Low Field Regime for the Relativistic Vlasov-Maxwell-Fokker-Planck System; the One and One Half Dimensional Case

We study the asymptotic regime for the relativistic Vlasov-Maxwell-Fokker-Planck system which corresponds to a mean free path small compared to the Debye length, chosen as an observation length scale, combined to a large thermal velocity assumption. We are led to a convection-diffusion equation, where the convection velocity is obtained by solving a Poisson equation. The analysis is performed in the one and one half dimensional case and the proof combines dissipation mechanisms and finite speed of propagation properties.


Introduction
We consider a population of charged particles interacting both through collisions and the action of their self-consistent electro-magnetic field. The evolution of such a system is governed by the relativistic Vlasov-Maxwell-Fokker-Planck (VMFP) equations. After a dimensional analysis (see the Appendix) we obtain the following equations where ε, δ, θ are dimensionless parameters and v(p) = ∇ p E(p) is the scaled relativistic velocity (see (81)). Here ρ ε = R 3 f ε and j ε = R 3 v(p)f ε are respectively the charge and current densities of the distribution f ε and D, J are the charge and current densities of a background particle distribution of opposite sign, ensuring the global neutrality condition The dimensionless parameter ε > 0 is proportional with the scaled thermal mean free path and also with the scaled macroscopic velocity. We are interested in the asymptotic regime 0 < ε << 1, δ = O(1), θ = O (1). By neglecting the magnetic field we obtain the Vlasov-Poisson-Fokker-Planck (VPFP) system The asymptotic behavior of the non relativistic system (4), (5) when ε goes to 0 was studied in [31], [23]. It was shown that the limit (ρ, Φ) := lim ε 0 (ρ ε , Φ ε ) solves the following drift-diffusion system Another interesting regime is obtained by taking as small parameter ε the square of the ratio of the thermal mean free path with respect to the Debye length and by assuming that the distance travelled by the light during the relaxation time is of order of the Debye length. In this case we obtain the equations Notice that in (7) the non linear term E ε · ∇ p f ε is of the same order of magnitude that the diffusion Fokker-Planck term. This asymptotic regime is called the highelectric field limit and the non relativistic case was studied recently in [7]. The following limit system was obtained The high-field limit of the VPFP system was studied in [28], [22]. We analyze here the parabolic limit of the relativistic one and one half dimensional VMFP system, i.e., f = f (t, x, p 1 , p 2 ), E = (E 1 (t, x), E 2 (t, x), 0), B = (0, 0, B(t, x)) for any (t, x, p 1 , p 2 ) ∈ [0, T ] × R 3 . We derive a limit system very similar to (6), which was obtained when analyzing the VPFP system. Our proofs rely on compactness arguments. One of the crucial point is to obtain L ∞ bounds for the electro-magnetic field, uniformly with respect to the small parameter ε > 0. This is why we restrict our analysis to solutions depending on only one spatial variable. We obtain the equations where D, J : [0, T ] × R → R are given functions satisfying D ≥ 0 and the continuity equation We prescribe initial conditions for the particle distribution and the electro-magnetic field After integration of (11) with respect to p ∈ R 2 we deduce that the charge and the current densities verify the continuity equation By using the continuity equations for positive/negative charges and by taking the derivative of (12) with respect to x we deduce that (15) is a consequence of (18). Notice that if initially the neutrality condition is satisfied i.e., R R 2 f ε 0 (x, p) dp dx = R D(0, x) dx, then we have R R 2 f ε (t, x, p) dp dx = R D(t, x) dx for any t ∈]0, T ]. We consider only smooth solutions. Unfortunately, to our knowledge, there are no mathematical results concerning the existence and uniqueness of strong solution for the VMFP system. For the VPFP system the situation is better : results concerning the existence of weak solutions can be found in [13], [34] while for existence and uniqueness results of strong solution we refer to [8], [9], [17], [29]. The existence of classical solutions in the collisionless case has been investigated by different approaches, see [20], [10], [25]. Recently global existence and uniqueness results have been obtained for reduced model for laser-plasma interaction, cf. [14], [6].
The analysis of such asymptotic regimes is motivated by applications in the theory of semiconductors, the evolution of laser-produced plasmas or description of tokamaks. High-field asymptotics for the kinetic theory of semiconductors have been analyzed in [30], [15]. Results for different physical models have been obtained in [1], [4], [18], [27]. Generally we appeal to usual compactness methods. Another approach uses the modulated energy method, as introduced in [35]. This method has been used for studying various asymptotic problems in plasma physics [11], [12], [21], [33], [5], [24].
The paper is organized as follows. In Section 2 we establish a priori estimates, uniformly with respect to the small parameter ε > 0. These bounds are obtained by performing classical computations involving the energy and the entropy of the VMFP system and by using also the hyperbolic structure of the Maxwell equations. In Section 3 we detail the passage to the limit. The dimensional analysis can be found in the Appendix.

