ASYMPTOTIC BEHAVIOR FOR THE VLASOV-POISSON EQUATIONS WITH STRONG UNIFORM MAGNETIC FIELD AND GENERAL INITIAL CONDITIONS

. We investigate the Vlasov-Poisson equations perturbed by a strong external uniform magnetic ﬁeld. We study the asymptotic behavior of the solutions, based on averaging techniques. We analyze the case of general initial conditions. By ﬁltering out the oscillations, we are led to a proﬁle. We prove strong convergence results and establish second order estimates.


MIHAÏ BOSTAN
For any particle density f = f (x, v), we denote by E[f ] the Poisson electric field associated to the charge density ρ[f ] = q R 2 f (·, v) dv. We also use the notation j[f ] = q R 2 f (·, v)v dv for the current density of f . We complete the above system by the initial condition Very recently, the Vlasov-Poisson equations with strong external non homogeneous magnetic field have been studied, when considering well prepared initial densities [9,14]. As usual, we are looking for quantities with small variations over a cyclotronic period (the guiding center), and we average with respect to the fast cyclotronic motion in order to obtain the effective Vlasov equation. Following this strategy, most of the previous studies provided formal or rigorous (based on compactness arguments) gyrokinetic approximations for the transport of charged particles under the action of strong external magnetic fields. The subject matter in [9] was to derive second order regular reformulations of the Vlasov-Poisson equations with strong magnetic field and to perform the error analysis. One of the key point was to split the advection field of the Vlasov equation into a fast and slow dynamics, such that the guiding center is left (exactly) invariant by the fast dynamics. It was shown that in this case the fast dynamics becomes periodic (even for a non homogeneous magnetic field), and therefore the homogenization procedure simplifies a lot : instead of taking ergodic means, it is enough to average over one period. When the magnetic field is uniform, it is possible to go further in our analysis, by considering smooth initial particle densities, not necessarily well prepared. We mention that most of the studies concentrate only on models with well prepared initial particle densities. We intend to extend the analysis in [9] for general initial conditions. The asymptotic behavior is more complicated because the particle densities (f ε ) ε>0 present fast oscillations in time. We appeal to a two scale approach by working in a extended phase space supplemented by a fast time variable s = t/ε. Up to a second order term, the oscillations of the family (f ε ) ε>0 can be described in terms of a profile solving a regular reformulation of the Vlasov-Poisson equations, see also [25].
be a non negative, smooth, compactly supported particle density. We denote by (f ε ) ε>0 the solutions of the Vlasov-Poisson equations (1), (2) with uniform external magnetic field B ε = 0, corresponding to the initial condition f in . The notation R(θ) stands for the rotation of angle θ ∈ R and ω c = qB/m is the rescaled cyclotronic frequency. Then for any T ∈ R + , there is a constant C T > 0 such that for any whereF is the solution of corresponding to the initial conditioñ Our paper is organized as follows. In Section 2 we derive formally the effective Vlasov-Poisson equations. We introduce the average operators along the fast dynamics and discuss their main properties. The effective model is studied in Section 3, and we summarize its properties. The well posedness of this model is a direct consequence of the well posedness of the vorticity formulation for the 2D incompressible Euler equations. Section 4 is devoted to the error analysis. We establish second order estimates, by constructing a suitable corrector on the extended phase space, supplemented by the fast time variable. Some generalizations are indicated in the last section.
2. Asymptotic analysis by formal arguments. The well posedness of the Vlasov-Poisson problem is well known, see [1] for weak solutions, and [26,21,23,2] for strong solutions. Essentially the same arguments provide the global existence and uniqueness for the strong solution of the Vlasov-Poisson problem with external magnetic field cf. Theorem 2.1 [9].
Theorem 2.1. Consider a non negative, smooth, compactly supported initial particle density f in ∈ C 1 c (R 2 × R 2 ) and a smooth magnetic field B ∈ C 1 b (R 2 ). There is a unique particle density f ∈ C 1 (R + × R 2 × R 2 ), whose restriction on [0, T ] × R 2 × R 2 is compactly supported for any T ∈ R + , whose Poisson electric field is smooth We assume that the initial particle density is smooth c (R 2 × R 2 ) and that the magnetic field is uniform B ε = B/ε = 0. We know by Theorem 2.1 that for every ε > 0, there is a unique strong solution f ε ∈ C 2 (R + × R 2 × R 2 ) (whose restriction on [0, T ] × R 2 × R 2 is compactly supported for any T ∈ R + ), for the Vlasov-Poisson problem with external magnetic field B ε = B/ε. By standard arguments, we also have uniform estimates with respect to ε > 0 for the L ∞ norm of the electric field E ε | [0,T ]×R 2 and the size of the support of the particle density f ε | [0,T ]×R 2 ×R 2 . Let us denote by (X ε , V ε )(t; t 0 , x, v) the characteristics associated to (1) The strong external magnetic field induces a large cyclotronic frequency with respect to the reciprocal advection time scale, and therefore a fast dynamics. Indeed, by introducing the characteristic scales (t, x, v) for time, length, velocity, we have t = x/v and ω ε c t ∼ 1/ε. We use the notations ω ε c = qB ε /m = ω c /ε, ω c = qB/m ∼ 1/t. It is well known that the guiding center, X ε (t) + ε ⊥ V ε (t)/ω c has small variations over one cyclotronic period [14,22,25]. Another quantity having small variations is ], see [20].
Lemma 2.2. The following quantities have small variations in time.
Proof. By direct computations we have B .
Notice also that Motivated by the above calculation, we introduce the relative velocity with respect to the electric cross field drift and the new particle densitỹ It is easily seen that Therefore the Poisson electric fields corresponding to the particle densities and we can use the same notation E ε (t) for denoting them. We introduce the notations Observe that the new particle densities (f ε ) ε>0 are smooth,f ε ∈ C 2 (R + × R 2 × R 2 ) and that the restrictions to [0, T ] × R 2 × R 2 are compactly supported, uniformly with respect to ε ∈]0, 1], for any T ∈ R + . We obtain the following problem in the new coordinates (x,ṽ) Thanks to the continuity equation the time derivative of the electric field E ε can be written in terms of the particle density f ε (orf ε ) The change of coordinates (5), (6) leads to the problem where b ε · ∇ x,ṽ = εṽ · ∇ x + ω c ⊥ṽ · ∇ṽ and for any particle densityf , a ε [f ] · ∇ x,ṽ stands for the vector field Notice that the vector fields It is periodic, and has the same period as the characteristic flow (X(s; The properties of these flows are summarized below. The proof details are left to the reader (see also Proposition 3.1 [9]).
and by (X(s; x,ṽ),Ṽ(s; x,ṽ)) the characteristic flow of the autonomous vector field 3. For any continuous function u ∈ C(R 2 × R 2 ) we define the averages along the For any R x , Rṽ ∈ R + we have

