A Vlasov-Poisson plasma with unbounded mass and velocities confined in a cylinder by a magnetic mirror

We study the time evolution of a single species positive plasma, confined in a cylinder and having infinite charge. We extend the result of a previous work by the same authors, for a plasma density having compact support in the velocities, to the case of a density having unbounded support and gaussian decay in the velocities.


Introduction
In this paper we study the behavior in time of a Vlasov-Poisson plasma, with infinite charge and velocities, confined in an infinite cylinder by an external magnetic field, the so-called magnetic mirror. To specify the problem, we consider a continuous distribution of positively charged particles, assembled at time zero in an infinite cylinder and kept inside it by means of an external magnetic field, which diverges at a distance A from its symmetry axis. The plasma evolves under the action of the auto-induced electric field plus the external force, and it is governed by the Vlasov-Poisson equations in which an extra term is added due to the external Lorentz force. Our aim is to investigate the existence and uniqueness of the time evolution of this system and its confinement over an arbitrary time interval [0, T ].
This problem has been studied by the same authors in [3,4,5], and in all these papers it is assumed that the density has compact support in the velocities. Moreover, in the first paper [3] it is considered a density having compact support in space; in the second one [4] this assumption is removed and it is only assumed that the density is bounded. In this case, however, it is considered an interaction potential of Yukawa type, that is Coulomb at short distance and exponentially decaying at infinity; finally in the third paper [5] it is considered the Coulomb potential, but it is assumed that the spatial density, even if not integrable, has to satisfy some decaying properties at infinity. Here we generalize this last result to a density without compact support in the velocities. More precisely, we consider a plasma having infinite charge and velocities, and we assume that its density is slowly decaying in space (not integrably) and gaussian in the velocities.
While the theory of the Vlasov-Poisson equation for integrable data is much developed (see for instance [8] and [11] for L 1 data in space and velocities, [12,16] for compactly supported densities and [7] for a nice review of results in this context), the difficulties are increasing as one tries to remove this assumption, since one has to match the divergence of the Coulomb interaction with the infinite charge and velocities of the system. We quote [10,13,14,15] as it regards results for infinite charge systems. Our method to control the infinite charge, and hence systems with infinite total mass and energy, is to introduce the local energy, which is the energy of a region interacting with all the rest of the plasma. In [1], [2] and successively in [4] and [5], it has been proved a bound on the local energy in terms of its initial value, which implies a control on the spatial density. In the setup of confined plasmas, from one side we have an extra difficulty, deriving from the singularity of the external confining Lorentz force, from the other side the system can be considered a one-dimensional system at infinity, which makes our job easier. We also quote [9] for a confined relativistic plasma in one dimension in space and two in the velocities, with bounded charge. An important feature of the results [4,5] is that, once we prove that the velocities are bounded, we have as a consequence the confinement of the plasma.
In the present paper we consider a case of unbounded total charge and unbounded velocities. We assume that the density has spatial support in the whole infinite cylinder, where it is slightly decaying (not integrably), and is fast decaying (gaussian) in the velocities. We introduce a regularized system, called the partial dynamics, in which the density has compact support in the velocities. In [5] it is proved the existence and uniqueness of the time evolution for such system, and our aim is to remove the compactness assumption, by letting the size of the velocity support go to infinity. By refining the estimate of the auto-induced electric field E made in [5], we prove some uniform bounds on the partial dynamics and in particular we prove that E is bounded. Since the Lorentz force does not affect the modulus of the velocities of the particles, this is sufficient to prove that any fixed particle has a bounded displacement and velocity, uniformly in the support. This allows us to state that the limit time evolution does exist unique globally in time and that the system remains confined.
We now discuss some related problems. While here we consider an external magnetic field parallel to the symmetry axis of the cylinder, we could consider other external magnetic fields as well, always divergent on the border of the cylinder and tangent to it (hence giving a formal confinement of the plasma); in this case the magnetic lines would not be straight lines, but lines with some curvature as, for instance, cylindrical helices. Moreover, we could consider domains whose boundaries have non-zero gaussian curvature. A problem within this context can be studied, for finite mass and bounded velocities, and in fact an explicit investigation has been done in [6], in the case of a torus. We remark that, at the moment, in other generic cases we are not able to reproduce results similar to those of the present paper. We sketch the plan of the paper at the end of the next section.