A priori estimates
In this section we establish a priori estimates for smooth solutions (f ε , E ε , B ε ) of the relativistic VMFP system in one and one half dimension. We will use the hypotheses H7) there is r > 1 such that sup ε>0 R R 2 (f ε 0 (x, p)) r e (r−1)E(p) dp dx < +∞.
We introduce the notations The following proposition states the usual bounds for the mass, energy and entropy (see Lemma 2.1 for the definition of the constant C 1/4 ).
be a smooth solution of the problem (11) − (17). Assume that the initial conditions satisfy H1, H2 and that H3, H4 hold. Then we have for any t ∈ [0, T ] The above estimates come by standard computations involving the energy conservation and the entropy dissipation. We use the following lemma, based on classical arguments due to Carleman.
is the scaled relativistic energy given by (81). Then for all k > 0 we have Proof. Therefore (|x|+E(p)) dp dx, and the conclusion follows easily.
Proof. (of Proposition 2.1) Integrating (11) with respect to (x, p) ∈ R × R 2 yields the charge conservation Similarly by H3 one gets R D(t, We multiply now the Vlasov equation by (1+ln f ε +E(p)) and integrate with respect to (x, p) ∈ R×R 2 . We obtain (20) Multiplying (12) by E ε 1 , (13) by E ε 2 and (14) by B ε yields after integration with respect to x ∈ R x)) dx. (21) By combining (20), (21) we deduce where In order to apply Lemma 2.1 let us multiply the Vlasov equation by |x| and integrate with respect to (x, p) ∈ R × R 2 . We deduce that (23) implying that Combining (22), (24) and Lemma 2.1 with k = 1 4 yields for any t ∈ [0, T ] which implies for any t ∈ [0, T ] and the first two statements follow easily by using Bellman lemma. For the last one write and we apply the Cauchy-Schwartz inequality.
Since we intend to use compactness arguments we need to estimate R ρ ε | ln ρ ε | dx. This can be done by using the standard result

Lemma 2.2 Assume that f is a non negative function satisfying
where C = sup{− √ y ln y : 0 < y < 1} and K = R 2 e −E(p) dp.

Corollary 2.1 Under the hypotheses of Proposition 2.1 we have for any
for some constant C T depending on T but not on ε.
Another way of estimating the solutions of the Fokker-Planck equation comes by multiplication with H (f e E(p) ), where H is a convex function, cf. [31].