For any function
and In order to filter out the fast oscillations corresponding to the vector field b ε (x,ṽ) ε · ∇ x,ṽ , we perform one more change of coordinates By applying the chain rule, we obtain where we have used the notations As the characteristic flow (X ε ,Ṽ ε ) in (8) is linear, the jacobian matrix simply writes for any ( and therefore (10) becomes We have obtained a two scale problem and we expect that the asymptotic behavior when ε becomes small will follow by averaging with respect to the fast time variable s = t/ε. As we are looking for second order approximations, we only need to average, with respect to s, when ε is small up to terms of order ε. By the second statement in Proposition 1, observe that and therefore we deduce that We concentrate now on the average of R(ω c s)a ε v [F ε (t) −s ] s · ∇ṼF ε . We only need to consider the contributions of order 1 in

MIHAÏ BOSTAN
Notice that, up to terms of order ε, the average of the first contribution writes ωc 2π By using the properties of the fundamental solution z → − 1 2π ln |z|, it is easily seen that and thus the above computations lead to Similarly, up to terms of order ε, the second contribution is, thanks to (11) ω c 2πB Combining (13), (14) we obtain Thanks to (12), (15) we are led to the model which is supplemented by the initial conditioñ 3. The effective Vlasov-Poisson equations. We expect that solving (16) together with the initial condition (17), will provide a second order approximation for (1), (2). Although the above solution depends on ε, we use the notationF , saying that it is an approximation, when ε becomes small. The well posedness of the limit model (16), (17) is a direct consequence of the well posedness of the vorticity formulation for the 2D incompressible Euler equations, see also Lemma 3.3 [25]. Indeed, integrating (16) with respect to the velocity leads to the Euler equations which allows us to determine ρ and E. Multiplying (16) byṼ and integrating with respect to the velocity give a transport equation for the current density as well and finally the particle densityF comes by solving the linear transport equation (16), with smooth advection field The proof details are left to the reader.
Theorem 3.1. Consider a non negative, smooth, compactly supported initial particle densityF in ∈ C 1 c (R 2 ×R 2 ) and a uniform magnetic field B ε = B ε = 0. There is a unique particle densityF ∈ C 1 (R + × R 2 × R 2 ) whose restriction on [0, T ] × R 2 × R 2 is compactly supported for any T ∈ R + , whose Poisson electric field is smooth E[F ] ∈ C 1 (R + × R 2 ), satisfying (16), (17). Moreover, if for some integer k ≥ 2 we The properties of the above limit model are summarized below : conservation of the total kinetic energy, conservation of the total electric energy, invariance of  1. The total kinetic energy is conserved. Indeed, multiplying (16) by m|Ṽ | 2 /2 and integrating with respect to velocity yields and our conclusion follows by integrating also with respect to X.

The total electric energy is conserved. Using the fundamental solution of the
Poisson equation in R 2 , we have d dt is an invariant of the transport operator in (16). Indeed, thanks to the mass and momentum balances, we deduce that 4. The model (16), (17) is invariant under rotation in the velocity space. More exactly, ifF solves (16), (17), thenF θ (t, X,Ṽ ) =F (t, X, R(θ)Ṽ ) solves (16) together with the initial conditioñ In particular, if the initial particle density satisfies b · ∇ X,ṼF in = 0, then, thanks to the uniqueness, we have b · ∇ X,ṼF (t) = 0 at any time t ∈ R + .

