Vanishing electron mass limit in the bipolar Euler–Poisson system

Nonlinear Analysis: Real World Applications 12 (2011) 1002–1012

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Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa

Vanishing electron mass limit in the bipolar Euler–Poisson system✩ Li Chen a,b , Xiuqing Chen c,∗ , Chunlei Zhang d a

Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

b

The Institute of Mathematical Sciences, Chinese University of Hongkong, China

c

School of Sciences, Beijing University of Posts and Telecommunications, Beijing, 100876, China

d

Department of Mathematics, Southern Utah University, Cedar City, UT 84720, USA

article

info

Article history: Received 22 May 2010 Accepted 19 August 2010 Keywords: Bipolar Euler–Poisson system Incompressible Euler equation Symmetric hyperbolic system Energy estimates

abstract The bipolar Euler–Poisson system in physics consists of the conservation laws for the electron and ion densities and their current densities, coupled with the Poisson equation for the electrostatic potential. The limit of vanishing ratio of the electron mass to the ion mass in the n-dimensional flat torus is proved in the case of well prepared initial data. The limiting system is composed of two separated equations, where the equation for electron is the incompressible Euler equation with damping, which means physically that the evolution for electrons and ions can be treated as separated motions in the small ratio case. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction The bipolar Euler–Poisson system is derived in semiconductor or plasma physics to study the time evolution of charged fluids. These models can be obtained from Boltzmann equation for charged particles, i.e., the electrons and ions, see [1,2]. The system consists of the conservation laws for the electron(ion) density and current density for electron(ion), coupled with the Poisson equation for the electrostatic potential. We will study the following (scaled) bipolar Euler–Poisson system for the electron density ne , the ion density ni , the electron’s and ion’s velocities ve , vi , and the electrostatic potential φ ,

∂t ni + ∇ · (ni vi ) = 0, mi ∂t (ni vi ) + mi ∇ · (ni vi ⊗ vi ) + ∇ pi (ni ) = −ni ∇φ − mi

ni v i

∂t ne + ∇ · (ne ve ) = 0, me ∂t (ne ve ) + me ∇ · (ne ve ⊗ ve ) + ∇ pe (ne ) = ne ∇φ − me

τi

,

ne ve

τe

(1.1)

,

−λ2 1φ = ni − ne for x ∈ Td , t > 0,

where pi,e are the pressure functions, usually given by pi,e (s) = ci,e sγi,e , s ≥ 0, with ci,e > 0 and γi,e ≥ 1. In this work, we only assume that pi,e is strictly monotone and smooth. d ≥ 1 is the space dimension. The initial conditions are given by ni,e (·, 0) = nI ,i,e ,

vi,e (·, 0) = vI ,i,e in Td . The particle mass mi,e , relaxation time τi,e , and Debye length λ are physical parameters. ✩ Supported by the National Natural Science Foundation of China (No. 10571101, 10871112 and 10771008), Research Fund for the Doctoral Program of Higher Education of China (No. 20090005120009), the Fundamental Research Funds for the Central Universities (BUPT2009RC0702) and the Talents Scheme Funds of BUPT. ∗ Corresponding author. E-mail addresses: [email protected] (L. Chen), [email protected] (X. Chen), [email protected] (C. Zhang).

1468-1218/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2010.08.023

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The wellposedness and different kinds of limit problems of the Euler–Poisson system have been widely studied in recent years. In the stationary case, Degond–Markowich [3] discussed the existence and uniqueness of the steady state solution in the subsonic case, while Gamba [4] studied the same problem in the transonic case. In the time evolution case, Zhang [5] and Marcati–Natalini [6] got the global existence of weak solutions of the initial boundary value problem and the Cauchy problem respectively by using compensated compactness. Later, the relaxation limit of the Euler–Poisson system to the driftdiffusion equations for the Cauchy and initial boundary value problem were studied separately by Marcati–Natalini [6] and Hsiao–Zhang [7]. In the consideration of smooth solution, Luo et al. [8], Hsiao–Yang [9], Li et al. [10] and Alì-Jüngel [11] investigated the asymptotic behavior of solutions to the Cauchy and initial boundary value problem respectively. Except the relaxation time limit (τi,e → 0) mentioned above, there are also two other kinds of limit problems, namely the quasineutral limit (λ → 0) and the zero mass limit (me /mi → 0). The quasineutral limit in the Euler–Poisson system has been analyzed for transient smooth solutions by Cordier and Grenier [12] in the one-dimensional case and independently in [13,14] in the multi-dimensional case. The zero mass limit in the Euler–Poisson system was firstly studied by Peng–Júngel in [15] and by Alì–Chen–Jüngel–Peng in [16] for a given ion density with well prepared initial data. In this paper, we will consider the zero mass limit in bipolar case, namely the case that ion density is not constant any more. The physical meaning of zero mass limit is in the following. Usually, the ions are much heavier than the electron, so we fix the ion mass mi to be a constant, say mi = 1 and let the electron mass me → 0. There are some physical papers on the zero mass limit simulation, such as [17,18]. The results of this paper will show the interesting phenomena that the evolution for electrons and ions can be treated as separated motions in the zero mass limit. In the following, we will let τi,e = λ = 1 for convenience. The bipolar Euler–Poisson system can be written into a symmetric hyperbolic system by introducing the enthalpy hi,e = hi,e (ni,e ) defined by h′i,e (ni,e ) = p′i,e (ni,e )/ni,e and hi,e (1) = 0. Let ε 2 = me , the system (1.1) is