Proposition 2.2
Assume that E ε , B ε are bounded smooth functions and that f ε is a smooth solution of (11), (16) with a non negative initial condition f ε 0 satisfying for some convex non negative function H. Then we have for any t ∈ [0, T ] Proof. After multiplication of (11) by H (f ε e E(p) ) we obtain After integration with respect to (x, p) ∈ R × R 2 one gets By Cauchy-Schwartz inequality and by taking into account that |v(p)| < 1/δ we obtain Combining (28), (29) yields Finally one gets for any t ∈ [0, T ] ) dp dx ds.
Corollary 2.2 Assume that E ε , B ε are bounded smooth functions and that f ε is a smooth solution of (11), (16) with a non negative initial condition f ε 0 satisfying Proof. By applying the previous proposition with the convex function H(s) = s r , s ≥ 0 we obtain We conclude by Gronwall lemma.
We are looking now for L ∞ bounds of the electro-magnetic field. We exploit the hyperbolic structure of the Maxwell equations and the entropy dissipation of the Fokker-Planck collision operator. We adapt the method used in [19], where L ∞ bounds of the electro-magnetic field have been obtained for the collisionless relativistic Vlasov-Maxwell system in one and one half dimension. Notice that the Maxwell equations (12), (13), (14) can be written Therefore the electro-magnetic field is given by where We need to find L ∞ bounds for the functions U ε , V ε ± . This can be done by using the local energy conservation and entropy dissipation.
For our further computations we will use the elementary results be the scaled relativistic energy, v(p) = ∇ p E(p) and δ = v th /(θc 0 ) (see the Appendix for the definitions of p th , v th , θ). Then we have the inequality In particular we have |v 1 Proof. For any p ∈ R 2 we obtain where q = p th mc 0 p. Obviously we have Combining (40), (41) yields Then for any (t, Proof. For any (t, x) ∈ [0, T ] × R consider the sets ∆ ε ± given by Integrating (42) with respect to (s, y) ∈ ∆ ε + yields and therefore we obtain The equality (44) follows by adding (45) Proof. Combining Proposition 2.3, and Lemma 2.4, (44) we obtain |h ε (s, y, p)| 2 dp dy ds |h ε (s, y, p)| 2 dp dy ds For any fixed (t, x) ∈ [0, T ] × R and s ∈ [0, t] we prove exactly as in Lemma 2.1 that where C = sup{− √ y ln y : 0 < y < 1}. We obtain the inequalities (|y|+E(p)) dp dy Combining (49), (51) yields |h ε (s, y, p)| 2 dp dy ds and finally by Lemma 2.3 one gets The estimate of U ε follows by applying Lemma 2.4 to the continuity equation Indeed, by (43) we have for any (t, f ε 0 (y, p) dp dy, and thus we deduce that ±U ε (t, x) ≤ M ε 0 .

Asymptotic analysis
We are now in position to perform the asymptotic analysis when ε goes to zero. The uniform estimates obtained in the previous section allow us to extract sequences as follows.
We focus our attention to the moment equations of (11). Integrating (11) with respect to p ∈ R 2 yields the continuity equation ∂ t R 2 f ε dp+ 1 ε ∂ x R 2 v 1 (p)f ε dp = 0. Let us multiply now by p 1 and integrate with respect to p ∈ R 2 .
We need to examine the limit as ε goes to zero of each term in the above equation. We identify easily the limits of all these terms, except for the term ∂ x R 2 v 1 (p)p 1 f ε dp.
In order to analyze formally this term, observe that by Proposition 2.1 we have where K = R 2 e −E(p) dp. In this case we obtain and thus we can guess that lim ε 0 ∂ x R 2 v 1 (p)p 1 f ε dp = ∂ x ρ. Multiplying now (11) by p 2 and integrating with respect to p ∈ R 2 yields ε∂ t Similarly one gets at least formally (p) ) dp = 0, and therefore lim ε 0 ∂ x R 2 v 1 (p)p 2 f ε dp = 0.

