A Vlasov-Poisson plasma of infinite mass with a point charge

We study the time evolution of the three dimensional Vlasov-Poisson plasma interacting with a positive point charge in the case of infinite mass. We prove the existence and uniqueness of the classical solution to the system by assuming that the initial density slightly decays in space, but not integrable. This result extends a previous theorem for Yukawa potential obtained in [ 10 ] to the case of Coulomb interaction.

1. Introduction. In this paper, we study the time evolution of the Vlasov-Poisson plasma interacting with a positive point charge in the case of infinite mass in R 3 .
We denote by f (t, x, v) the distribution of the particles in the plasma at time t ≥ 0 and position x ∈ R 3 , moving with velocity v ∈ R 3 , then the spatial density is defined by ρ(t, x) = R 3 f (t, x, v)dv and the time evolution of the plasma-charge system is governed by the following Vlasov-Poisson type system [9]: x−y |x−y| 3 ρ(t, y)dy, F (t, x) = x−ξ(t) |x−ξ(t)| 3 , ξ(t) = η(t),η(t) = E(t, ξ(t)), f (0, x, v) = f 0 (x, v), (ξ(0), η(0)) = (ξ 0 , η 0 ). (1) Here, E(t, x) indicates the self-consistent electrostatic field, which is induced by the positively charged particles of the plasma. The position and velocity at time t of the point charge is denoted by ξ(t) and η(t) respectively. F (t, x) is the Coulomb's force field, which models the repulsive interactions between the plasma and the positive point charge.
When considering the Vlasov-Poisson system with infinite mass, we can refer to [3,4,5,6,7,8,12,19,24,29,30,31]. In [3], by ignoring the singularities in the interaction potential, which is assumed to be positive, bounded and short-range in R 3 , the authors obtained the existence and uniqueness of the solution to the system. In [12], the Yukawa potential was considered to deal with the well-posed problem with infinite charge in R 2 . Moreover, the problem that the plasma of infinite mass is confined by an external magnetic field has been studied in [4,5] for the Yukawa and Coulomb potentials respectively. When considering the plasma with charged particles mutually interacting via the Coulomb force in R 3 , the authors of [6] proved the existence and uniqueness of the solution to the system under the assumption that the initial density is slowly decaying at infinity. Recently, the Vlasov-Poisson plasma with infinite mass and velocities confined in a cylinder and in the whole space respectively has been studied in [7,8].
The study of the system (1) is relatively few. For attractive case, the authors of [11] found a convenient condition on support of the initial density f 0 of the plasma, under which they succeed in establishing a global-in-time existence of solutions in two dimensional case. Global existence of weak solutions in three space variables was obtained in [14] by the theory of Diperna-Lions flow and compactness argument. For repulsive case, the first global existence and uniqueness result for compactly supported classical solution was established in [9] in two dimensional case, which was then extended to the case of three space variables [22]. Recently, higher order velocity moments were shown to be propagated by weak solutions [17]. This result is important and extends partially the Lions-Perthame theory [21] for the classical Vlasov-Poisson system to the case of the Vlasov-Poisson plasma interacting with a positive point charge (see also [13] for a better upper bound).
We stress that the initial density is assumed to belong to L ∞ ∩ L 1 in the papers above. In the present paper, we consider the case that the initial density is not in L 1 . In fact, in [10] the assumption of the initial density having spatial compact support is removed and it is only assumed that the initial density is bounded. However, it is only investigated that the mutual interaction of the charged particles of the plasma is Yukawa type in R 2 . There is no result about the three dimensional Vlasov equation with the Coulomb mutual interaction for the case that the plasma of the infinite charge distribution interacts with a positive point charge.
In this paper, we extend the result of [10] to the case that the charged particles of the plasma mutually interact by the Coulomb force in R 3 . To overcome the difficulty induced by infinite mass, we use two fundamental tools: one is the local energy introduced in [10], and the other is a new energy function h(t, x, v), which consists of the pointwise energy for a plasma particle and the kinetic energy of the point charge.
We denote the position and velocity of the plasma particles beginning from (x, v) at time t = 0 by (X(t, 0, x, v), V (t, 0, x, v)), the characteristic equations corresponding to the system (1) is given by and the evolution of the point charge is governed by Along the characteristics, we have We introduce the set where δ 0 and V 0 are given positive constants.
In the following, positive constants denoted by C, which may change from line to line, depend only on the initial data and the time T . Some constants indexed by the number 1, 2, 3 will be quoted in the sequel. Then, we give the main result of the paper.
for some fixed > 1 8 and C 1 > 0. Then, for arbitrary time T > 0 , there exists a unique solution (f (t, x, v), ξ(t), η(t)) to (1) on [0, T ] such that: and |i−x|≤1 where C 2 and C 3 are positive constants depending only upon T and the initial data.
As pointed out above, we introduce the energy function where k > 1 is a suitably large positive constant which will be given in (24). Then, we have |v| ≤ 2 √ h. Along the characteristic of the plasma particles, we obtain that The maximal velocity and the maximal displacement of the plasma particles are respectively defined by 2. The estimate of the local energy. To prove Theorem 1.1, we consider the following initial condition: is the characteristic function of the set A. And we introduce the cut-off system: Then, the system (8) has a unique global solution (f N , ξ N , η N ) for any fixed N , which has been proven in [22]. The characteristic equations corresponding to the system (8) is given by and the evolution of the point charge is governed by Along the characteristics, we have For any given time interval [0, T ], the main purpose of the paper is to prove that P N (t) is bounded uniformly in N . Now, we introduce the local energy, which has been used to deal with the unboundedness of the charged particles in [3,4,5,6,7,8,10,12]. For any given µ ∈ R 3 and R > 0, we define the function where ϕ ∈ C ∞ (R) such that In this paper, similar to [10], the local energy is defined as follows: Furthermore, for any fixed N , the maximal velocity and the maximal displacement corresponding to the cut-off system (8) are respectively defined by For the local energy, we have the following fundamental estimate, Proposition 1. In the hypotheses of Theorem 1.1, there exists a positive constant C, independent of N, such that The proof of Proposition 1 can be deduced by Lemma 2.1 and Lemma 2.2.
Lemma 2.1. There exists a positive constant C, independent of N, such that Proof. Firstly, we set Then, by the definition of the R N (t), we get According to the definition of the local energy, we have
By the change of variables (x, v) → (X N (s), V N (s)) and (y, w) → (Y N (s), W N (s)), we obtain that The time derivative of W N (µ, R N (t, s), s) can be divided into three parts [10]: For the term C, similar to [6,10], we deduce that For the term A, the estimate has been established in [6], we repeat it here for the completeness of the proof, according to the system (8), we have where we have used the change of the variables (y, w) → (x, v). By the mean value theorem and the definition of ϕ µ,R , we get Hence, By the change of variables ( Then, by (11) and the definition of the local energy, we get Hence,