∂t hi + vi · ∇ hi + p′i (ni )∇ · vi = 0, ∂t vi + vi · ∇vi + ∇ hi = −∇φ − vi , ∂t he + ve · ∇ he + p′e (ne )∇ · ve = 0, ∂t ve + ve · ∇ve + 1φ = ne − ni ,

1

(1.2)

1

∇ he = 2 ∇φ − ve , ε2 ε x ∈ Td , t > 0,

with initial conditions hi,e (·, 0) = (hεi,e )I ,

vi,e (·, 0) = (viε,e )I in Td .

(1.3)

We also require that Td φ dx = 0 for the Poisson equation. Since the pressure function is invertible, so does hi,e (ni,e ) and we will use the notation ni,e (hi,e ). By a similar formal asymptotic analysis as in [16], we can expect that if the initial data of the electron and ion densities are perturbations of a constant, i.e., (ne )I , (ni )I = n0 + O(ϵ) > 0, the limiting velocities of electron and ion, ve0 , vi0 satisfy the equations



∇ · ve0 = 0,

∂t ve0 + ve0 · ∇ve0 + ve0 = ∇ P ,

(1.4)

∇ · vi0 = 0,

∂t vi0 + vi0 · ∇vi0 + vi0 = 0,

(1.5)

and

respectively. (1.4) is the incompressible Euler with damping. To simplify the notations, let u0 = vi0 in the above equations. In general, (1.5) doesn’t have a solution for arbitrary initial data vi0,I (x). A necessary condition for u0 to be a solution of (1.5) is that ∇ · (u0 · ∇ u0 ) = 0, which is equivalent to

∂i u0j ∂j u0i = 0 since u0 is divergence free. There are some special cases of the initial data with which (1.5) has a unique local smooth solution. For example, (1.5) has a unique solution when vi0,I (x) = vi0,I is constant velocity or vi0,I (x) satisfies ∇ · vi0,I (x) = 0 and vi0,I (x) · ∇vi0,I (x) = 0. In this case, (1.5) is an ODE ∑d

i ,j = 1

∂t vi0 + vi0 = 0, which has solution u0 (x, t ) = vi0,I (x)e−t → 0 as t → ∞.

(1.6)

In the u0I (x) is the initial data such that (1.5) has a unique smooth solution ∑following, we assume  d 0 0 u0 ( x , t ) i,j=1 ∂i uj ∂j ui = 0 .

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Now motivated by the formal analysis, we will use the transformation of variables, (let h0e and h0i be constants such that ne (h0e ) = ni (h0i ) = n0 .) h˜ i = h˜ e =

hi − h0i

ε

,

he − h0e

ε

u=

,

v i − u0 , ε

v = ve ,

φ˜ =

φ ε

then system (1.2) changes into

 Ai (ε h˜ i )∂t h˜ i + ε Ai (ε h˜ i )u · ∇ h˜ i + Ai (ε h˜ i )∇ h˜ i · u0 + ∇ · u = 0,     ∂t u + ε u · ∇ u + ∇ h˜ i = −∇ φ˜ − u − (u0 · ∇ u + u · ∇ u0 ),     A (εh˜ )∂ h˜ + A (εh˜ )v · ∇ h˜ + 1 ∇ · v = 0, x ∈ Td , t > 0, e e t e e e e ε (1.7) 1 1   ˜ ˜  ∂ v + v · ∇v + ∇ h = ∇ φ − v, t e   ε ε   1  0 1φ˜ = (n (εh˜ + h ) − n (εh˜ + h0 )), e e i i e i ε where we have used the notation Ai,e (ε h˜ i,e ) = 1/p′i,e (ni,e (ε h˜ i,e + h0i,e )). The main point of this paper is to perform the limit ε → 0 in (1.7). We will first get a uniform (in ε ) local existence of smooth solution, then pass to the limit by the uniform estimates with the help of well prepared initial data. Local uniform existence is Theorem 1.1. Let ni,e ∈ C ∞ (R) be strictly increasing and s > d/2 + 1 and let the initial data (h˜ εi )I , (h˜ εe )I , uεI , vIε satisfy

   ˜ε  (hi )I , (h˜ εe )I  + ‖uεI , vIε ‖s ≤ M0 , s

where M0 > 0 is a constant independent of ε . Then there exist constants T0 > 0 and M0′ > 0, independent of ε , and ε0 (M0 ) > 0 such that for all 0 < ε < ε0 (M0 ), the problem (1.7) has a classical solution (h˜ εi , h˜ εe , uε , v ε , φ˜ ε ) in [0, T0 ] satisfying

|||h˜ εi , h˜ εe |||s,T0 + |||uε , v ε |||s,T0 + |||∇ φ˜ ε |||s,T0 ≤ M0′ . Here we have used the notations:

‖ · ‖s = ‖ · ‖H s (Td ) ,

||| · |||s,T = sup ‖ · ‖s for s ∈ R, 0
‖ · ‖∞ = ‖ · ‖L∞ (Td ) .

