On global solutions to the Vlasov-Poisson system with radiation damping

In this paper, the dynamics of three dimensional Vlasov-Poisson system with radiation damping is investigated. We prove global existence of a classical as well as weak solution that propagates boundedness of velocity-space support or velocity-space moment of order two respectively. This kind of solutions possess finite mass but need not necessarily have finite kinetic energy. Moreover, uniqueness of the classical solution is also shown.


Introduction.
As is known to all, the Vlasov-Poisson system is a typical nonlinear kinetic equation modeling the time evolution of a plasma or a galaxy at the mesoscopic level [13]. The research of the system culminated in the 1990s when two different proofs for global existence of classical solutions with general data were obtained almost simultaneously but independently by P. L. Lions and B. Perthame [23] and by K. Pfaffelmoser [33]. Before this, the first global existence result of a modified system was proved [2] and global existence of spherically or cylindrically symmetric solutions was established in [3,16,17]. In recent years a significant interest has been focused on the research of this system. Many research papers on the mathematical aspects of this theory have been published about uniqueness, propagation of moments and large time behavior of compactly supported classical solution (see e.g., [6,10,13,19,20,24,26,27,28,29,30,31,34,35,36,37,38] and the references therein).
Recently, based on the Vlasov-Poisson theory a new model for plasma physics by considering radiation damping effects has been introduced by M. Kunze and A. D. Rendall in [21], namely, where ε > 0 is a given small constant, f ± (t, x, v) are microscopic densities of two species of charged particles (i.e., ions ("+") and electrons ("−")) at time t ≥ 0 and The radiation reaction force in (1) is described by D [2] (t). Actually, effect of radiation damping should is characterized by the second derivativeD(t) = D [2] (t) + ε ... D(t) (ρ + + ρ − )dx of the dipole moment D(t) = R 3 xρ(t, x)dx. Nevertheless, the corresponding mathematical model becomes more intricate and there has been no any progress up to now (see [21,4] for detailed discussion). For another simplified model proposed in [4] which is slightly more complex than (1), S. Bauer established local existence of classical solutions for general smooth initial data f ± 0 (x, v), at present existence and asymptotic behavior of global classical solutions are obtained only for small initial data [7].
Here, (w) ∼ L 1 (R 6 ) denotes the topological vector space L 1 (R 6 ) equipped with its weak topology. Furthermore, large time behaviors for weak solutions like estimates (2)−(4) were also proved to hold true. All the above discussions and results concern for solutions with finite kinetic energy. In this paper, inspired by a recent work [9] we are aimed at constructing classical as well as weak solutions to system (1) which allow for infinite kinetic energy and establishing their large time behaviors. Through out the paper α is a given positive number, and for the sake of convenience we denote by C 1 0 (R 3 × R 3 ) the function class consisting of continuously differentiable functions f (x, v) defined on R 3 × R 3 vanishing at infinity. Now, we are in a position to give the first result concerning classical solutions with unbounded supports.
then there exists a unique global classical solution for any T > 0. Moreover, there is positive constant C depending only upon Remark 1. Compared with [21], we use weaker initial data, but obtain the global existence and uniqueness of classical solutions. The proof of the existence of global classical solutions does not depend on an a-priori bound of the kinetic energy. Actually, the solution constructed in Theorem 1.1 need not necessarily have finite kinetic energy. To overcome this difficulty we explore dispersive effect associated with potential energy and velocity-space moment of order two and deduce a-priori estimates for various macroscopic quantities.
The second result in this paper concerns for global weak solutions to system (1). A pair of nonnegative functions f ± (t, x, v) are said to be a global weak solu- x−y |x−y| 3 ρ(t, y)dy and D [2] x))dx are well-defined, and for any test function Then, we have , t > 0. Moreover, there exists a positive constant C depending only upon the initial data such that Throughout the paper, C denotes a generic positive constant which may depend on the initial data, but not on T . If a constant depends on T , we will give an explanation. C 0 , C 1 , · · · denote numerical positive constants and · p represents the norm of the usual Lebesgue space L p over R n with n = 3 or n = 6 as the case may be. Finally, B r = {x ∈ R 3 : |x| < M } denotes the ball in R 3 with radius M and centered at the origin.
