v (f −g)|

p−2·∂αx ∂
β
v (f −g)w on both sides of (4.20), and then integrating

over Rnx × R
n
v yields that

1

p

d

dt
‖∂αx ∂

β
v (f − g)‖

p

L
p
x,v(w)

+
∑

β1<β

C
β1
β

〈

∂β−β1v v · ∂
β1
v ∇x∂

α
x (f − g), |∂

α
x ∂

β
v (f − g)|

p−2 · ∂αx ∂
β
v (f − g)w

〉

︸ ︷︷ ︸

J̃1

=
〈

∇xφf · |∂
α
x ∂

β
v (f − g)|

p, ∇vw
〉

︸ ︷︷ ︸

J̃2

−
∑

06=α1≤α

Cα1α

〈

∂α1x ∇xφf · ∂
α−α1
x ∂

β
v ∇v(f − g), |∂

α
x ∂

β
v (f − g)|

p−2 · ∂αx ∂
β
v (f − g)w

〉

︸ ︷︷ ︸

J̃3

−
∑

0≤α1≤α

C
α1
α

〈

∂
α1
x (∇xφf − ∇xφg) · ∂

α−α1
x ∂

β
v ∇vg, |∂

α
x ∂

β
v (f − g)|

p−2 · ∂αx ∂
β
v (f − g)w

〉

︸ ︷︷ ︸

J̃4

.

(4.21)

We could repeat the estimates in the proof of the existence except for some special term.

Thus we would like to write down the estimates directly without the details.

For J̃1, we have

J̃1 .
∑

β1<β

‖∂β1v ∇x∂
α
x (f − g)‖Lpx,v(w) · ‖∂

α
x ∂

β
v (f − g)‖

p−1
L

p
x,v(w)

.

For J̃2, we get

J̃2 .
∑

|i|≤2

‖∂ixf‖Lpx,v(w) · ‖∂
α
x ∂

β
v (f − g)‖

p

L
p
x,v(w)

.



222 Cong He & Jingchun Chen CUBO
24, 2 (2022)

For J̃3, since 0 < |α1| ≤ m − 2, we have

J̃3 .
∑

0<|α1|≤m−2

∑

|i|≤2

‖∂i+α1x f‖Lpx,v(w) · ‖∂
α−α1
x ∂

β
v ∇v(f − g)‖Lpx,v(w) · ‖∂

α
x ∂

β
v (f − g)‖

p−1
L

p
x,v(w)

,

where |α − α1| + |β| + 1 ≤ |α| + |β| ≤ m − 1.

For J̃4, note that −∆x(φf − φg) =

∫

Rn

(f − g) dv. We consider two cases separately.

Case 1: 0 ≤ |α1| ≤ m − 3

J̃4 .
∑

0≤|α1|≤m−3

∑

|i|≤2

‖∂i+α1x (f − g)‖Lpx,v(w) · ‖∂
α−α1
x ∂

β
v ∇vg‖Lpx,v(w) · ‖∂

α
x ∂

β
v (f − g)‖

p−1
L

p
x,v(w)

,

where |i| + |α1| ≤ 2 + m − 3 = m − 1 and

|α − α1| + |β| + 1 ≤ |α| + |β| − |α1| + 1 ≤ m − 1 − |α1| + 1 ≤ m.

Case 2: |α1| = m − 2

J̃4 .
∑

|α1|=m−2

∑

|i|≤m−2

‖∂α1x (f − g)‖Lpx,v(w) · ‖∂
i
x∂

α−α1
x ∂

β
v ∇vg‖Lpx,v(w) · ‖∂

α
x ∂

β
v (f − g)‖

p−1
L

p
x,v(w)

,

where |i| + |α − α1| + |β| + 1 ≤ m − 2 + |α| − |α1| + |β| + 1 ≤ m.

Collecting all the estimates of J̃j, j = 1, 2, 3, 4, we have

d

dt
‖∂αx ∂

β
v (f − g)‖

p

L
p
x,v(w)

.
∑

β1<β

‖∂β1v ∇x∂
α
x (f − g)‖Lpx,v(w) · ‖∂

α
x ∂

β
v (f − g)‖

p−1
L

p
x,v(w)

+
∑

|i|≤2

‖∂ixf‖Lpx,v(w) · ‖∂
α
x ∂

β
v (f − g)‖

p

L
p
x,v(w)

+
∑

0<|α1|≤m−2

∑

|i|≤2

‖∂i+α1x f‖Lpx,v(w) · ‖∂
α−α1
x ∂

β
v ∇v(f − g)‖Lpx,v(w) · ‖∂

α
x ∂

β
v (f − g)‖

p−1
L

p
x,v(w)

