BOUNDARY LAYER PROBLEM AND QUASINEUTRAL LIMIT OF COMPRESSIBLE EULER-POISSON SYSTEM

. We study the boundary layer problem and the quasineutral limit of the compressible Euler-Poisson system arising from plasma physics in a domain with boundary. The quasineutral regime is the incompressible Euler equations. Compared to the quasineutral limit of compressible Euler-Poisson equations in whole space or periodic domain, the key diﬃculty here is to deal with the singularity caused by the boundary layer. The proof of the result is based on a λ - weighted energy method and the matched asymptotic expansion method.

The boundary conditions of the system read as: 1, in such case the density of electrons almost equals to the density of ions, and the zero Debye length limit λ → 0 is also called quasineutral limit.
Many results have been obtained about the quasineutral limits of corresponding Euler-Poisson systems. These results may be divided into two classes according to their boundary conditions: the case of without the boundary layer and the case of with the boundary layer. In the first case, there are many mathematical studies on quasineutral limits of the corresponding Euler-Poisson system in a domain without boundary. In Peng [20], Wang [29], Loeper [16], Peng et al. [23] and Peng et al. [24], the authors considered the systems describing electrons with fixed ions. Cordier et al. [3] and Pu [25] studied the models of ions with electrons following a Maxwell-Boltzmann law. Jiang et al. ( [10] and [11]), Peng et al. [21] and Li et al. [14] considered the quasineutral limits in two-fluid isentropic Euler-Poisson systems, where the quasineutral regime is governed by compressible Euler equations.
In the second case, some authors studied the quasineutral limits of Euler-Poisson systems in domains with boundaries. Slemrod et al. [26] considered the quasineutral limit for the steady-state Euler-Poisson system on a bounded interval in R 1 , proved that the quasineutral limit is obtained untill the ion velocity reaches the ion-sound speed. Peng [19] studied the zero-electron mass limit, the quasineutral limit and the zero-relaxation-time limit for one dimensional steady-state Euler-Poisson system, proved the strong convergence of a sequence of solutions and gave the corresponding convergence rate. Peng and Wang [22] and Violet [28] studied the quasineutral limit in the steady-state Euler-Poisson system for a potential flow, justified the asymptotic expansions up to first order in one-dimensional case and to any order in multidimensional case respectively. Suzuki [27] was concerned with boundary layers of a multicomponent plasma which consists of electrons and several positive ion species. Gérard-Varet et al. ([4] and [5]) studied the quasineutral limit of the isothermal Euler-Poisson system for the ions in a domain with boundaries, and proved the quasineutral regime is given by the compressible Euler equations. The corresponding result on the quasineutral limit is extended to the case of the twofluid isothermal Euler-Poisson system in a bounded domain of R 3 by Ju et al. [12]. Recently, the quasineutral limit for a model of a three dimensional Euler-Poisson system with linear enthalpy function has been studied in Liu et al. [15].
In this paper, we study the boundary layer problem and the quasineutral limit of nonstationary isentropic Euler-Poisson system (1)-(3) for a rotational flow of the electron (the density of ions being prescribed as a constant background) in a domain of R 3 with the boundary condition (4).
The rest of our paper is organized as follows. In section 2 we state the main result. In section 3 we construct the boundary layer approximations. Section 4 is devoted to the convergence to the incompressible Euler equations.
In this paper, the operator ∂ i stands for ∂ xi = ∂ ∂xi , i = 1, 2, 3. For convenience, we shall omit the spatial domain R 3 + in integrals.