4.
The error analysis. The solution of (16), (17) will allow us to describe the asymptotic behavior of the family (f ε ) ε>0 corresponding to the initial condition f in , when ε becomes small. Let us introduce the particle densitỹ whereF solves (16), (17) and (X,Ṽ) is the characteristic flow associated to the vector field b(x,ṽ) · ∇ x,ṽ = ω c ⊥ṽ · ∇ṽ, see Proposition 1. The idea is to comparẽ f ε (t, x,ṽ) with respect to f (t, x + ε ⊥ṽ /ω c ,ṽ), where f (t, x,ṽ) =f (t, t/ε, x,ṽ). By direct computation we check that We introduce the application T ε (x,ṽ) = (x + ε ⊥ṽ /ω c ,ṽ). Notice that f (0) • T ε is a second order approximation off ε (0) in L 2 (R 2 × R 2 ) By introducing a corrector term, we will prove that f • T ε is a second order approximation off ε in L ∞ loc (R + ; L 2 (R 2 × R 2 )). We mention that the asymptotic behavior for a very similar problem has been investigated in Theorem 1.2 [25], but without indicating the convergence rate. Our goal is to complete the asymptotic analysis by justifying the second order approximation. The arguments developed for well prepared initial particle densities in [9] apply for general initial particle densities as well, justifying the robustness of our method. For any smooth particle densitỹ For constructing the corrector, we need essentially to invert the transport operator ∂ s + b · ∇ x,ṽ on the subspace of functions with zero average with respect to the characteristic flow of the vector field ∂ s + b · ∇ x,ṽ , see [7,8]. The expression of the corrector is explicit and follows by direct computations. Its smoothness and uniform boundedness with respect to the fast time variable will be crucial when establishing the error estimate. We consider particle densities depending also on the fast time variable and therefore we work in the phase space (s, x,ṽ) ∈ R × R 2 × R 2 .
Proposition 2. Assume thatf =f (s, x,ṽ) ∈ C 1 (R × R 2 × R 2 ) is S = 2π |ωc| periodic and uniformly compactly supported in (x,ṽ) with respect to s ∈ R, such that Then we have the equality Proof. The particle densityf satisfies the constraint (∂ s +b·∇ x,ṽ )f = 0 and therefore we havef Observe Notice that As already observed in (13), the average with respect to s of whose zero average primitive with respect to s is Coming back to (20), and taking into account that ∇ṼF (x, R(ω c s)ṽ) belongs to the kernel of ∂ s + b · ∇ x,ṽ , since it depends only on the invariants x, R(ω c s)ṽ of Similarly we obtain Our conclusion follows by combining (21), (22).
Proof. (of Theorem 1.1) Estimating the error betweenf ε (t), f (t) • T ε is not of all obvious. A direct comparison between the above densities does not lead to the desired error estimate. We have to use a corrector term. More exactly, for any t ∈ R + , the particle density (s, x,ṽ) →f (t, s, x,ṽ) =F (t, (X,Ṽ)(−s; x,ṽ)) belongs to C 2 (R × R 2 × R 2 ), is S = 2π |ωc| periodic with respect to s and uniformly compactly supported in (x,ṽ), with respect to s ∈ R and t ∈ [0, T ], T ∈ R + . By definition, for any t ∈ R + , the particle densityf (t) satisfies the constraint (∂ s + b · ∇ x,ṽ )f (t) = 0. Thanks to Proposition 2, there isf 2 (t, s, x,ṽ) such that It is easily seen, by the explicit formula off 2 , that the correctorf 2 belongs to |ωc| periodic with respect to s, and uniformly