The equation and the main result
The equation we consider is a Vlasov-Poisson equation, with an extra Lorentz force acting on the particles, in order to keep the plasma confined in a cylinder. Denoting by x = (x 1 , x 2 , x 3 ) and v = (v 1 , v 2 , v 3 ) the position and velocity vectors in R 3 and by f (x, v, t) the plasma density in the phase-space at time t, we have: (2.1) We consider an infinite, open cylinder D of radius A and symmetry axis x 1 , that is and the confining vector field B, diverging on the boundary of D, is defined as where θ has been chosen large enough for further purposes (see eqn. (3.55)). Letting X(t) = X(x, v, t) and V (t) = V (x, v, t) represent position and velocity at time t of a particle starting at time t = 0 from x with velocity v, the related characteristics equations are We also consider the sub-cylinder D 0 ⊂ D : The following theorem states the main result of this paper.
Theorem 1. Let us fix an arbitrary positive time T. Let f 0 (x, v) be supported on D 0 × R 3 and satisfy the two following assumptions: and, for any i ∈ Z/{0}, (2.7) for some positive constants C 0 , C 1 and λ.
Then there exists a solution to system (2.4) in [0, T ] such that, for any Moreover, there exist positive constants C 2 ,λ and C 3 such that and, for any i ∈ Z/{0}, This solution is unique in the class of those satisfying (2.8) and (2.9).
Remark 1. We put in evidence the fact that assumption (2.7) does not imply that ρ 0 belongs to any L p space, as it can be satisfied even in case ρ 0 is only bounded, but supported over suitably sparse sets. The fact that ρ 0 ∈ L ∞ (and also ρ(t)) is a direct consequence of (2.6) (and (2.8) respectively). We also note that the case α > 1 deals with finite total mass, whereas the main difficulties we face concern the infinite mass case. Hence the more interesting case for the present paper is realized for 5/9 < α ≤ 1.
The strategy of the proof of this result is the following: we introduce a truncated dynamics, denoted as partial dynamics, in which we assume that the initial datum f 0 has (infinite) spatial support in D 0 and compact support in the velocities. This allows us to make use of the results in [5] to find a bound on the electric field. Then we are ready to prove that the partial dynamics is converging, as the size of the velocity support goes to infinity, to some limit dynamics, which satisfy the equations and the confinement globally in time. The theorem is proved in the next Section 3: in 3.1 we introduce the partial dynamics, and we state the main estimate on the electric field, which is an improvement of the analogous in [5]; in 3.2 we show that the partial dynamics is converging as the compactness of the velocity support of f 0 is removed; in 3.3 we prove that the displacement and the velocity of a single particle are proportional to its initial velocity, independently of the initial support. This allows to complete the demonstration, proving that the limit dynamics satisfy the equations and the confinement. For the sake of clearness in the presentation of the result, we postpone to Section 4 some estimates concerning the local energy, and to Section 5 the proof of the estimate of the electric field. In Section 6 it is proved the main estimate on the local energy, together with other technicalities. We report here the bound on the local energy for completeness, even if its proof does not differ from the one given in [5].
3 The proof of the Theorem