Next is our main result on taking the zero mass limit. Theorem 1.2. Assume that u0I (x) is the initial data such that

∇ · u0 = 0, ∂ t u0 + u0 · ∇ u0 + u0 = 0 , (1.8) 0 0 d u (·, 0) = uI (·) in T ∑d has a unique smooth solution u0 (x, t ) with i,j=1 ∂i u0j ∂j u0i = 0. Let ni,e ∈ C ∞ (R) be strictly increasing and s > d/2 + 1 and ∇ · veε,I = 0. h0e and h0i are constants such that ne (h0e ) = ni (h0i ) = n0 > 0. Suppose  ε   ε  0  (hi )I − h0i     +  (he )I − he  ≤ M1 , (1.9)     ε2 ε2 s s    vi,I − u0I    + ‖ve,I ‖s ≤ M0 ,   ε s where M0 , M1 > 0 does not depend on ε . Furthermore, let (hεi,e , viε,e , φ ε ) be a classical solution to (1.2) in [0, T0 ] with T0 > 0 independent of ε . Then, as ε → 0, hεi,e → h0i,e ,

viε → u0 ,

∇φ ε → 0 strongly in L∞ (0, T0 ; H α (Td )) ∩ C 0,1 ([0, T0 ]; L2 (Td )),

veε → v 0 strongly in C 0 ([0, T0 ]; H α (Td )) for all α < s, where u0 and v 0 are the classical solution of (1.8) and following equation separately,

∇ · v 0 = 0, (∂t + v 0 · ∇)v 0 + v 0 = ∇π , 0 0 v (·, 0) = vI in Td ,

x ∈ Td , t > 0,

(1.10)

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and π is the limit of

∇φ ε − ∇ hεe ∗ ⇀ ∇π weakly* in L∞ (0, T0 ; L2 (Td )), ε where u0I and vI0 are the limits of initial data (viε )I and (veε )I = vIε in H s (Td ). We mention here that in the fixed ion density case, i.e., the unipolar Euler Poisson system, the vanishing electron mass limit has some similarity with the low Mach number limit in the Euler system, except an additional singular term from electrostatic potential. Thus the singular terms in the symmetric hyperbolic main part can be controlled by energy estimates for the symmetric hyperbolic system used in [19], while the other singular term with respect to the potential cannot. This additional singular term requires more attention towards the uniform estimates, details on this method can be found in [16]. In the bipolar case, there are conservations of mass and balances of velocity for both electron and ion. It is obvious that the main structure of (1.7) is still symmetric hyperbolic. But the equations for electron and ion are not similar due to the singular terms that only exist in the equations for the electron. The ion and electron are weakly coupled through the Poisson equation. The fact that a singular potential in the electron velocity equation determines that the electron mass and ion mass must be of the same order. Thus we can expect that the ion velocity must correspond to the particle with constant mass. Since in the equations for ion, no singular terms could contribute a pressure, the limiting velocity must satisfy an equation with constant pressure, which gives a reasonable explanation of (1.10). The difficulties however arise from singular electrostatic potential, there needs to be more attention paid to the coupling of the equations for the bipolar system, i.e., the balance between these two systems through the Poisson equation. So far, there are no results on the vanishing mass limit for the Euler–Poisson system with ill prepared initial data. One can expect that based on the uniform estimates obtained in [16] for the unipolar case or the result in Section 2 of this paper for the bipolar case, it is possible to have it proved using some ideas introduced by Ukai [20], Grenier [21], Métivier and Schochet [22] for singular perturbation problems of fluid dynamic systems. This will be studied in our future works. We arrange the paper as follows. In Section 2, we will give the uniform local existence by using the uniform energy ˜ , see the estimates, where we will show the key idea on obtaining the uniform estimates for the most singular term ∇ φ/ε estimate for K3 , which also includes the unavoidable difficulty from the coupling of electron and ion. In Section 3, for well prepared initial data, we will describe the uniform estimates for a time derivative of the unknowns, which will give us the possibility to take limit ε → 0. 2. Uniform local existence We will employ the basic theory of smooth solutions to hyperbolic systems of Majda [23]. The key result is contained in the following lemma. Lemma 2.1. Suppose that it holds, for some T ∗ > 0 (maybe depend on ε ) and M > 0 (independent of ε ),

|||h˜ i , h˜ e |||s,T ∗ + |||u, v|||s,T ∗ ≤ M .

