Quasineutral limit of the Euler-Poisson system under strong magnetic fields

The quasineutral limit of the three dimensional compressible Euler-Poisson (EP) system for ions in plasma under strong magnetic field is rigorously studied. It is proved that as the Debye length and the Larmor radius tend to zero, the solution of the compressible EP system converges strongly to the strong solution of the one-dimensional compressible Euler-equation in the external magnetic field direction. Higher order approximation and convergence rates are also given and detailed studied.

1. Introduction. We consider the following Euler-Poisson system where n(t, x) and u(t, x) are the density and the velocity vector of the ions in a plasma with magnetic field, φ(t, x) is the electric potential at time t > 0 and x ∈ R d , d ≤ 3. Here e = (0, 0, 1) T and e/ε is the magnetic field with magnitude 1/ε such that if u = (u 1 , u 2 , u 3 ) then e × u = (−u 2 , u 1 , 0), and λ > 0 is the scaled Debye length, which is a small quantity compared to the characteristic length of physical interest for typical plasma applications. The parameters in this equation have obvious physical meanings. The parameter T i is the temperature of the ions, which are called cold when T i = 0. The parameter ε is proportional to the Larmor radius, which is physically understood as the typical radius of the helix around the axis x 3 that the particles follow, due to the intense magnetic field. The Euler-Poisson system and its related models (Vlasov-Poisson, two fluid Euler-Poisson and so on), and their asymptotic behaviors have attracted the interests of mathematicians and physicists during the past two decades, possibly due to their applications to plasmas and semiconductors. In this paper only the quasineutral limit for (1) is concerned.
Since the Debye length is small, the plasma is electrically neutral, i.e., there is no charge separation or electric field, and this limit is widely used in practice. Therefore it is important to justify this 'zero Debye length' limit mathematically rigorously, also known as the quasineutral limit. We say the magnetic field is strong in the sense that as the Debye length λ → 0, the magnitude of magnetic field 1/ε → ∞.
where (n j , u j , φ j ) denotes the velocity vector field of the j th approximation. Assume accordingly that the initial data (n ε 0 , u ε 0 , φ ε 0 ) admit such an asymptotic expansion where (n j 0 , u j 0 , φ j 0 ) j≥0 are sufficiently smooth in T 3 .

Derivation of the Euler equation.
Inserting the ansatz (2) into (1) and balance the powers of ε, we obtain the following systems of equations. At the order O(ε −1 ), we obtain At the order O(1), we obtain thanks to (3). From (3), (4b) and (4c), we obtain This leads us to consider the compatibility condition of the initial data from (4) It follows from (4a), (4d) and (4e) that the leading profiles satisfy the following one dimensional compressible Euler equation in the x 3 -direction This implies that there is no more dynamics in the (x, y) plane, which can be interpreted as a confinement result. We also assume that the plasma is uniform and electrically neutral near infinity. More precisely, letñ be a smooth strictly positive function, constant outside x ∈ [−1, 1], satisfyingñ → n ± > 0 and u → 0 as x → ±∞. We also assume that the initial conditions (n 0 0 , u 0 3,0 ) satisfy for some constant σ > 0 and s > 1 + d 2 sufficiently large. Then the limiting system (7) is hyperbolic symmetrisable, whose classical result for the existence of sufficiently regular solutions in small time can be stated as (see [9]) Theorem 2.1. Let s > d/2 + 1 and (n 0 0 , u 0 3,0 ) be initial data such that (8) holds. Then there exists T * > 0, maximal time of existence and a unique solution (n 0 (t), u 0 3 (t)) to the Cauchy problem (7) with initial data (n 0 0 , u 0 3,0 ) such that (n 0 −ñ, u 0 In what follows, we will work on a time interval [0, T ] with T < T * , arbitrary close to T * in order to insure 0 < σ < n 0 (t, x) < σ for all (t, x) for some positive constants σ and σ .