2. Main result. For Poisson equation (3), letting λ = 0 , we first get the neutrality: From (1)-(2) in the limit λ → 0 and the fact that n = 1, we have the following incompressible Euler system: We expect that the Euler-Poisson system (1)-(3) converges to this system. System (5)- (6) and initial data is well-posed with the boundary condition However, the solution (u 0 , φ 0 ) of (5)-(6) cannot in general satisfy the boundary condition φ 0 | x3=0 = φ b . In order to correct this boundary condition, we expect the formation of a boundary layer. This will be given in the following section 3. Let us state our main result of this paper as follows: The proof of Theorem 2.1 is based on the λ-weighted energy method. This is a boundary layer problem. The main difficulties are to deal with oscillation terms like u · ∇n a , caused by the boundary layer function n a = n 0 (x 1 , x 2 , x 3 , t) + N 0 (x 1 , x 2 , x3 λ , t) + λ(n 1 + N 1 ) + · · · , given in section 3, and the general γ-law pressure term ∇h(n a + n) = h (n a + n)(∇n a + ∇n). Here our key point is to control the oscillating behavior of the electric field due to the loss of the damping term in the Poisson equations. These are required to overcome by some new techniques different from the papers in Gérard-Varet et al. [4], [5] and Ju et al. [12]. Also, the quesineutral regime here is incompressible Euler flow, which is different from the previous results on quasineutral limits in a domain with boundary.
3. Construction of boundary layer approximations. We establish the approximate solution of the Euler-Poisson system (1)-(3) under the form of an asymptotic expansion of a power series in λ: where K is an arbitrarily large integer. The coefficients (n i , u i , φ i ) of the first sum and (N i , U i , Φ i ) of the second sum should describe the macroscopic behaviour and a boundary layer of size λ near the boundary of the solutions respectively.
not only on the independent variables (t, y), but also on a stretched variable z = x3 λ ∈ R + . At the same time, we shall assume that According to the boundary conditions (4) of the system (1)-(3), for the whole approximation, the coefficients shall further satisfy the following boundary conditions: Thus the leading terms (n 0 , u 0 , φ 0 ) will determine the dynamics of these approximate solutions. From the previous section, we need to prove that (u 0 , φ 0 ) is the solution of the incompressible Euler system with n 0 = 1 and the non-penetration condition u 0 3 (t, y, 0) = 0.
We have the following result: Theorem 3.1. Assume the initial velocity satisfies u 0 0 satisfy u 0 0 ∈ H m+2K+3 , K ∈ N + , m ∈ N + , m ≥ 3, divu 0 0 = 0 , and some compatibility conditions on boundary. Then there exists T > 0 and an approximate solution formed (8) of the Euler-Poisson system (1)-(3) such that 1) (u 0 , φ 0 ) is a solution to the incompressible Euler system (5)-(6) with initial da- and their derivatives are uniformly exponentially decreasing functions with respect to the last variable z. In addition, let us assume (n λ , u λ , φ λ ) be a solution to Euler-Poisson system (1)-(3) and define Then (n, u, φ) satisfies the system of the error equations: where R n , R u , R φ are remainders satisfying: The rest of the section is the proof of Theorem 3.1.
To obtain the approximate solution (8) of (1)-(4), we solve the equations which the leading terms of (8) satisfy. By substituting the expansion (8) into the Euler-Poisson system (1)-(4) and considering the same amplitude terms, we get series of equations on the coefficients (n i , u i , φ i ) and We divide the domain of variabilities R 3 + into two zones: one is the inner zone and the other is the outer zone (the boundary layer zone). In the inner zone , as λ → 0, z = x3 λ → +∞, the boundary layer correctors (N i , U i , Φ i ) → 0. In the outer zone , by using Taylor expansion (the same for u i and φ i ), we receive the boundary layer equations in variables (t, y, z). For brevity, we denote f | x3=0 = Γf .
In the outer zone, by collecting the O(λ −1 ) amplitude terms, we get the relations Equation (21) implies that (Γn 0 + N 0 )(Γu 0 3 + U 0 3 ) does not depend on z. From boundary condition (10), we have Then we obtain that (n 0 , u 0 , φ 0 ) satisfies the incompressible equations (12)-(13) together with n 0 = 1 and the boundary condition (14). Now, equation (22) can be rewritten as follows By using (9), we get Still in the outer zone, collecting the O(λ 0 ) terms in the Poisson equation (3), we get Combining (23) with (24), we have with the boundary conditions This should determine completely Φ 0 . Once Φ 0 is determined, then N 0 can be solved by (24). For the nonlinear boundary layer system (25)- (26), in view of the definition of enthalpy h(n), we get an ordinary differential equation in z.