The outline of the remainder of this paper is as follows. In section 2, we address the estimates of field and its derivative, meanwhile, establish a new conservation law which is different from [21]. Section 3 is devoted to establishing global existence and uniqueness of classical solutions to system (1). In section 4, we use the results in section 3 to investigate global existence of weak solutions to system (1). In appendix, we prove Proposition 2.

2.
Preliminaries. In this section, we collect some useful tools. The first one is the following interpolation inequality established in [9].
, and m ≥ l > −3, α > 0. There exists a constant C depending only on m, l and f ∞ such that for all t ≥ 0 and From Hardy-Littlewood-Sobolev inequality and Calderón-Zygmund inequality, we can show (see [21] for details): In addition, for smooth and compactly supported vector fields Γ : 3), and define Then we have where the constant C 0 > 0 depends only on p. If we further assume Proof. The proof of (10) is the same as that in [21], we only sketch the proof of (11). Combining the definition of E(x) with classical differentiability give that for any x ∈ R 3 , d > 0 and i, k = 1, 2, 3, we have With 0 < d 1 < d 2 < ∞, the above identity implies that Choosing d 2 = 1 and d 1 = 1/(1 + j=± Lip x (ρ j )), we get (11).
Under the framework of classical solutions, it is convenient to use characteristic curves of the Vlasov equation. Suppose E(t, x) ∈ C(R + ; C 1 b (R 3 )), then the characteristics generated by the field E(t, x) is defined by the ODE system By the Cauchy-Lipschitz theorem we know that there exists a unique global solution (X ± (s, t, x, v), V ± (s, t, x, v)) to (12), in the following we shall use the shorthand: (X ± (s), V ± (s)) = (X ± (s, t, x, v), V ± (s, t, x, v)).
It follows from Liouville theorem that the characteristics (X ± (s), V ± (s)) defines a C 1 homeomorphism from R 3 × R 3 onto itself which preserves Lebesgue measure. As usual, we can rewrite the Vlasov-Poisson system (1) along the characteristics and obtain By (12), it is easy to find that [2] (s).
Suppose that f ± (t, x, v) is a classical solution to (1) and verifies (6), its kinetic energy E kin (t) is not necessary finite although its potential energy E pot (t) remains bounded. Therefore, the energy dissipation equatioṅ obtained in [21] for compactly supported solutions is not available in the present situation. On the other hand, it seems to be reasonable to choose the inertia as an alternative for the kinetic energy in (15) due to (6). In fact, inspired by [9], we can really extend (15) to some extent. To this end, we need to introduce some notation. For any k > 0, we define Then, we are going to show and Moreover, there exists a constant C > 0 depending only on for any t > 0.
Proof. (16) follows from (13) and measure preserving of the characteristics. Now we are in a position to prove (17). Using the Vlasov equation in (1) and integration by parts, we have that d dt where On the other hand, we know from [21] that andĖ Inserting (20), (21) into (19) we then get (17). Finally, we show (18).
In both cases we obtain (22). Note that ε > 0 is a small constant (might as well let 0 ≤ ε ≤ 1), from (22) we get by Hölder's inequality that d dt Integration of this inequality from 0 to t ≥ 0 yields Consequently, the Gronwall's inequality deduce that Inserting this inequality into (23), we get 3. Proof of Theorem 1.1. To prove Theorem 1.1, we start with constructing local solution and establishing its continuation criterion. Actually, we have the following result.
Proposition 2. Assume that the initial data f ± 0 (x, v) satisfy the conditions as stated in Theorem 1.1.
then T max = ∞, i.e., the solution is global in time.