(2.11)

Then there exist ε0 = ε0 (M ) > 0 and c (M ) > 0 such that for all 0 < ε < ε0 , it holds

˜ s,T ∗ ≤ ec (M )T (M0 + c (M )T ∗ ). |||h˜ i , h˜ e |||s,T ∗ + |||u, v|||s,T ∗ + |||∇ φ||| ∗

(2.12)

So in the following we will focus on the proof of Lemma 2.1. First we recall for convenience some Moser-type inequalities which we will use in the subsequent analysis. Let α = α α (α1 , . . . , αd ) be a multi-index. The differential operator Dα is defined by Dα = ∂x11 · · · ∂xdd ; Ds for s ∈ N denotes the sth derivative.

• Let s ≥ 0, f , g ∈ H s (Td ) ∩ L∞ (Td ), and α a multi-index with |α| ≤ s. Then, for some constant cs > 0, ‖Dα (fg )‖0 ≤ cs (‖f ‖∞ ‖Ds g ‖0 + ‖g ‖∞ ‖Ds f ‖0 ). • Let s ≥ 1, f ∈ H s (Td ) with Df ∈ L∞ (Td ), g ∈ H s−1 (Td ) ∩ L∞ (Td ), and |α| ≤ s. Then, for some constant cs > 0, ‖Dα (fg ) − fDα g ‖0 ≤ cs (‖Df ‖∞ ‖Ds−1 g ‖0 + ‖g ‖∞ ‖Ds f ‖0 ).

(2.13)

(2.14)

Proof of Lemma 2.1. Let f be a smooth function and let (2.11) hold. Then there exist constants c (M ) such that sup ‖∇ f (ε h˜ i,e ), ∂t f (ε h˜ i )‖∞ ≤ ε c (M ),

(0,T )

sup ‖∂t f (ε h˜ e )‖∞ ≤ c (M ).

(0,T )

(2.15)

These estimates can be obtained by (1.7) and (2.11), i.e.

|||∂t f (εh˜ e )|||∞,T ≤ |||f ′ (εh˜ e )|||∞,T |||ε∂t h˜ e |||∞,T ≤ c (M )(|||εv · ∇ h˜ e |||∞,T + |||A−1 (εh˜ e )∇ · v|||∞,T ) ≤ c (M ). We will use the following notations. Let α with |α| ≤ s be a multi-index. Then we define |D|α| u| = supα |Dα u|.



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Taking derivatives Dα on (1.7), we have Ai (ε h˜ i )∂t Dα h˜ i + ε Ai (ε h˜ i )u · ∇ Dα h˜ i + Ai (ε h˜ i )∇ Dα h˜ i · u0 + ∇ · Dα u = Fα ,

∂t Dα u + ε u · ∇ Dα u + ∇ Dα h˜ i = −∇ Dα φ˜ − Dα u − u0 · ∇ Dα u − Dα u · ∇ u0 + Gα , Ae (ε h˜ e )∂t Dα h˜ e + Ae (ε h˜ e )v · ∇ Dα h˜ e + 1

1

ε

∇ · Dα v = Hα ,

x ∈ Td , t > 0,

(2.16)

1

∂t Dα v + v · ∇ Dα v + ∇ Dα h˜ e = ∇ Dα φ˜ − Dα v + Lα , ε ε α˜ ′ 0 α˜ ′ ˜ ˜ 1D φ = ne (ε he + he )D he − ni (ε hi + h0i )Dα h˜ i + Nα , where Fα = ε Ai (ε h˜ i )(u · ∇ Dα h˜ i − Dα (u · ∇ h˜ i )) + Ai (ε h˜ i )(u0 · ∇ Dα h˜ i − Dα (u0 · ∇ h˜ i )) 1 ˜ + ∇ · Dα u − Ai (ε h˜ i )Dα (A− i (ε hi )∇ · u),

Gα = ε u · ∇ Dα u − ε Dα (u · ∇ u) + u0 · ∇ Dα u − Dα (u0 · ∇ u) + Dα u · ∇ u0 − Dα (u · ∇ u0 ), 1 1 ˜ Hα = Ae (ε h˜ e )(v · ∇ Dα h˜ e − Dα (v · ∇ h˜ e )) + (∇ · Dα v − Ae (ε h˜ e )Dα (A− e (ε he )∇ · v)),

ε

Lα = v · ∇ Dα v − Dα (v · ∇v), 1 Nα = Dα (ne (ε h˜ e + h0e ) − ni (ε h˜ i + h0i )) − (n′e (ε h˜ e + h0e )Dα h˜ e − n′i (ε h˜ i + h0i )Dα h˜ i ).