2.2.
Determinacy of the profiles (n j , u j , φ j ) j≥1 . Proceeding as above, we obtain at the order O(ε) that Since u 1 1 and u 1 2 have already been solved from (5), we need only to solve n 1 , u 1 3 and φ 1 in (9), which reduces to the one-dimensional linearized compressible Euler equation for (n 1 , u 1 3 ) From (10), we can solve n 1 , u 1 3 and φ 1 , and then by inserting the result to (9), we can further solve u 2 1 and u 2 2 . We also note that the system (10) is self-contained and do not depend on the (n j , u j , φ j ) for j ≥ 2. 3 ) to (10) with initial data (n 1 0 , u 1 Generally, we obtain the coefficients at order O(ε k ) for k ≥ 1 as follows We note that to the k th approximations, in (11), ) has already known in the (k − 1) th step, and thus the initial conditions for (11) is only given by (n k 0 , u k 3,0 ).

Main results.
To show that (n ε , u ε ) converges to the solution of the compressible Euler equation, we need to make the above procedure rigorous. Let (n ε , u ε , φ ε ) be a solution of (1) of the following expansion where (n app , u app , φ app ) denotes the approximation of the solution (n ε , u ε , φ ε ) to the M th order, (n R , u R , φ R ) denotes the remainder terms of the approximation, . Inserting the expression (12) into (1), and subtracting the equations satisfied by (n app , u app , φ app ) ((4), (11) for 1 ≤ k ≤ M ), we obtain the remainder system for (n R , u R , φ R ): where To state the main results, we define Let The main result in this paper is the following be initial data such that (8) holds and (n 0 (t), u 0 3 (t)) be the solution of the Euler system (7) on [0, T ) given by Theorem 2.1, and (n j (t), u j (t), φ j (t)) are solutions of (11) with initial data (n j 0 , u j 3,0 ) given by Theorem 2.3 for 1 ≤ j ≤ M . Assume that the initial data (n ε 0 , u ε 0 ) satisfy the compatibility conditions (4), (9) and (11) for all 0 ≤ k ≤ M and Then there exists ε 0 > 0 and solutions (n ε (t, x), u ε (t, x), φ ε (t, x)) of (1) with initial data (n 0 (t), 0, 0, u 3 (t)) on [0, T ε ) with lim inf ε→0 T ε ≥ T . Moreover, for every T < T and for all 0 < ε < ε 0 , there holds where the constant C is independent of ε and {n j , u j , ψ j } 0≤j≤M are solutions to the problems (7) and (11).
Remark 1. When M = 1, we obtain This implies that as ε → 0, the solution to the Euler-Poisson system (1) converges in the H s -norm to the solution of the Euler equation (7) with the same initial data. (15) is independent of the temperature T i for T i small. Precisely, there exists T em > 0 such that the constant C is independent of T i ∈ [0, T em ]. This implies that the quasineutral limit result in Theorem 2.4 also applies for all T i ≥ 0.

Remark 2. The estimate in
3. Rigorous justifications. This section is dedicated to proof of Theorem 2.4. Before we prove the main results, we first give two calculus inequalities, which will be frequently used throughout.
Lemma 3.1. Suppose that s > 0 and p ∈ (1, +∞). There hold and with p 2 , p 3 ∈ (1, +∞) such that and f, g are such that the right hand side terms make sense.
In particular, when s ≥ 2, H s is an algebra. Then if f, g ∈ H s then f g ∈ H s with where we may take p 1 = 3, p 2 = 6, p 3 = 2, p 4 = ∞ and p = 2 for example.
Let N = n R , U = u R and Φ = φ R . We can rewrite (13) in terms of (N, U, Φ) as For fixed T i ≥ 0, we will give uniform (in ε) estimates for the solution (N, U, Φ) to (16). To simplify the representation slightly, we assume that (1) has smooth solutions in very small time T ε dependent on ε. This fact can be proved by classical arguments by adapting the following proof. LetC be a constant to be determined later, much larger than the bound of (N 0 , U 0 ) s , such that Hence, there exists some ε 1 > 0 depending possibly onC such that σ /2 < n ε < 2σ , ∀0 < ε < ε 1 .
In what follows, we will simplify (n ε , u ε , φ ε ) as (n, u, φ) by omitting the superscript ε. The readers should not be confused.
Proof. We only consider the case M = 1, while the cases M ≥ 2 can be proved exactly in the same spirit. From the construction of