GANG LI AND XIANWEN ZHANG
For the term B, according to the system (8), we have Now, we estimate each term on the right hand side of the above inequality separately. Firstly, we have Indeed, by the change of variables (x, v) → (X N (s), V N (s)), we obtain that where we have used the fact that Then, we obtain that where we have used the fact that By the assumption of (4), Hence, (15) is proved. Secondly, similar to the proof of (2), we have

GANG LI AND XIANWEN ZHANG
Then, by (15), (16) and the fact that Q N (R N (t, s), s) ≥ 1, we deduce that By (13), (14), (17), we obtain that Hence, we get By the Gronwall lemma, we get which is implied by the estimate (12) and the proof is completed.
In the hypotheses of Theorem 1.1, we have Proof. Due to assumptions on the initial data and the definition of ϕ, we have where C is a constant only depending on V 0 , δ 0 and the initial data. Firstly, by the hypotheses of Theorem 1.1, we get Due to by the assumption of (4), we deduce that Secondly, by the hypotheses of Theorem 1.1, if |µ| ≤ 5R N (t), we get If |µ| > 5R N (t), by the assumption of (4), we deduce that where we have used the fact that combining (18), (19) and (20), we get by (11) and the definition of R N (t), we have This is the desired result.
3. The estimate of the electric field.
Proposition 2. There exists a positive number C such that Proof. For any given µ ∈ R 3 and 0 < ν ≤ 3R N (t), we have where Because for the term (21), we deduce that For the term (22), .
By (11) and the definition of the local energy, we deduce that Then, by Hölder inequality, we have ρ N (y, t) By the definition of V N (t), we have by Proposition 1, we can choose k to be suitably large such that Then, we deduce that For the term (23), by the change of variables (ȳ,w) → (Y N (t, 0, y, w), W N (t, 0, y, w)), we obtain that where we have used the fact that Then, we have where we have used the fact that |µ − y| ≥ |µ − i| − |i − y| ≥ |µ − i| − R N (t) and (4).
To proceed further, we choose We partition the interval [0, T ] into n subintervals [t j , t j+1 ], j = 0, 1, · · · , n − 1, we have Then, we have Proof. By the estimate of (7), we get The proof is completed.
The following proposition, whose proof is similar to that of [6,8], is important in the proof of Theorem 1.1. For the sake of completeness of this paper we give the detailed proof of it.
Proposition 3. In the hypotheses of Theorem 1.1, there exist positive constants C and β < 1 such that Proof. We consider the system (9) on [t j , t j+1 ] for a fixed j. By the change of variables (ȳ,w) → (Y N (t, t j ,ŷ,ŵ), W N (t, t j ,ŷ,ŵ)), we obtain that