ε

The Friedrichs energy estimate for (2.16) (i.e., multiplying the equations in (2.16) by Dα h˜ i , Dα u and Dα h˜ e , Dα v , integrating over Td , taking the sum and integrating by parts) gives 1 d

∫

2 dt

Td

B

(Ai (ε h˜ i )|Dα h˜ i |2 + |Dα u|2 + Ae (εh˜ e )|Dα h˜ e |2 + |Dα v|2 )dx +

∫

1

(|Dα h˜ i |2 + |Dα u|2 + |Dα h˜ e |2 + |Dα v|2 )dx + 2 Td 2 ∫ ∫ 1 α˜ α α˜ α − ∇ D φ · D udx + ∇ D φ · D v dx ε Td Td = I1 + I2 + I3 + I4 , ≤

∫ Td

∫ Td

|Dα u|2 + |Dα v|dx

(|Fα |2 + |Gα |2 + |Hα |2 + |Lα |2 )dx

where B is a constant bounded by c (M ) due to (2.15). We are left to control I1 , I2 , I3 and I4 . The inequalities (2.15) show that I1 ≤ c ( M )

∫ Td

(|D|α| h˜ i |2 + |D|α| u|2 + |D|α| h˜ e |2 + |D|α| v|2 )dx.

The integral I2 can be estimated in a similar way as done by Klainerman and Majda [19,24]. By employing the Moser-type estimates (2.14) and (2.13), we obtain for |α| ≥ 1 (noting that F0 = G0 = H0 = L0 = 0)

  ‖Fα ‖0 ≤ c (M ) ‖D|α| h˜ i ‖0 + ‖D|α| u‖0 + 1   ‖Gα ‖0 ≤ c (M ) ‖D|α| u‖0 + 1   ‖Hα ‖0 ≤ c (M ) ‖D|α| h˜ e ‖0 + ‖D|α| v‖0 + 1   ‖Lα ‖0 ≤ c (M ) ‖D|α| v‖0 + 1 and for |α| ≥ 2,

‖Nα ‖0 ≤ ‖Dα−1 (n′e (ε h˜ e + h0e )Dh˜ e ) − n′e (εh˜ e + h0e )Dα h˜ e ‖0 + ‖Dα−1 (n′i (εh˜ i + h0i )Dh˜ i ) − n′i (εh˜ i + h0i )Dα h˜ i ‖0 ≤ c (‖Dn′e (εh˜ e + h0e )‖∞ ‖D|α|−2 Dh˜ e ‖0 + ‖Dh˜ e ‖∞ ‖D|α|−1 n′e (ε h˜ e + h0e )‖0 + ‖Dn′i (ε h˜ i + h0i )‖∞ ‖D|α|−2 Dh˜ i ‖0 + ‖Dh˜ i ‖∞ ‖D|α|−1 n′i (εh˜ i + h0i )‖0 )   ≤ ε c (M ) ‖D|α|−1 h˜ i ‖0 + ‖D|α|−1 h˜ e ‖0 + 1 ,

(2.17) (2.18) (2.19) (2.20)

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we have also

     1  1 0 0 ′ 0 ˜  0 0 ′ ˜ 0 ˜    ˜ ˜ ˜ ‖N0 ‖0 ≤  (ne (εhe + he ) − ne (he )) − ne (εhe + he )he  +  (ni (εhi + hi ) − ni (hi )) − ni (εhi + hi )hi  ε ε 0 0 ≤ ‖n′e (εθ1 h˜ e + h0e ) − n′e (εh˜ e + h0e )‖∞ ‖h˜ e ‖0 + ‖n′i (εθ2 h˜ i + h0i ) − n′i (ε h˜ i + h0i )‖∞ ‖h˜ i ‖0 ≤ ε c (M )(‖h˜ e ‖0 + ‖h˜ i ‖0 ), ‖N1 ‖0 = 0. From the Poisson equation, we have

˜ 0 ≤ c (M )(‖Dα h˜ e ‖0 + ‖Dα h˜ i ‖0 + ‖Nα ‖0 ) ‖∇ Dα φ‖ ≤ c (M )(‖h˜ e ‖|α| + ‖h˜ i ‖|α| ), consequently,

∫ I3 = − Td

∇ Dα φ˜ · Dα udx ≤ c (M )(‖h˜ e ‖|α| + ‖h˜ i ‖|α| + ‖D|α| u‖0 ).

Now we turn to the delicate integral I4 . This term cannot be controlled directly from the Riesz transformation (derived from the coupling with the Poisson equation), as done in [11], for instance. Our idea is to replace the singular term ε −1 ∇ Dα φ˜ . After integration by parts directly in I4 , by Eq. (2.16),

∫ I4 = − Td

∫ =− Td

1 Dα φ˜ ∇ · Dα v dx

ε

Dα φ˜ Hα dx +

∫

= K1 + K2 + K3 .