The proof of this Proposition is postponed to the Appendix. Based on Proposition 2, in order to prove global existence of classical solution as stated in Theorem 1.1 it is sufficient to show that condition (28) holds true. To this end, we shall adopt the method developed in [9,23].
) be the solution constructed in Proposition 2 with T max being its maximal life span. Then for any k > 3 Proof. Due to estimate (25), we know that for any fixed t ∈ [0, T max ) and k ≥ 0. Therefore, what we need to show is to prove that H ± k (t) does not blow up at T max . Similar to [9,23], the proof is divided into several steps.
Step 1. An inequality of H + k (t). In view of the Vlasov equation in (1), integration by parts, Hölder's inequality, and Lemma 2.1 we obtain /k and without loss of generality we may assume that H + k (t) > 1, then we deduce that Thus, we get that It is obvious that the last two inequalities also hold true if we replace H + k by H − k . Consequently, they hold true with H + k replaced by H k .
Step 2. The estimates of E(t) k+3 and the term of D [2] (t). Based on references [9,21,23], we have 3 , we get by Hölder's inequality that where C > 0 depends only on f ± 0 1 . By Lemma 2.1, we get Combining the above two estimates with Minkowski's inequality, we then have Next, we estimate σ(t) k+3 . For some t 0 ∈ (0, T max ), which will be specified later, we have by the definition of σ(t), We firstly compute (Ef To estimate J 1 , we choose r close enough to 3 such that max{ 3(k+3) k+4 , 6−k} ≤ r < 3, then by (35) and (26) we have that In the above, we have used the fact E(t) r ≤ C. Actually, from weak Young's inequality (see [35]) we have E(t) r ≤ C ρ(t) q for 1+ 1 r = 1 q + 2 3 and r, q ∈ (1, ∞), then by (26) we also have ρ(t) q ≤ C for any q ∈ [1, 5 3 ].
Similarly, replacing f + and H + k by f − and H − k respectively, the above inequality remains true. Then we obtain that 1 k+3 .
Inserting (41), (42), (34) and (33) into (32) we then get that Step 3. Gronwall estimate of H k (t). Combining (43) and (30), we get that By the definition of m we have 2− 3 Now, assume that sup{H k (t) : t ∈ [0, T max )} = ∞. By monotonicity there exists a unique time t * ∈ (0, T ) such that Then putting this t 0 into (44), we have for any t ≥ t * . Integrating it against t we get Thus, Gronwall's inequality implies that H k (t) is bounded on [0, T max ), which is a contradiction to the assumption. This concludes the proof of (29).

4.
Proof of Theorem 1.2. Taking a sequence of smooth function f ± 0,n (n = 1, 2, . . .), each of which satisfies the assumption (5) and (6) of Theorem 1.1 such that where s ∈ [1, ∞]. Using these regularized initial data, we construct approximate equations of system (1) as follows According to Theorem 1.1, system (47) has a unique nonnegative solution f From Proposition 1 and Theorem 1.1, we know that for t ≥ 0 Proof of Theorem 1.2. By virtue of the reflexivity of L p space (1 < p < ∞), the Banach-Alauglu theorem for L ∞ and the Dunford-Pettis theorem for L 1 , as well as estimate (48) and (49), there exists a pair of nonnegative functions f ± (t, x, v) defined on [0, ∞) × R 6 such that as n → ∞, for T < ∞ and up to a subsequence. Next, we show that f ± n is equicontinuous. As f ± n is a classical solution, for any φ(x, v) ∈ C ∞ c (R 6 ), due to (50) we get The second term on the right hand side can be handled according to the discussion in [18], thus, we get where C is a constant and only depends on f ± 0 1 , f ± 0 ∞ , ∇ x,v φ ∞ and suppφ. Then we obtain the desired equicontinuity. Thus, following from [18], there exists a nonnegative function f ± (t, x, v) defined on R 6 such that for any t ∈ [0, ∞) as Then, a further computation shows that f ± = f ± . Thus, the weakly (weakly * ) lower semicontinuous of norms implies To proceed further, especially to prove the continuity of nonlinear terms in (1), we need the following velocity averaging lemma. and where the positive constant C depends only on φ and its support.