The partial dynamics
We introduce the partial dynamics, that is a sequence of differential systems in which the initial densities have compact velocity support: where χ is the characteristic function of the set b(N ) = {v ∈ R 3 : |v| < N }, In [5] it is proven the existence of a solution to system (3.1), provided f 0 satisfies the assumptions of Theorem 1, since f N 0 has compact support in the velocities. Moreover, it is proven the confinement of the solution, that is sup From now on all constants appearing in the estimates will be positive and possibly depending on the initial data and T , but not on N . They will be denoted by C and some of them will be numbered in order to be quoted elsewhere in the paper.
We introduce the maximal velocity of a characteristic where C 4 is a constant that will be chosen large enough.
We premise the following result on the partial dynamics, which is fundamental for the proof of Theorem 1 and will be proved in Section 5.
As a consequence, the following holds: being γ the exponent in (3.5) and θ the one in (2.3).
Proof. To prove (3.6) we observe that the external Lorentz force does not affect the modulus of the particle velocities, being d dt This fact, by Proposition 1 and the choice of the initial data such that v ∈ b(N ), implies Hence, since γ + 1 < 2, by taking the sup t∈[0,T ] we obtain the thesis. Now we prove (3.7). Putting by using the invariance of the density along the characteristics we have We notice that, putting from (3.10), Proposition 1 and (3.6) it follows Hence, we decompose the integral as follows Since the inequality holds for any t ∈ [0, T ], it holds also at the time in which V N reaches its maximal value over [0, t], that is Hence from (3.13) it follows (3.14) We prove (3.8) by an analogous argument to the one used in [5] to prove the confinement of the plasma. Writing by components equations (3.1), after elementary manipulation we get, omitting for simplicity the argument t and the index N, after putting r(t) = X 2 (t) 2 + X 3 (t) 2 : Let H be a primitive of h. By integrating in time by parts over [0, t], for any t ∈ [0, T ], we obtain by (3.15), Recalling that the initial data are such that H(r 2 (0)) < C, by (3.6), (3.3) and Proposition 1 we get From this, by the definition of the field B, it follows the thesis.

Convergence of the partial dynamics
Remark 2. We stress that estimate (3.8) allows us to make explicit the constant in (3.3), that is, for a fixed N, we have that This implies that all the spatial integrals in the sequel of this section have to be intended over the infinite cylinder D, that is three-dimensional over small sets and one-dimensional over large sets.
We fix a couple (x, v) ∈ D 0 × b(N ) and we consider X N (t) and X N +1 (t), that is the time evolved characteristics, both starting from this initial condition, in the different dynamics relative to the initial distributions f N 0 and and Let us start by estimating the term F 1 . We will prove a quasi-Lipschitz property for E N . Let us put |x − y| := d. In case d ≥ 1, recalling Remark 2, by (3.7) we have In case d < 1, we definez = x+y 2 and decompose the integral as follows: where If |z −z| ≤ 2d then |x − z| ≤ 3d and |y − z| ≤ 3d. Hence, always by (3.7), For the term I 2 we have, by the Lagrange theorem: recalling again the Remark 2. Hence by (3.22) and the estimates of the terms I i , i = 1, 2, 3, we have shown that Now we draw our attention to the term F 2 . We putX = X N +1 (t) and we have: (3.30) Now we pass to the term F ′′ 2 . We put Since by the Liouville theorem dydw = dY N (t)dW N (t), by the invariance of the density f N along the characteristics, putting and By the Lagrange theorem where ξ N (t) is a point of the segment joining Y N (t) and Y N +1 (t). Note that if y ∈ S N (t) then, by the definition of δ N (t), which implies that |X − ξ N (t)| is certainly bigger than 1 2 |X − Y N (t)|. Hence by (3.7) it follows recalling that the integral is over the cylinder D. It is easily seen that, for any positive a < 1 and ǫ < 1 it holds Hence, if δ N (t) < 1, we have If δ N (t) ≥ 1, estimate (3.37) gives us We choose ǫ = e −λN 2 , and in both cases it results Let us now estimate the term I 2 . By the choice of the initial condition it follows where we have used the bound (3.6) on the maximal velocity and again the fact that the integral is over the cylinder D.
Let us estimate the term I 3 . Formula (3.36) implies that Now we observe that Hence, putting (Y N (t), W N (t)) = (ȳ,w) and recalling once again the definition of δ N (t), we have This together with estimate (3.7) implies that Going back to (3.31), from estimates (3.40), (3.41) and (3.45) it follows so that this last estimate, (3.27) and (3.30) imply Finally we estimate the term F 3 . We have By applying the Lagrange theorem we have This, together with the bounds (3.6) and (3.8), imply where in (3.26) we have taken into account the bound (3.6), which gives |δ N (t)| ≤ CN. On the other side, by using the same method to estimate the quantity η N (t), we get, analogously we have, summing up (3.51) and (3.53): (3.54) Putting (3.56) We insert in the integrals the same inequality for σ N (t 1 ) and σ N (t 2 ) and iterate in time, up to k iterations. By direct inspection, using in the last step the estimate sup t∈[0,T ] σ N (t) ≤ CN, we arrive at (3.57) We observe that the i-th iterate consists in a sum of integrals of increasing order j ∈ [i, 2i] and that the binomial coefficients represent all the possible ways to arrange the integrals of the same order. By putting We start by estimating S ′′ k . Recalling that i j < 2 i we get The use of the Stirling formula a n n n ≤ n! ≤ b n n n for some constants a, b > 0 yields: from which it follows, again by the Stirling formula, For the term S ′ k , putting j + k = ℓ, we get The hypothesis (2.3) on the external field B and the range of the parameter γ given by Proposition 1 guarantee that ν < 2, so that, choosing k = N ζ with ζ > 4, we have, for sufficiently large N, Going back to (3.58), by (3.61) and (3.63) we have seen that Hence, being ν < 2, we can conclude that there exists a positive number c such that