Td

Dα φ˜ Ae (ε h˜ e )v · ∇ Dα h˜ e dx +

∫ Td

Dα φ˜ Ae (ε h˜ e )∂t Dα h˜ e dx

We will control K1 , K2 , and K3 term by term. With the help of the estimate (2.19), it is not difficult to see that the first integral K1 can be controlled by

˜ 20 + ‖D|α| h˜ e ‖20 + ‖D|α| v‖20 + 1) K1 ≤ c (M )(‖Dα φ‖ ≤ c (M )(‖D|α| h˜ i ‖20 + ‖D|α| h˜ e ‖20 + ‖D|α| v‖20 + 1). We integrate by parts in the second integral since ∇ Dα h˜ e cannot be estimated by D|α| h˜ e ,

∫ K2 = − Td

∇ Dα φ˜ · v Ae (εh˜ e )Dα h˜ e dx −

∫ Td

˜ Ae (ε h˜ e ) · v Dα h˜ e dx − Dα φ∇

∫ Td

Dα φ˜ Ae (ε h˜ e )∇ · v Dα h˜ e dx

˜ 20 + ‖Dα h˜ e ‖20 + ‖Dα φ‖ ˜ 20 ) ≤ c (M )(‖∇ Dα φ‖ ≤ c (M )(‖D|α| h˜ i ‖20 + ‖D|α| h˜ e ‖20 + 1). The difficult term is now K3 . In order to control K3 , we cannot use (2.16) since this would (again) give a term containing ε −1 ∇ · Dα v . Our strategy is to employ the Poisson equation in the following way. Taking the time derivative of the reformed Poisson equation in (2.16), we have

1∂t Dα φ˜ = n′e (ε h˜ e + h0e )∂t Dα h˜ e − n′i (ε h˜ i + h0i )∂t Dα h˜ i + ∂t (n′e (εh˜ e + h0e ))Dα h˜ e − ∂t (n′i (ε h˜ i + h0i ))Dα h˜ i + ∂t Nα , and from which we can deduce an expression for ∂t Dα h˜ e and put it in K3 . We arrive at Ae (ε h˜ e )

∫ K3 =

′ ˜e Td ne (ε h

Td

∫ − Td

)

Ae (ε h˜ e )

∫ −

+

h0e

Dα φ˜ 1∂t Dα φ˜ dx +

n′ (ε h˜ e

+ ) Ae (ε h˜ e ) e

h0e

n′e (ε h˜ e + h0e )

α

∫ Td

n′i (ε h˜ i + h0i )Ae (ε h˜ e ) α ˜ t Dα h˜ i dx D φ∂ n′e (ε h˜ e + h0e )

˜ t (n′e (εh˜ e + h0e ))Dα h˜ e dx + D φ∂ ˜ t Nα dx Dα φ∂

= J1 + J2 + J3 + J4 + J5 .

Ae (ε h˜ e )

∫ Td

n′ (ε h˜ e

e

+ h0e )

˜ t (n′i (εh˜ i + h0i ))Dα h˜ i dx Dα φ∂

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L. Chen et al. / Nonlinear Analysis: Real World Applications 12 (2011) 1002–1012

After integration by parts we obtain J1 = −

Ae (ε h˜ e )

∫

1 2

Td

∫

e

e

˜ 2 dx ∂t |∇ Dα φ|

+ h0e ) 

Td

∂t ∇ D φ˜ · ∇

2 dt

Td

Td

∂t ∇ D φ˜ · ∇

˜ 2 dx + |∇ D φ|

1



∫

2

Td

∂t



Ae (ε h˜ e ) n′e (ε h˜ e + h0e )

˜ 2 dx |∇ Dα φ|



Ae (ε h˜ e )

α

−

α

n′e (ε h˜ e + h0e )



∫

Dα φ˜ dx

n′e (ε h˜ e + h0e )

Ae (ε h˜ e )

∫

1 d



Ae (ε h˜ e )

α

=−

=−

n′ (ε h˜

n′e (ε h˜ e + h0e )

Dα φ˜ dx.

The bound (2.15) allows to estimate the second and third integral: J1 ≤ −

1 d

Ae (ε h˜ e )

∫

2 dt

Td

n′e (ε h˜ e + h0e )

˜ 2 dx + c (M ) |∇ Dα φ|

∫ Td

˜ 2 dx + c (M ) |∇ Dα φ|

∫ Td

˜ 2 + |Dα φ| ˜ 2 )dx. (|ε∂t ∇ Dα φ|

In order to estimate the integral over |ε∂t ∇φα |2 , we employ again the Poisson equation in (2.16), now written in the form

ε 1∂t Dα φ˜ = ∂t Dα (ne (εh˜ e + h0e ) − ni (εh˜ i + h0i )). For |α| = 0, we have, observing (2.15),

˜ 0 ≤ c ‖∂t (ne (ε h˜ e + h0e ) − ni (ε h˜ i + h0i ))‖0 ≤ c (M ). ‖ε∂t ∇ φ‖ For 1 ≤ |α| ≤ s, we proceed by induction. Elliptic estimates give

  ˜ 0 ≤ c ‖Dα ∂t (ne (ε h˜ e + h0e ) − ni (ε h˜ i + h0i ))‖−1 + ‖ε∂t Dα φ‖ ˜ 0 ‖ε∂t ∇ Dα φ‖ ˜ 0 ). ≤ c (‖D|α|−1 (n′e (εh˜ e + h0e )ε∂t h˜ e )‖0 + ‖D|α|−1 (n′i (εh˜ i + h0i )ε∂t h˜ i )‖0 + ‖ε∂t Dα φ‖ The last term is bounded by the induction hypothesis. The first two terms are controllable since the time derivative provides the factor ε in front of ∂t h˜ α and ε∂t h˜ α can be estimated. Therefore, by Moser-type calculus,