We rewrite (27) as the corresponding Hamiltonian system, d dz where By linearizing the Hamiltonian system at point (p, Φ 0 ) = (0, 0), we have d dz Since the determinant of the Jacobi matrix is a negative number, (p, Φ 0 ) = (0, 0) is a saddle fixed point and the stable manifold is locally a curve which is tangent to . As H(p, Φ 0 ) = 0, we obtain the stable manifold , Obviously, combining equation (27), on this branch, the solutions and all their derivatives decay exponentially to 0. From the above statement, (n 0 , u 0 , φ 0 ), and (U 0 3 , Φ 0 , N 0 ) have been derived. Next, we obtain the equations for (n 1 , u 1 , φ 1 ) and (U 1 3 , U 0 y , Φ 1 , N 1 ). In the inner zone, collecting the amplitude O(λ 1 ) terms in (1)-(3), we have with n 1 = 0. To solve the system, we need a boundary condition.
In the outer zone, collecting the O(λ 0 ) terms in equation (2), we have We notice that U 0 1 = 0 and U 0 2 = 0 are the trivial solutions to the above system by combining (13) , (23) and the fact that h(Γn 0 ) ≡ const. Collecting the O(λ 0 ) terms in equation (1), we obtain where . Note that (9) requires the following compatibility conditions: . Integrate above equation from 0 to z, by using (10), we get By using the decay condition (9), we can get So we get the boundary condition of system (29)-(30). From system (29)-(30), we can determine (n 1 , u 1 , φ 1 ) together with the initial value and the boundary condition (36). Using (35), we get U 1 3 . Collecting the O(λ 1 ) terms in equation (3), we obtain Combining (33) and (37), together with the decay condition (9) and the Dirichlet condition we can get Φ 1 and N 1 eventually. More generally, we derive for all i ≥ 2: In the inner zone, collecting amplitude O(λ i ) terms in (1)-(3) , we obtain systems of type: where In the outer zone, collecting O(λ i−1 ) terms in equations (1)-(2), we get: The source terms = 0 are the trivial solutions to the system (42)-(45) by combining (13) and (23).
Collecting terms with amplitude O(λ i ) in (3), we have: where F i Φ depends on (n k , φ k ) and on (N k , Φ k ), k ≤ i − 1. Note that (9) requires the following compatibility conditions: Integrate above equation from 0 to z, by using (10), we get From the decay condition (9), we deduce that So we get the boundary condition of system (39)-(41). From system (39)-(41), we can determine (n i , u i , φ i ) together with a good initial value and the boundary condition (48). Using (47), we get U i 3 . Combining (45) and (46), together with the decay condition (9) and the Dirichlet condition we can get Φ i and N i eventually.
As regards the linear hyperbolic systems (29)-(36) and (39)-(48), it is easy to get the well-posedness. For brevity, we won't cover them in this article.
4. Stability estimates. In this section, we study the stability of the boundary layer approximations built in the previous section.
Assume that (n 0 , u 0 ) ∈ H m (Ω) for m ≥ 3, then there exists T λ > 0 such that problem (50)-(54) admits a unique solution defined on [0, T λ ] and there exists M > 0 independent of λ such that where χ (x)3 := χ( x3 δ ) and χ is a smooth compactly supported function equal to 1 around zero and δ > 0 is chosen so that 4.1. L 2 estimate for the suitably linearized equation. First, we linearize the error system (50)-(52) and establish an L 2 estimate for the solution (ṅ,u,φ) to the linearized equation.