Conclusion of the proof
We have shown that the sum σ N (t) converges uniformly in t ∈ [0, T ], and then the sequences X N (t) and V N (t) are Cauchy sequences, uniformly on [0, T ]. Hence, for any fixed (x, v) they converge to limit functions which we call X(t) and V (t). Now it remains to prove To this aim, we start by giving a N -uniform estimate for |V and, by the choice of N 0 , From this it follows |X N (t) − x| ≤ C(|v| + 1). (3.68) We use these estimates to prove properties (2.8) and (2.9) for the density.
To prove the first one, we observe that the bound (3.67) implies To prove the decay of the spatial density we partition the velocity space by the sets S 1 and S c 1 , with Hence, for any i ∈ Z/{0}, (3.71) By (3.69) we get On the other side, by a change of variables and (3.68) we have Since |k| ≤ Ca i , for large |i| it is |i + k| ≥ |i| 2 . Hence previous formula gives which, together with (3.72), implies (2.9). By the estimate (2.8) we obtain that the field E N is uniformly bounded in N. Indeed: (3.75) Moreover, it can be seen that E N (x, t) → E(x, t) uniformly on [0, T ]. In fact, the term |E N (x, t) − E(x, t)| can be estimated in the same way as we did with F 2 in the proof of the convergence (subsection 3.2), by using the bound (3.70) on the density, yielding which proves the confinement. Hence, we have proved that the limit functions (X(t), V (t)) satisfy the integral version of the characteristics equation

The local energy
Since this system has unbounded energy thus, in order to have a control on the spatial density, and then proceed in proving the estimate (3.5) on the electric field E, we introduce the following local energy. For µ ∈ R and R > 0 we define the mollified function, where ϕ, assumed to be smooth for technical purposes, is defined as: The local energy is the following function: (4.5) The function W N is a kind of smoothed energy of a bounded region, in which the interaction with the rest of the system has been taken into account. Note that it does not contain the effects of the magnetic force, as it does not contribute to energy variations. We put The assumptions (2.6) and (2.7) on f N 0 imply that Q N is finite at time t = 0 and has the following bound: Proof. The proof is quite similar to that in [5]. We consider R integer for simplicity. It is easily seen that Indeed, it is: Now, if |µ| ≤ 2R, by (2.7) it is: Now it is: This, together with (4.7), proves the proposition.
We define the maximal displacement of a plasma particle as and put Q N (t) = sup We state the most important result on the local energy, whose proof is given in Section 6: Proposition 3. There exists a constant C independent of N such that As consequence of Propositions 3 and 2 we have: We give now a first estimate on the electric field E, in terms of the local energy. We will make use of it in the proof of Proposition 1 in the next section.

Proposition 4.
(4.12) Proof. We premise an estimate on the spatial density: for any µ ∈ R and any positive number R it is: In fact: By minimizing over a, taking the power 5 3 of both members and integrating over the set {x : |µ − x 1 | ≤ R} we get (4.13). Now we choose a sequence of positive numbers A 0 , A 1 , A 2 , ...A k , ... such that A 0 = 0, A 1 < A has to be chosen suitably in the following and A k = (k − 1)R N (t) for k = 2, 3, ... Then we have We estimate the terms in (4.14). We have: Moreover by (4.13) we get: The minimum value of J 0 (x, t) + J 1 (x, t) is attained at For the remaining terms, for any k = 2, 3, ... we observe that from the definition (4.9) of the maximal displacement it follows that if Then by a change of variables we get: since the volume of the set {y ∈ D : The proof is achieved by (4.14), (4.16) and (4.18).

