Focusing Solutions of the Vlasov-Poisson System

We study smooth, spherically-symmetric solutions to the Vlasov-Poisson system and relativistic Vlasov-Poisson system in the plasma physical case. We construct solutions that initially possess arbitrarily small C^k norms for the charge densities and the electric fields, but attain arbitrarily large L^1 norms of them at some later time.

Here, f (t, x, v) ≥ 0 is the density distribution of the particles. In the equation, x ∈ Ω ⊂ R 3 is the particle position, v ∈ R 3 is the particle momentum, and E is the electric field. Moreover, the charge density ρ is defined as We also consider the Vlasov-Poisson System in the relativistic setting (RVP): x − y |x − y| 3 ρ(t, y)dy .
1+|v| 2 is the velocity. The systems VP and RVP enjoy the conservation of the total mass (1.6) Here f 0 is the initial particle density. We assume spherical symmetry in the problem. It is known that spherically symmetric initial data give rise to global-in-time, spherically symmetric solutions to the two systems, see [11], [13], [14]. Also, [11] tells us that the solutions must be finite in the L ∞ sense.
The behavior of the solution to VP has being an important topic that caught wide attention. In two papers [1] by J. Ben-Artzi, S. Calogero and S. Pankavich, it is shown that one can construct solutions of VP such that the particle density and the electric field are initially as small as desired, but become large as desired at some later time, as stated in the following theorem: Theorem (J. Ben-Artzi, S. Calogero and S. Pankavich) For any positive constants η, N, there exists a smooth, spherically symmetric solution of VP, such that while for some T > 0, An analogous result for RVP is proved in [2] by the same authors.
In contrast to their results, there is the classical estimate from E. Hörst in 1990 that any spherically symmetric solution must decay for t sufficiently large. Namely, there exists C > 0 and T > 0, such that for all t ≥ T (see [11]).
However, in the examples provided in [1] and [2] the initial data are actually large in C 1 sense. In this paper, we consider initial data supported on arbitrary shell that have small C k norm (k ≥ 1) and obtain solutions that become large and are concentrated near the origin at some later time. We call them "focusing solutions". Specifically, we prove, for VP: Theorem 1.1. For any positive integer k and positive constants η, N, b, ǫ 0 , there exists a smooth, spherically symmetric solution of VP, such that (1.9) The contents of the paper are arranged as follows. In Section 2, we give some lemmas that describe the particle trajectories, which allow us to observe the focusing phenomena. Section 3 is devoted to the proof of Theorem 1.1, which involves a careful selection of parameters and computation of the norms of ρ(t, x) and E(t, x). At the end of Section 3 we also give the corollary on the setting with a bounded domain. The analogous result on RVP to Theorem 1.1 will be stated (see Theorem 4.1) and proved in Section 4.

Characteristics and Useful Lemmas
A spherically symmetric solution to VP or RVP can be described as (f (t, r, w, l), E(t, r)), where the spatial radius r, radial velocity w and square of the angular momentum l are defined as follows: for the system VP, and similarly, for RVP the Vlasov equation is reduced to Here The electric field is then and curl m(t, r)x This is enough to verify that the formula E(t, x) = m(t,r)x r 3 gives an E that satisfies the Vlasov-Maxwell system with B = 0. To see it matches the expression E(t, x) = R 3 x−y |x−y| 3 ρ(t, y)dy, note that they could only differ by the gradient of a harmonic function g(x). However, since we assume E(t, x) has finite L ∞ norm, g(x) must be finite too. By Liouville's Theorem, g must be a constant, which implies that the two E(t, x) expressions matches each other.
The total mass of the plasma is Next we introduce the characteristics for VP and RVP, as well as the lemmas that give detailed information for the particle trajectories.
then we can take the square root and obtain We rewrite this inequality as Integrating yields