GANG LI AND XIANWEN ZHANG
Firstly, we give the estimate of |E N (t, ξ N (t))|, setting By Lemma 3.1 and by the change of variables (Y N (t, t j ,ŷ,ŵ), W N (t, t j ,ŷ,ŵ)) → (ȳ,w), we have Then, we deduce that for i = 1, 2, and For the term Ξ 1 , similar to the proof of (25), we can deduce that For the term Ξ 2 , we define Then, we have For the term (30), by the assumption of Theorem 1.1, we obtain that For the term (31), along the characteristics, we obtain that where we have used the fact that If |ξ N (t j )| ≤ 5R N (T ), by the assumption of Theorem 1.1 and the definition of R N (t), we obtain that where we have used the fact that |ξ N (t j ) + i| ≥ |ξ N (t j )| − |i| ≥ |i| 4 , and the assumption of (4). Hence, by (33) and (34), we obtain that For the term (28), if |ξ N (t j ) −ȳ| ≥ 3R N (T ), we deduce that By the change of variables (ȳ,w) → (Y N (t, 0, y, w), W N (t, 0, y, w)), we obtain that where we have used the fact that we obtain that Then, we deduce that where we have used the fact that Hence, By (29) Secondly, analogous to the way above, we give the estimate of |E N (t, X N (t, 0, x, v))|. Setting Similar to the estimate of |E N (t, ξ N (t))|, we have for j = 1, 2, and with For the term A 1 , analogous to the proof of (25), we deduce that For the term A 2 , we also define the sets as follows: Then, we have For the term (40), by the assumption of Theorem 1.1, we obtain that For the term (41), analogous to the proof of (31), we get Now, we give the estimate (38). If |X N (t j , 0, x, v) −ȳ| ≥ 3R N (T ), we deduce that By the change of variables (ȳ,w) → (Y N (t, 0, y, w), W N (t, 0, y, w)), we obtain that where we have used the fact that we obtain that
4. The proof of Theorem 1.1. By the estimate (7) and (27), we obtain that by the definition of P N (t), we get Hence, we obtain that where C 2 is a positive constant depending only on the initial data and the time T .
Since the estimates of the fields and (46) are uniform in N , then the solutions of system (9) and (10) converge to the solutions of system (2) and (3) in the limit N → ∞.
Indeed, for any positive integer K, we define that (X N (t), V N (t), ξ N (t), η N (t)) and (X N +K (t), V N +K (t), ξ N +K (t), η N +K (t)) are the two solutions to the system (9) and (10), with the initial distributions f N 0 and f N +K Then, we have the following estimate: Proposition 4. For any t ∈ [0, T ], we have The proof of Proposition 4 will be given in the last section. Now, we give the proof of the main result.
Proof of Theorem 1.1. By the assumption of (4) and the Proposition 4, we deduce that . Then, there exist some limit functions uniformly on [0, T ] as N → ∞, and Moreover, by Proposition 1, Proposition 2 and the estimate (46), we know that the fields E N and F N are uniformly bounded in N . Analogous to the estimate of (57), we get Thus, we have By the estimate of (46), we get Hence, (f (t, x, v), ξ(t), η(t)) satisfy system (1) on [0, T ]. Now, we give the proof of (5) and (6). Firstly, by the estimate (46), we deduce that Secondly, by the change of variables (x, v) → (X(t, 0, x, v), V (t, 0, x, v)), similar to [6], we obtain that where i ∈ Z 3 : |i| ≥ 1. If |i| ≤ 2R(t), we get µ∈Z 3 : If |i| > 2R(t), we get where we have used the fact that By the definition of R(t), we have Hence, we obtain that Furthermore, the uniqueness of the solution to the system (2) and (3) Proof. The proof is similar to [6]. We define |x − y| = D. In case D ≥ 1, by Proposition 1, Proposition 2 and the estimate (46), we deduce that In case D < 1, setting λ = x+y 2 , we have where we have used the fact that |µ − x| ≤ 3D and |µ − y| ≤ 3D. For the term I 2 , by the the mean value theorem, we have we have For the term I 3 , we have where we have used the fact that similar to the estimate (22) and (23), we get Hence, by (48), (49) and (50), we obtain that The proof of Lemma 5.1 is completed.
Proof. According to the system (10), we have For the term E 1 , by Lemma 5.1, we have For the term E 2 , we have by the change of variables (y, w) → (Y N (s), W N (s)) and (y, w) → (Y N +K (s), W N +K (s)), we obtain that where For the term E 2 , setting, we have

GANG LI AND XIANWEN ZHANG
For the term F 1 , we get For the first term on the right hand side of the above inequality, due to where we have used the estimate (46). Similar to the estimate above, we have Thus, we get For the term F 2 , by the mean value theorem, we deduce that Due to a N, Then, we obtain that By the change of variables (Y N (s), W N (s)) → (y, w), we get similar to the estimate of (22)and (23), we have For the first term on the right hand side of the above inequality, we have By the estimate (46), we have Thus, we deduce that

GANG LI AND XIANWEN ZHANG
By (52)and (53), we obtain that For the term E 2 , we have where we choose (y, w)|dydw 1 3 .
Firstly, for the first term on the right hand side of the inequality (55), by the change of variables (Y N +K (s), W N +K (s)) → (y, w), we have where we have used the estimate (46).
Finally, we obtain that Hence, we deduce that The proof of Proposition 4 is completed.