+
∑

0≤|α1|≤m−3

∑

|i|≤2

‖∂i+α1x (f − g)‖Lpx,v(w) · ‖∂
α−α1
x ∂

β
v ∇vg‖Lpx,v(w) · ‖∂

α
x ∂

β
v (f − g)‖

p−1
L

p
x,v(w)

+
∑

|α1|=m−2

∑

|i|≤m−2

‖∂α1x (f − g)‖Lpx,v(w) · ‖∂
i
x∂

α−α1
x ∂

β
v ∇vg‖Lpx,v(w) · ‖∂

α
x ∂

β
v (f − g)‖

p−1
L

p
x,v(w)

.

(4.22)

Step 2. β = 0, |α| ≤ m − 1, applying ∂αx on both sides of (4.19)1 yields

(∂t + v · ∇x + ∇xφf · ∇v)∂
α
x (f − g) = −

∑

06=α1≤α

Cα1α ∂
α1
x ∇xφf · ∂

α−α1
x ∇v(f − g)

−
∑

0≤α1≤α

Cα1α ∂
α1
x (∇xφf − ∇xφg) · ∂

α−α1
x ∇vg.

(4.23)



CUBO
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Vlasov-Poisson equation in weighted Sobolev space W m,p(w) 223

Repeating the process of step 1, we get

d

dt
‖∂αx (f − g)‖

p

L
p
x,v(w)

.
∑

|i|≤2

‖∂ixf‖Lpx,v(w) · ‖∂
α
x (f − g)‖

p

L
p
x,v(w)

+
∑

0<|α1|≤m−2

∑

|i|≤2

‖∂i+α1x f‖Lpx,v(w) · ‖∂
α−α1
x ∇v(f − g)‖Lpx,v(w) · ‖∂

α
x (f − g)‖

p−1
L

p
x,v(w)

+
∑

|α1|=m−1

∑

|i|≤m−2

‖∂α1x f‖Lpx,v(w) · ‖∂
i
x∂

α−α1
x ∇v(f − g)‖Lpx,v(w) · ‖∂

α
x (f − g)‖

p−1
L

p
x,v(w)

+
∑

0<|α1|≤m−3

∑

|i|≤2

‖∂i+α1x (f − g)‖Lpx,v(w) · ‖∂
α−α1
x ∇vg‖Lpx,v(w) · ‖∂

α
x ∂

β
v (f − g)‖

p−1
L

p
x,v(w)

+
∑

m−2≤|α1|≤m−1

∑

|i|≤m−2

‖∂α1x (f − g)‖Lpx,v(w) · ‖∂
i
x∂

α−α1
x ∇vg‖Lpx,v(w) · ‖∂

α
x (f − g)‖

p−1
L

p
x,v(w)

.

(4.24)

Note f(0, x, v) = g(0, x, v),

sup
0≤s≤t

‖∂i+α1x f(s)‖Lpx,v(w) ≤ M, sup
0≤s≤t

‖∂α−α1x ∂
β
v ∇vg(s)‖Lpx,v(w) ≤ M,

and

sup
0≤s≤t

‖∂ix∂
α−α1
x ∂

β
v ∇vg(s)‖Lpx,v(w) ≤ M, sup

0≤s≤t
‖∂ixf(s)‖Lpx,v(w) ≤ M.

Integrating (4.22) and (4.24) over [0, t], then summing over |α|+|β| ≤ m−1, we deduce

E1((f − g)(t)) . (1 + M)

∫ t

0

E1((f − g)(s)) ds,

where

E1(f(t)) =:
∑

|α|+|β|≤m−1

‖∂αx ∂
β
v f(t)‖

p

L
p
x,v(w)

.

By Gronwall’s inequality, we have E1((f − g)(t)) ≡ 0 implying f ≡ g, which completes

the proof of uniqueness. Thus we end the proof of Theorem 2.1.

Remark 4.2. All in all, we improved the results in [4] to the more general function space W m,p(Rn)

which does not have to be C2(Rn) (too strong). Our results also shed light on exploring solutions

in Sobolev spaces. We are very confident that our method could be applied in fractional Sobolev

spaces, even the supercritical spaces which are far from being understood yet.

5 Appendix

For the sake of completeness, we cite some known results about the estimate for the Riesz potential.

First of all, we give the pointwise estimate of the Riesz potential, for more details, see chapter 3,

section 1, page 57 in [1].