˜ 0 ≤ c (M )(‖D|α| h˜ e ‖0 + ‖D|α| h˜ i ‖0 + ‖D|α| v‖0 + 1). ‖ε∂t ∇ Dα φ‖ Hence, the integral J1 is bounded by J1 ≤ −

1 d 2 dt

˜ 2 Ae (ε h˜ e )|∇ Dα φ|

∫ Td

n′e (ε h˜ e + h0e )

dx + c (M )

∫ Td

(|D|α| h˜ e |2 + |D|α| v|2 + |D|α| h˜ i |2 + |D|α| u|2 )dx + c (M ).

Integrating by parts and using the equation in (2.16), we can get the estimate for J2 ,

∫ J2 = − Td

Dα+1 φ˜

n′i (ε h˜ i + h0i )Ae (ε h˜ e ) n′e (ε h˜ e + h0e )

∂t Dα−1 h˜ i −



∫ Td

Dα φ˜ D

n′i (ε h˜ i + h0i )Ae (ε h˜ e ) n′e (ε h˜ e + h0e )

 ∂t Dα−1 h˜ i

˜ 20 + ‖D|α| φ‖ ˜ 20 + ‖∂t Dα−1 h˜ i ‖20 ) ≤ c (M )(‖D|α|+1 φ‖ ≤ c (M )(‖D|α| h˜ e ‖20 + ‖D|α| h˜ i ‖20 + ‖Dα u‖20 + ‖Fα ‖20 ) ≤ c (M )(‖D|α| h˜ e ‖20 + ‖D|α| h˜ i ‖20 + ‖Dα u‖20 + 1). We could obtain the estimate of J3 and J4 by using again the fact that ‖ε∂t h˜ i,e ‖∞ can be controlled. More precisely, we have J3 + J4 ≤ c (M )‖∂t (n′e (ε h˜ e + h0e ))‖∞

∫ Td

˜ 2 )dx (|Dα h˜ e |2 + |Dα φ|

∫ ˜ 2 )dx + c (M )‖∂t (n′i (εh˜ i + h0i ))‖∞ (|Dα h˜ i |2 + |Dα φ| Td ∫ ˜ 2 )dx. ≤ c (M ) (|Dα h˜ e |2 + |Dα h˜ i |2 + |Dα φ| Td

For the control of J5 , first we calculate ∂t Nα

∂t Nα = Dα (n′e (εh˜ e + h0e )∂t h˜ e ) − Dα (n′i (ε h˜ i + h0i )∂t h˜ i ) − (n′e (ε h˜ e + h0e )Dα ∂t h˜ e − n′i (ε h˜ i + h0i )Dα ∂t h˜ i ) − (n′′e (εh˜ e + h0e )ε∂t h˜ e Dα h˜ e − n′′i (εh˜ i + h0i )ε∂t h˜ i Dα h˜ i ).

L. Chen et al. / Nonlinear Analysis: Real World Applications 12 (2011) 1002–1012

1009

Obviously,

‖∂t N0 ‖0 ≤ ‖n′′e (εh˜ e + h0e )ε∂t h˜ e h˜ e ‖0 + ‖n′′i (ε h˜ i + h0i )ε∂t h˜ i h˜ i ‖0 ≤ c (M ) and ‖∂t N1 ‖0 = 0. By Moser’s type inequality, we have for |α| ≥ 2,  ‖∂t Nα ‖0 ≤ c (M ) ‖∇ n′e (ε h˜ e + h0e )‖∞ ‖D|α|−1 ∂t h˜ e ‖0 + ‖∂t h˜ e ‖∞ ‖D|α| n′e (εh˜ e + h0e )‖0 + ‖∇ n′i (εh˜ i + h0i )‖∞ ‖D|α|−1 ∂t h˜ i ‖0 + ‖∂t h˜ i ‖∞ ‖D|α| n′i (εh˜ i + h0i )‖0  + ‖ε∂t h˜ e ‖∞ ‖Dα h˜ e ‖0 + ‖ε∂t h˜ i ‖∞ ‖Dα h˜ i ‖0 ≤ c (M )(‖D|α| h˜ e ‖20 + ‖D|α| h˜ i ‖20 + ‖D|α| u‖20 + ‖D|α| v‖20 + ‖F|α|−1 ‖20 ‖H|α|−1 ‖20 ) ≤ c (M )(‖D|α| h˜ e ‖20 + ‖D|α| h˜ i ‖20 + ‖D|α| u‖20 + ‖D|α| v‖20 + 1). Thus we have the estimate for J5 ,