The estimate of E N : proof of Proposition 1
The proof of Proposition 1 follows the same lines of the analogous in [5]. The difference consists in the fact that here we need a more refined estimate of E N in terms of the maximal velocity, since the exponent γ has to be smaller than 2 3 . To this aim we need to control the time average of E N over a suitable time interval. Setting we have the following result, which is the core of this Section: There exists a positive number∆, depending on N, such that: In the proof of Proposition 1, we need to introduce several positive parameters, all depending only on the parameter α fixed in the hypothesis (2.7) which we list here, for the convenience of the reader; From now on we will skip the index N through the whole section. Moreover, we put for brevity V := V(T ) and Q := Q(T ).
Proof. Let us define a time interval where C 7 is the constant in (4.12). We remark that, since we have to prove a more refined estimate on E than that in [5], we choose a smaller ∆ 1 than the one chosen there. For a positive integer ℓ we set: being Intg(a) the integer part of a.
Assume that the following estimate holds (it will be established in the next subsection 5.1), for any positive integer ℓ : then, since R(t) ≤ CV(t), the choice of the parameters made in (5.2) and Corollary 2 imply with γ < 2 3 . Hence, definingl as the smallest integer such that .
This argument shows that, in order to prove Proposition 1, we have to prove that (5.6) holds, which we will do in the next subsection. For the moment, we observe that Proposition 5 is sufficient to achieve the proof of Proposition 1, which can be done by dividing the interval [0, T ] in n subintervals [t i−1 , t i ], i = 1, ..., n, with t 0 = 0, t n = T, such that∆/2 ≤ t i−1 − t i ≤∆, and using Proposition 5 on each of them.

Proof of (5.6)
To prove (5.6) we need some preliminary results, which are stated here and proved in Section 6.
We consider two solutions of the partial dynamics, (X(t), V (t)) and (Y (t), W (t)) , starting from (x, v) and (y, w) respectively. Let η be the parameter introduced in the definition (5.3) of ∆ 1 . Then we have: .
such that for any s ∈ [t, t + ∆ ℓ ] it holds:

Lemma 4.
There exists a positive constant C such that, for any µ ∈ R and for any couple of positive numbers R < R ′ we have: Estimate (5.6) will be proved analogously to what has been done in [5], the difference being that the time interval ∆ 1 has been chosen smaller. We use an inductive procedure, that is: step i) we prove (5.6) for ℓ = 1; step ii) we show that if (5.6) holds for ℓ − 1 it holds also for ℓ.
Step i) is the fundamental one, while step ii) is an almost immediate consequence, as it will be seen after.

Proof of step i ).
All the parameters appearing in this demonstration have been listed in (5.2).
We have to show that the following estimate holds: We fix any t ∈ [0, T ] such that t + ∆ 1 ≤ T , and we consider the time evolution of the characteristics over the time interval [t, t + ∆ 1 ]. For any s ∈ [t, t + ∆ 1 ] we set (X(s), V (s)) := (X(s, t, x, v), V (s, t, x, v)), X(t) = x (Y (s), W (s)) := (Y (s, t, y, w), W (s, t, y, w)), Y (t) = y. Then We decompose the phase space in the following way. We define where for any s ∈ [t, t + ∆ 1 ] and Let us start by the first integral. By the change of variables (Y (s), W (s)) = (ȳ,w), and Lemma 1 we get