For any function
|p|f k dp dx dt , and therefore (64) holds provided that for any R > 0 we have By Cauchy-Schwartz inequality we deduce that , and therefore we are done if we prove that sup k∈N, x, p) dp dx < +∞. Indeed, we obtain by Hölder inequality where r is the conjugate exponent of r, i.e., 1/r + 1/r = 1. Consider now the term It remains to analyze the term E k and therefore lim k→+∞ E k . As before we have and for the term where and therefore, by using the strong convergence of (E k 1 ) k in L 1 loc ([0, T ] × R) and the weak convergence of (E k 2 ) k in L 2 (]0, T [×R) we deduce that lim k→+∞ Q k 1 = 0. By using (14), (12) we have ) and J belongs to L 1 (]0, T [; L ∞ (R)) we deduce that lim k→+∞ Q k 2 = 0. Thus we proved that lim k→+∞ E k 2 ∂ x E k 1 = 0 in D (]0, T [×R) and therefore the second convergence in (65) holds. The convergence (66) follows easily since (ε k δB k ) k is bounded in and thus we obtain for any (t, The convergences of Proposition 3.2 are sufficient for passing to the limit with respect to k in (61). We obtain the equations We have already proved that lim k→+∞ E k 2 = 0 weakly in L 2 (]0, T [×R). Under the hypothesis Take R large enough such that |x|>R ( and thus we have Therefore the first and second term in the right hand side of (70) vanish as k → +∞. For the last two terms observe that we have where for any (t, By using (62) we can write We are done if we prove that lim k→+∞ T k ±,l = 0, l ∈ {1, 2, 3, 4}. Observe that and After integration by parts, by taking into account that ψ k ± (T, ·) = 0 we find implying that lim k→+∞ T k ±,1 = 0. Similarly one gets by using (74) Notice that ϕ has compact support and then we deduce by (67) that lim k→+∞ T k ±,2 = 0. The convergence lim k→+∞ T k ±,4 = 0 follows by (66). Let us concentrate our attention on the convergence of (T k ±,3 ) k . Consider the functions Here Therefore we can write for any k ≥ k 1 (η) Take now k 2 large enough such that 1 dε k δ > R + d for any k ≥ k 2 . Observe that for saying that for any (t, Thus for any (t, Combining (75), (76) yields for any k ≥ max{k 1 (η), and we deduce that lim k→+∞ (35)). We have proved the theorem Theorem 3.1 Let (f ε , E ε , B ε ) be smooth solutions of the problem (11) − (17). Assume that H1-H7 hold and consider (ε k ) k the sequence constructed in Proposition 3.1. Then we have ρ k ρ ≥ 0 weakly in L 1 (]0, T [×R),

The limits ρ, E 1 satisfy in distribution sense
, uniformly on compact sets of R.
Moreover if H8 holds we have It is possible to show also that (f k ) k converges towards ρ(t, x) e −E(p) R R 2 e −E(q) dq in some sense. We need to establish first that (ρ k ) k converges towards ρ in C 0 ([0, T ]; w−L 1 (R)). As in [22] we can prove Then (ρ ε ) ε>0 is relatively compact in C 0 ([0, T ]; w−L 1 (R)).
By using the dominated convergence theorem we have x))M (p)ϕ(x) dx dp dt = 0.
It remains to discuss f k − ρ k M . By logarithmic Sobolev inequality (see [3], [2]) we Let f (t, x, p) denote the particle distribution function, which depends on the time t > 0, space coordinates x ∈ R 3 and impulsion coordinates p ∈ R 3 . The evolution of f is described by the Fokker-Planck equation where the relativistic Fokker-Planck collision operator is given by .
Here M(p) = e −E(p)/µ is the relativistic maxwellian where E(p) = mc 2 0 ( 1 + |p| 2 /(m 2 c 2 0 )− 1) is the relativistic energy, µ = KT th and the thermal impulsion p th > 0 is given by which is equivalent to p th = µ 2 /c 2 0 + 2µm. The evolution of the electro-magnetic field (E, B) is given by the Maxwell equations where ρ = q R 3 f dp, j = q R 3 v(p)f dp are respectively the charge and current densities. We denote by v th > 0 the thermal velocity given by By direct computation we check that p th v th µ = µ+2mc 2 0 µ+mc 2 0 =: θ ∈]1, 2[. We introduce a length unit L, a time unit T and the parameters α = T v th L , β = τ v th L . As impulsion unit we take P = p th . We define dimensionless variables and unknowns by the relations t = T t , x = Lx , p = p th p , where N is the total number of particles, U th is the thermal potential given by qU th = µ. After changing variables and unknowns, we obtain dropping the primes