224 Cong He & Jingchun Chen CUBO
24, 2 (2022)

Proposition 5.1 ([1]). For any multi-index ξ with |ξ| < α < n, there is a constant A such that

for any f ∈ Lp(Rn), 1 ≤ p < ∞, and almost every x, we have

|Dξ(Iα ∗ f(x))| ≤ AMf(x)
|ξ|
α · (Iα ∗ |f|(x))

1−
|ξ|
α , (5.1)

where Iα =
γα

|x|n−α
, γα =

Γ(n − α
2
)

π
n
2 2αΓ(α

2
)
.

Remark 5.2. In our paper, we consider −∆φ =

∫

Rn

fdv =: g, n ≥ 3. Thus, in our context, Iα

can be taken

I2(x) =
1

(n − 2)ωn−1
·

1

|x|n−2
, i.e. α = 2, (5.2)

where ωn−1 =
2π

n
2

Γ(n
2
)
is the (n − 1)−dimensional area of the unit sphere in Rn, then we have

|Dξ(I2 ∗ g(x))| ≤ cMg(x)
|ξ|
2 · (I2 ∗ |g|(x))

1−
|ξ|
2 . (5.3)

Next, we give the off-diagonal estimate of the Riesz potential I2. For the details, see chapter V,

section 1 and page 119 in [14].

Lemma 5.3 ([14]). If −∆φ = g ∈ Lp(Rn), then φ = I2 ∗ g and

‖I2 ∗ g‖Lq̃(Rn) ≤ c‖g‖Lp(Rn), (5.4)

where 1 < p < n
2
, c = c(p, q̃) and

1

q̃
=

1

p
−

2

n
. (5.5)



CUBO
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Vlasov-Poisson equation in weighted Sobolev space W m,p(w) 225

References

[1] D. R. Adams and L. I. Hedberg, Function spaces and potential theory, Grundlehren der math-

ematischen Wissenschaften, Berlin: Springer-Verlag, 1996.

[2] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yong, “Regularizing effect and local

existence for the non-cutoff Boltzmann equation”, Arch. Ration. Mech. Anal., vol. 198, no. 1,

pp. 39–123, 2010.

[3] A. A. Arsen’ev, “Global existence of a weak solution of Vlasov’s system of equations”, USSR

Comp. Math. Math. Phys., vol. 15, no. 1, pp. 131–143, 1975.

[4] C. Bardos and P. Degond, “Global existence for the Vlasov-Poisson equation in 3 space vari-

ables with small initial data”, Ann. Inst. H. Poincaré Anal. Non Linéaire, vol. 2, no. 2, pp.

101–118, 1985.

[5] J. Batt, “Global symmetric solutions of the initial value problem of stellar dynamics”, J.

Differential Equations, vol. 25, no. 3, pp. 342–364, 1977.

[6] Y. Guo, “The Vlasov-Poisson-Boltzmann system near Maxwellians”, Comm. Pure Appl.

Math., vol. 55, no. 9, pp. 1104–1135, 2002.

[7] C. He and Y.-J. Lei, “One-species Vlasov-Poisson-Landau system for soft potentials in R3”,

J. Math. Phys., vol. 57, no. 12, 121502, 25 pages, 2016.

[8] C. He and H.-J. Yu, “Large time behavior of the solution to the Landau equation with specular

reflective boundary condition”, Kinet. Relat. Models, vol. 6, no. 3, pp. 601–623, 2013.

[9] E. Horst, “On the classical solutions of the initial value problem for the unmodified non-linear

Vlasov equation (Parts I and II)”, Math. Methods Appl. Sci. vol. 3, no. 2, 1981, pp. 229–248

and vol. 4, no. 1, pp. 19–32, 1982.

[10] E. Horst, “On the asymptotic growth of the solutions of the Vlasov-Poisson system”, Math.

Methods Appl. Sci., vol. 16, no. 2, pp. 75–85, 1993.

[11] P.-L. Lions and B. Perthame, “Propagation of moments and regularity for the 3-dimensional

Vlasov-Poisson system”, Invent. Math., vol. 105, no. 2, pp. 415–430, 1991.

[12] K. Pfaffelmoser, “Global classical solutions of the Vlasov-Poisson system in three dimensions

for general initial data”, J. Differential Equations, vol. 95, no. 2, pp. 281–303, 1992.

[13] J. Schaeffer, “Global existence of smooth solutions to the Vlasov-Poisson system in three

dimensions”, Comm. Partial Differential Equations, vol. 16, pp. 1313–1335, 1991.



226 Cong He & Jingchun Chen CUBO
24, 2 (2022)

[14] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathe-

matical Series, No. 30, Princeton, N. J.: Princeton University Press, 1970.


	Introduction 
	Preliminaries and main theorem
	Notations and definitions

	 Lp-Lq  estimates
	Proof of main theorem
	Appendix