compactly supported in (x,ṽ) with respect to s ∈ R and t ∈ [0, T ], T ∈ R + . Taking s = t/ε in (24), multiplying by ε and combining with (18) yield Using the correctorf 2 led to a model similar to (7), nevertheless the transport operators appearing in the terms b ε · ∇ x,ṽ f (t) + ε 2f 2 (t, t/ε) , b ε ε · ∇ x,ṽf ε are different, and the key point is to take advantage of the map T ε . After composition with T ε , the above equation becomes, thanks to (23) Combining with (7) and using the notation r ε (t) =f ε (t)−f (t)•T ε −ε 2f 2 (t, t/ε)•T ε , lead to As the magnetic field is uniform, the vector fields a ε [f ε ], b ε are divergence free and therefore Multiplying (25) by r ε (t, x,ṽ) and integrating with respect to (x,ṽ) imply THE VLASOV-POISSON EQUATIONS WITH STRONG MAGNETIC FIELD   545 and by Bellman lemma one gets We are working for t ∈ [0, T ], T ∈ R + , and we denote by C any constant depending on m, ε 0 , q, T, B and the initial particle density f in , but not on ε. Thanks to (19) we have and thus clearly r ε (0) L 2 ≤ Cε 2 . Using the C 1 c regularity off 2 which comes from the C 2 c regularity ofF , it is straightforward that We claim that This part of the proof relies on the smoothness and the uniform estimates of the densities (f ε ) ε>0 , together with the elliptic regularity results. Thanks to the uniform bounds sup Thanks to elliptic regularity results, the quantity is bounded by the L 2 norms of the charge and current densities and thus by the L 2 norm of the particle densities f ε (t) − f (t) • T ε L 2 (R 2 ×R 2 ) . For any ε > 0 we obtain the inequality (a[f ε (t)]−a[f (t)•T ε ])·∇(f (t)•T ε ) L 2 ≤ C f ε (t)−f (t)•T ε L 2 (R 2 ×R 2 ) , t ∈ [0, T ]. The inequality (27) follows immediately, noticing that a[f (t) • T ε ] · ∇(f (t) • T ε ) − (a[f (t)] · ∇f (t)) • T ε L 2 (R 2 ×R 2 ) ≤ Cε, t ∈ [0, T ], ε > 0. Finally (26) writes By Gronwall lemma we deduce f ε (t) − f (t) • T ε L 2 ≤ Cε 2 exp(Cεt), t ∈ [0, T ], ε > 0. Clearly we have

5.
Conclusions. We presented regular reformulations for the Vlasov-Poisson equations with uniform magnetic fields and general initial conditions. The effective model comes by averaging over one cyclotronic period, once we have determined a periodic fast dynamics. Certainly, in the framework of the magnetic confinement, a much more interesting case is that of curved magnetic fields. The above results extend to the three dimensional Vlasov-Poisson system, with strong external curved magnetic field. The same arguments lead to regular reformulations and second order estimates for both well prepared and general initial particle densities. In the three dimensional setting, we emphasize a fast periodic dynamics leaving invariant not only the guiding center and the modulus of the perpendicular velocity, but also the parallel velocity. Averaging over one period allows us to obtain completely explicit effective models. Nevertheless the analysis is much more elaborated due to the combination between the parallel and perpendicular dynamics and to the curvature effects. These studies will be the topic of future works [10].
Another interesting issue will be to handle models with self-consistent magnetic field, that is the Vlasov-Maxwell equations, perturbed by a strong external magnetic field.