Rearranging the inequality gives
Noticing that r 2 l+M r r 2 w 2 +l+M r is actually the minimum of the parabola (r + wt) 2 + (lr −2 + Mr −1 )t 2 in t, so the conclusion (3) follows. This completes the proof of the lemma.
For RVP the forward characteristics (R(s), W (s), L(s)) of the Vlasov equation are described by dR ds for s ≥ 0, with the initial conditions (2.10). We introduce the following lemma from [2]: Let r > 0, l > 0, w < 0 be given, and let (R(t), W (t), L(t)) be a solution to (2.13) and (2.10) for all t ≥ 0, and define Then we have: (3) There holds Proof.
Since 0 ≤ m(t, R(t)) ≤ M and W (t) ≤ 0, we have From the first inequality, we obtain . Evaluating this at t = T 0 and using W ( , we notice that this can be rewritten as Integrating over [0, t] yields Multiplying both sides by the conjugate and using for all t ∈ [0, T 0 ]. Evaluating this inequality at t = T 0 and rearranging gives R(T 0 ) ≤ R + , where R + := r D r 2 w 2 +D . Also, using this lower bound for W (t) 2 , we have We denote A = w 2 +Dr −2 1+w 2 +lr −2 , B = D 1+w 2 +lr −2 , and multiply the previous inequality by Integrating over [0, t] and rearranging gives Hence we have Lastly we establish the bound for T 0 . Direct calculation yieldsR(t) ≥ 0. Hencė Therefore, since w < 0, there holds Indeed, we can show T 0 < +∞ using argument by contradiction.
and hence for all t ≥ 0, Take t sufficiently large, we arrive at W (t) > 0. A contradiction. Hence T 0 is finite. This completes the proof of the lemma.
In particular, Also, we denote We choose T 0 > 0 as in Lemma 2.1 and 2.2 in the VP and RVP settings, respectively. We present the following lemma given in [1] and [2] in order to describe the concentrating phenomenon: 3. Let f (t, r, w, l) be a spherically-symmetric solution of RVP or VP with associated charge density ρ(t, r) and electric field E(t, x). Let (R(t, r, w, l), W (t, r, w, l), L(t, r, w, l)) be a solution to the equations of the particle trajectories with the initial condition (2.10). If at some time T > 0 we have Proof. Let f (t, r, w, l) be a given spherically-symmetric solution with initial data f 0 (r, w, l). The total mass M is conserved for all t. Also, we have S(T ) ⊂ [0, K] × R × [0, +∞). Using the radial form of the particle density, we have

This proves the first inequality.
For the second inequality, we denote (R(t, s, r, w, l), W (t, s, r, w, l), L(t, s, r, w, l)) to be the particle trajectory at time t of the particle that takes the trajectory value (r, w, l) at time s. We define For all s ≥ 0, f 0 (r, w, l) = f (s, R(s, r, w, l), W (s, r, w, l), L(s, r, w, l)) .

Concentrating Solution to the Nonrelativistic VP System
Now we are ready to establish Theorem 1.1.
We now prove that T (b) can be constructed as an increasing function for b > 1. Indeed, for b > 1, we want to take a 0 and ǫ such that (3.21) and (3.22) are satisfied. From the definition of d k , we notice that there exists n being large enough and independent of b, such that (3.24) nb ≥ 20d k η −1/3 C 1/3 0 holds for all b > 1. Hence we can take a 0 = nb so that (3.21) holds. Also, from (3.21) and the definition (3.22) of β 0 , we observe that for b > 1 there exists a constant C independent of b, such that Therefore we can take ǫ = C, so that (3.22) is satisfied. Using these chosen values of a 0 and ǫ, we have, for b > 1, which is an increasing function of b ∈ (1, +∞).
We choose T 0 = T 0 (r, w, l) > 0 for (r, w, l) ∈ S + as in Lemma 2.2. From the lemma we have
Next we prove Remark 1.2 i).
Proof. We assume 1 ≤ k ≤ 5. Recall that we want to take a 0 and ǫ such that (3.21) and (3.22) are satisfied. For b ∈ (0, 1), from the definition of d k , there existsñ independent of b, such that Hence we can take a 0 =ñb −k so that (3.21) holds. Also, recall (3.21) and the definition (3.22) of β 0 , we observe that for b ∈ (0, 1), there existsC independent of b, such that Hence we can take ǫ =Cb 3 . With these chosen values of a 0 and ǫ, we have When 1 ≤ k ≤ 5, we first pickñ large, then pickC small, such that when b ∈ (0, 1), the inequalities (3.35) and (3.34) as well asñC 2 > 9C 3 +1000ñ 2C 4 are satisfied. With this we can verify that when b ∈ (0, 1), T ′ (b) ≥ 0, and the termñC 2 b 6−k is dominant in the expression of T (b). Hence T (b) is an increasing function of b ∈ (0, 1).
Moreover, we construct T (b) for b > 1 as described in the proof of Theorem 1.1, so T (b) is an increasing function of b ∈ (1, +∞). Recall that the constraints on n and C are (3.24) and (3.25). We take n large enough such that not only (3.24) holds, but also Then (3.26) gives a function T = T (b) such that T (b) ≥ T (1) for all b > 1. Combining this together with (3.36), we obtain a function T (b) that is increasing on (0, +∞) for 1 ≤ k ≤ 5. This completes the proof of Remark 1.2 i).
The following corollary follows easily from Theorem 1.1. It tells us that the concentrating effect for the system (2.2), (2.4), (2.5), (2.6) also happens on certain type of bounded domains. 3 inf x∈∂Ω r(x), there exists some T > 0 and a smooth, spherically symmetric solution which has lifespan no less than T , such that Proof. The proof is identical as the one to Theorem 1.1.
We choose the initial data to be where Γ(w) = 0 for w ≥ 0, and Γ(w) = 1 for w < 0. Hence S(0)\S + is a zero-measure set.

Acknowledgement
The author thanks her advisor, Walter Strauss for all the guidance, encouragement and patience. Also, she thanks Jonathan Ben-Artzi for helpful discussions.