Velocity averaging lemmas was discovered at the middle of 1980s and have been proved to be a greatly useful tool in kinetic theory (see e.g.: [11,12,14,15,22,32]). The above one is a variant among others which was deduced in [8] by the method of [5].
). In view of (48)−(50), we get where C is a constant depending only on f ± 0 1 , f ± 0 ∞ , M ± 2 (0), ψ ∞ and ∇ v ψ ∞ . By Lemma 4.1, we then have that On the other hand, we further have that Actually, from the Vlasov equation in (1), we deduce We estimate the two terms on the right hand side separately: In order to finish the proof, we also need the following lemma (see e.g. [1]).
In particular, we have extracting a subsequence if necessary Obviously, the above conclusion also holds true for f − n . Now, we show In fact It follows from (49) that there exists a positive constant C such that Q 2 ≤ C/R 2 0 . For any > 0, we choose R 0 so large such that Q 2 < /2. For such fixed R 0 , we get Due to this estimate and (51), we have Q 1 → 0 as n → ∞. Thus, the 2 argument shows that (52) holds true. Consequently, we have (up to a subsequence) Using Gronwall's inequality and (45), we get for 0 ≤ t ≤ T , where C is a constant depending only on f ± 0 1 , f ± 0 ∞ , M ± 2 (0) and |x|f ± 0 1 . Furthermore, by (53) it comes
Since for any ϕ(t, Passing to the limits n → ∞ and using the above estimates, we get for any So f ± is a global nonnegative weak solution to (1). Taking limit n → ∞ in (49) we have This completes the proof of Theorem 1.2.

Appendix.
Proof of Proposition 2. Let S ± be subsets of x, v) which satisfy the following conditions: j=± where t ∈ [0, T ] and T > 0. R(t) = R + (t) + R − (t) and R ± (t), L(t) > 0 are continuous functions, which will be fixed later. It is clear that S ± is a closed and bounded convex subset of C b ([0, T ] × R 3 × R 3 ). For any g ± (t, x, v) ∈ S ± , let g = g + − g − . Then by virtue of the definition of E(t, x) and D [2] (t), we have x))dx.
Let (X ± g (s), V ± g (s)) = (X ± g (s, t, x, v), V ± g (s, t, x, v)) be the solution to the ordinary differential system then the solution to the linear partial differential equation ). In this way we shall assign a pair of functions (f + , f − ) to the given a pair of functions (g + , g − ), which are denoted by (f + , f − ) = F(g + , g − ). Therefore, F is a map defined on S + × S − . Clearly, if we can show that F maps S + × S − into itself and is continuous and compact, then the Schauder's fixed point theorem ensures that F has a fixed point First, we display F maps S + × S − into itself. It is only to verify (56). Note that Combining (10) with the definition of S ± , we obtain where the maximal solution of the integral equation Combining the definition of D [2] g (t), (14) with (58), we get that for any 0 ≤ s ≤ t < δ and (x, v) ∈ suppf ± 0 . By virtue of f ± (t, X ± g (t, 0, x, v), V ± g (t, 0, x, v)) = f ± 0 (x, v) and (59), we finally get that If δ 0 ∈ (0, δ), we have that for any t ∈ [0, δ 0 ], R(t) ≤ C and E g (t) ∞ ≤ C, where the constants C only dependent on f ± 0 1 , f ± 0 ∞ , R ± 0 and δ 0 . We note that . We fix x, v ∈ R 3 and write (X ± g , V ± g )(s) instead of (X ± g , V ± g )(s, t, x, v). If we differentiate the characteristic system (57) with respect to x and integrate with respect to time, we get By Gronwall's inequality,

and hence
j=± Inserting the uniform bound on [0, δ 0 ] of ρ ± g (t) ∞ and the above estimate into (11), we find that .
Next, we estimate J 3 and we have