Now it is:
Now we give a bound on the spatial density ρ(ȳ, s). Setting we have: where K(y, s) = dw|w| 2 f (y, w, s). Minimizing in a we obtain Hence from (5.22) we get where we have also applied Lemma 4. Going back to (5.21), this bound implies Minimizing in ε we obtain: We observe that, since by (4.9) it is R N (t) ≤ C(1 + V N (t)), Corollary 2 and the lower bound for η in (5.2) imply that I 1 (X(s)) is bounded by a power of V less than 2 3 . Now we pass to the term I 2 . Proceeding as for the term I 1 , defining S ′ 2 = {w : |w r | ≤ 2V ξ }, by Lemma 2 and the Holder inequality we get: The bound (4.13) implies By minimizing in ε we get: Analogously to what we have seen before for the term I 1 (X(s)), also in this case the upper bound for the parameter ξ in (5.2) implies that I 2 (X(s)) is bounded by a power of V less than 2 3 . Now we estimate I 3 (X(s)). It will be clear in the sequel that it is because of this term that we are forced to bound the time average of E, and then to iterate the bound, from smaller to larger time intervals.
We cover T 1 ∩S 3 by means of the sets A h,k and B h,k , with k = 0, 1, 2, ..., m and h = 1, 2, ..., m ′ , defined in the following way: where: with β chosen in (5.2). Consequently we put Since we are in S 3 , it is immediately seen that By adapting Lemma 1 and Lemma 2 to this context it is easily seen that ∀ (y, w, s) ∈ A h,k it holds: and Hence, setting By the choice of the parameters α k and l h,k made in (5.28) we have: h .
The choice of β is such that Now we pass to I ′′ 3 (h, k). Setting we have: we have: (5.41) Moreover: Now it is: Hence, where c positive, the estimate (4.12) of the electric field in Proposition 4 would be improved, at least on a short time interval ∆ 1 . Indeed, the choice of the parameters ensure that it is so, provided that α > 5 9 .

Proof of step ii ).
In the previous step we have seen that, starting from estimate (4.12) on [0, T ], we arrive at (5.13) on ∆ 1 . Let us now assume that (5.6) holds at level ℓ − 1. Since it is uniform in time, it holds over [0, T ] and, in particular, over ∆ ℓ = G∆ ℓ−1 . Hence, we can assume (5.6) at level ℓ − 1 to arrive to an improved estimate over ∆ ℓ .
We recall that the term I 3 was the one for which we needed to do the time average. Hence, proceeding in analogy to what we have done above, we arrive at the analogous of estimate (5.45), and consequently which proves the second step. Hence (5.6) is proved for any ℓ.

Some technical proofs
Also in this section we will skip the index N in the estimates, but it has to be reminded that the following estimates concern the partial dynamics.
Proof of Lemma 4. It follows from the definition of the function ϕ µ,R that, for any µ ∈ R and any couple R, R ′ such that 0 < R < R ′ , it is: Hence, since both terms in the function W are positive, we have: Proof of Proposition 3.
For any s and t such that 0 ≤ s < t ≤ T we define Then, Let (X(t), V (t)) and (Y (t), W (t)) be the two characteristics starting at time t = 0 from (x, v) and (y, w) respectively. We have: from which it follows, by deriving the function W with respect to s, and The term A 2 (t, s) is negative. Indeed the quantity in square brackets is positive, whereas, by the definition of the function ϕ, it is since ∂ s R(t, s) = −V(s). Hence Thus, being ϕ ′ ≤ 0, we have proved that A 2 (t, s) ≤ 0. (6.7) In the term A 1 , we observe ∇|x − y| −1 is an odd function. Hence, recalling (3.9), by the change of variables (x, v) → (y, w) we obtain: By symmetry we have: Since it is r ∇(|r| −1 ) = 1 r , we have, By the change of variables (X(s), V (s)) = (x,v) and (Y (s), W (s)) = (ȳ,w) we get, after integrating out the velocities  Proof of Lemma 1.
We give first the proof for ℓ = 1, that is ∆ ℓ = ∆ 1 . Since the magnetic force gives no contribution to the first component of the velocity, by (4.12) and (5.3) we get, for any s ∈ [t, t + ∆ 1 ], Analogously we prove the second statement: We show now that Lemma 1 holds true also over a time interval ∆ ℓ , ℓ > 1, supposing for the electric field the estimate (5.6) at level ℓ − 1 (that is, only Lemma 1 at level less than ℓ is needed to establish (5.6) at level ℓ). Since the estimate is uniform in time, it holds also over [0, ∆ ℓ ]. Hence, proceeding as before we get for any s ∈ [t, t + ∆ ℓ ], by (5.8) and the choice of the parameters made in (5.2). We proceed analogously for the lower bound.
By the same argument used at the end of the proof of Lemma 2, we see that the same proof works also considering the interval [t, t + ∆ ℓ ], ℓ > 1 and assuming for the electric field the estimate (5.6) at level ℓ − 1.