XUEKE PU
The last term II is independent of the unknown remainder Φ and is classified into F φ . The first term I depends on Φ and is put into √ εR φ ( √ εφ R ) and can be bounded by where we suppress the dependence of C on φ 1 L ∞ . By taking the L 2 norm, we . The same treatments will lead to the estimates for all M ≥ 2. The greater M is, the easier the proof is.
On the other hand, repeating this process, by Moser's inequality and Sobolev inequalities, we can obtain the estimates in higher Sobolev norms With this estimate, the following elliptic estimates for the Poisson equation (16c) can be easily achieved. Lemma 3.3. Let M ≥ 1 and any multi-indices α ≥ 0, there exists some ε 1 > 0 such that for any 0 < ε < ε 1 , We note that the two norms ∂ α φ R + √ ε ∂ α ∇φ R + ε ∂ α ∆φ R and ∂ α φ R + ε ∂ α ∆φ R are equivalent by interpolation.
3.2. First estimates. We will make estimates by taking inner product of (16b) with U − ε∆U. But since the proof is too long to be readable in that way, we will split the estimates into two subsections. First, we make estimates with U in this subsection and then with −ε∆U in the next subsection.
Let s ≥ 4 be an integer, and α be a multi-index with |α| ≤ s and set Differentiating the remainder system (16b) with ∂ α , taking inner product with U α and by integration by parts, we obtain For the term I 1 ∼ I 3 and I 6 , it is easy to show that thanks to the calculus inequalities in Lemma 3.1 and R u is of O(1).
Estimate of I 4 . By integration by parts, I 4 in (18) can be rewritten as where R p (N ) = {∇N + B − bN }/n and n = n app + ε M N . We also note that B and b are both of order O(1) and hence For the term I 41 , we have by integration by parts From calculus inequalities in 3.1, the last two terms can be bounded by In what follows, we treat the first term on the RHS of I 41 . By (20), we have From Lemma 3.1, it is easy to show that Moreover, by integration by parts, we have Therefore, we have the estimates for I 4 in (18)

XUEKE PU
Estimates of I 5 . Differentiating the equation (16a), we obtain Inserting this equation into I 5 , we obtain We first treat the term I 52 in (21). By integration by parts, which implies that thanks to the fact that H s is an algebra. We now treat I 53 . Differentiating (16c) with ∂ α , we have which enables us to rewrite I 53 as =I 531 + · · · + I 536 .
Since by continuity assumption, σ /2 < n L ∞ < 2σ , we obtain by integration by parts Similarly, by calculus inequalities in Lemma 3.1

Moreover, by integration by parts twice, we obtain
Hence, by Hölder inequality, we obtain Similarly, we have which follows from Lemma 3.2 that For the last two term, it is easy to show that Therefore, the term I 53 in (21) is bounded by Since R n is of O(1), it is easy to show that Finally, we treat the term I 51 in (21). Using (16c) again, we have On the other hand, since 0 < σ < n 0 < σ is bounded from below and above For the last term on the right hand side, we have by integration by parts and hence From (16a), we obtain Moreover, since we have The estimate of I 511 is left to the next subsection.
Therefore, we arrive at the following Proposition 1. We arrive at the following 3.3. Second estimates. Now, we differentiate the remainder system (16b) with ∂ α , then take the inner product of (16b) with −ε∆U α , and by integration by parts twice, we obtain where By integration by parts, it is easy to show that For the term J 2 , we have The estimate of J 4 . The term J 4 in (25) can be treated exactly as I 4 since For J 4112 , it is easy to show thanks to (16a). Moreover, from (20), we have which implies by Hölder inequality that