(60) Proposition 1. Let (n a , u a , φ a ) be the approximate solution constructed in Theorem 3.1 and some smooth (n, u, φ) such that u 3 | x3=0 = 0 and Then there exist C(M ) and C(C a , M ) independent of λ such that we have on [0, T ] the estimate d dt ((n a + n) Proof. In the proof, we shall denote by C a a number which may change from line to line but which is uniformly bounded for λ ∈ (0, T 0 ] where T 0 is the interval of time on which the approximate solution is defined. Since the leading boundary layer term of u a vanishes, we have For n a ,φ a , we have Let us now prove the energy estimate. Multiplying the velocity equation by (n a + n)u and performing standard manipulations, we obtain: (n a + n) |u| 2 2 = (n a + n)u∂ tu + ∂ t (n a + n) |u| 2 2 = ∂ t (n a + n) |u| 2 2 + r u · (n a + n)u − u · ∇u a · [(n a + n)u] − O(n)ṅ∇n a · (n a + n)u + I 1 + I 2 + I 3 , where I 1 = − (u a + u) · ∇u · [(n a + n)u], The first four terms at the r.h.s. of (66) can be easily estimated by using (63), (64) and (65): u · ∇u a · [(n a + n)u] ≤C(C a , M ) u 2 − O(n)ṅ∇n a · (n a + n)u ≤C(C a , M ) Let us turn to the treatment of I 1 .
Relying on (63), (64) and u · ∇n a L 2 ≤ C a ∇u L ∞ , we infer that: Next, we estimate I 2 . Integrating by parts, we first have By using equation (57) to express div[(n a + n)u], we have By applying ∂ t to the Poisson equation (59), we get that So I 2 can be expressed as follows: Integrating by parts and multiplying the Poisson equation by (u a + u), we get For I 1 2 , integrating by parts, we have Using (61) and (63), we get For I 2 2 , thanks to (63), we obtain Combining the previous inequalities, we have Finally, we need to estimate I 3 . Since div((n a + n)uh (n a + n)ṅ) =(n a + n)uh (n a + n) · ∇ṅ + (n a + n)u · ∇h (n a + n)ṅ + div((n a + n)u)h (n a + n)ṅ =(n a + n)uh (n a + n) · ∇ṅ + div((n a + n)u)h (n a + n)ṅ we have So, Integrating by parts, we obtain: For ṅ 2 h (n a + n)[∂ t (n a + n) + ∇(n a + n) · (u a + u)], since we have the equation ∂ t n a + ∇n a · u a = −n a divu a − λ K R n , together with (61), we have |∂ t n a + ∇n a · u a | ≤ C a .
Using directly the estimate of (62), we get that where we have set We deal with the term r n 2 L 2 and the last two terms in the right hand side of (79) with integrating by parts and using the Poincaré inequality as follows: Therefore, (79) is proved by using the estimate in Proposition 1. The proof of Proposition 2 is complete.
We shall also use the norms: where the vector fields Z i are defined by and For the sake of brevity, we will also use the following notation:
Estimate of the H m λ norm. Let us recall a classical estimate for products in dimension 3: with s 1 + s 2 = 3 2 , s 1 = 0, s 2 = 0. We shall first prove that by using the equation, we can estimate the H m λ norm of the solution of (50)-(52) on [0, T λ ): for some C > 0 independent of λ and where C stands for a continuous non-decreasing function with respect to all its arguments which does not depend on λ.
By using similar arguments as above, we shall also get: For λ ∈ (0, 1], we have the estimate Normal derivatives estimate.
In the same way, we have For r n Z α φ, using the Poincaré inequality, we have With the help of the regularity theory of the Poisson equation (104), we easily get that In order to eliminate the singular terms in (110), we define the following norms: f H θm co,λ (R 3 Q θm (t) = (n, u, λ∇φ) H θm co,λ (R 3 + ) + w(t) H θ(m−1) λ (R 3 + )