˜ 20 + ‖∂t Nα ‖20 ) J5 ≤ c (M )(‖Dα φ‖ ≤ c (M )(‖D|α| h˜ e ‖20 + ‖D|α| h˜ i ‖20 + ‖D|α| u‖20 + ‖D|α| v‖20 + 1). Now the bounds for Ji , i = 1, . . . , 5 above show that the integral K3 can also be controlled. The control of K1 , K2 , and K3 then controls I4 . Finally, the bounds for I1 , I2 , I3 and I4 give the inequality ∫ 1 d (Ai (εh˜ i )|Dα hi |2 + |Dα u|2 + Ae (εh˜ e )|Dα he |2 + |Dα v|2 2 dt Td

+

Ae (ε h˜ e ) n′e (ε h˜ e + h0e )

≤ c (M )

∫ Td

˜ 2 )dx + |∇ Dα φ|

∫

(|Dα u|2 + |Dα v|2 )dx

Td

(|D|α| h˜ e |2 + |D|α| v|2 + |D|α| h˜ i |2 + |D|α| u|2 )dx + c (M ).

Summing up over all multi-indices α with the same norm gives 1 d

∫ 

2 dt

Td

+

Ai (ε h˜ i )|D|α| hi |2 + |D|α| u|2 + Ae (ε h˜ e )|D|α| he |2 + |D|α| v|2



Ae (ε h˜ e ) n′e (ε h˜ e + h0e )

≤ c (M )

∫ Td

|∇ D φ|

|α| ˜ 2

∫ dx + Td

(|D|α| u|2 + |D|α| v|2 )dx

˜ 2 + |D|α| φ| ˜ 2 )dx + c (M ). (|D|α| h˜ i |2 + |D|α| h˜ e |2 + |D|α| v|2 + |D|α| ∇ φ|

By Gronwall’s inequality, we obtain

˜ sup ‖(h˜ i , h˜ e , u, v, ∇ φ)(·, t )‖s + ‖(u, v)‖L2 (0,T ∗ ;H s (Td )) ≤ (M0 + 1)ec (M )T − 1 ∗

0
≤ ec (M )T (M0 + c (M )T ∗ ). ∗

This gives the assertion of the lemma. 3. Limit for well prepared initial data We first provide the estimates for the time derivative of h˜ i , h˜ e , u, v , and ∇ φ˜ . Lemma 3.1. Let the assumptions of Theorem 1.2 hold, then there exists ε1 ∈ (0, ε0 ) such that for all 0 < ε < ε1 , it holds

|||∂t h˜ i,e |||0,T ∗ + |||∂t u, ∂t v|||0,T ∗ + |||∇ φ˜ t |||0,T ∗ ≤ c (M , M0 , M1 , T ∗ ). Proof. Taking the time derivative of (1.7), we obtain Ai (ε h˜ i )∂tt h˜ i + ε Ai (ε h˜ i )u · ∇∂t h˜ i + Ai (ε h˜ i )∇∂t h˜ i · u0 + ∇ · ∂t u = Ft ,

∂tt u + ε u · ∇∂t u + ∇∂t h˜ i = −∇∂t φ˜ − ∂t u − u0 · ∇∂t u − ∂t u · ∇ u0 + Gt , Ae (ε h˜ e )∂tt h˜ e + Ae (ε h˜ e )v · ∇∂t h˜ e + 1

1

1

ε

∇ · ∂t v = Ht ,

∂tt v + v · ∇∂t v + ∇∂t h˜ e = ∇∂t φ˜ − ∂t v + Lt , ε ε 1∂t φ˜ = n′e (ε h˜ e + h0e )∂t h˜ e − n′i (ε h˜ i + h0i )∂t h˜ i ,

x ∈ Td , t > 0,

(3.21)

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L. Chen et al. / Nonlinear Analysis: Real World Applications 12 (2011) 1002–1012

where 1 ˜ Ft = −Ai (ε h˜ i )∂t (u0 + ε u) · ∇ h˜ i − Ai (ε h˜ i )∂t (A− i (ε hi ))∇ · u

Gt = −∂t (u0 + ε u) · ∇ u − u · ∇∂t u0 1 1 ˜ Ht = −Ae (ε h˜ e )∂t v · ∇ h˜ e − Ae (ε h˜ e )∂t (A− e (ε he ))∇ · v,

ε

Lt = −∂t v · ∇v.

˜ 20 ≤ c (M ) The Poisson equation for ∂t φ˜ gives the estimates for ‖∇∂t φ‖ Friedrichs estimates for (3.21) give



Td

(|∂t h˜ i |2 + |∂t h˜ e |2 )dx.

∫ 1 (Ai (ε h˜ i )|∂t h˜ i |2 + |∂t u|2 + Ae (ε h˜ e )|∂t h˜ e |2 + |∂t v|2 )dx + (|∂t u|2 + |∂t v|2 )dx 2 dt Td 2 Td ∫ ∫ 1 ≤ (|Ft |2 + |Gt |2 + |Ht |2 + |Lt |2 )dx + c (M ) (|∂t h˜ i |2 + |∂t u|2 + |∂t h˜ e |2 + |∂t v|2 )dx d 2 Td T ∫ ∫ 1 ˜ ˜ − ∇∂t φ · ∂t udx + ∇∂t φ · ∂t v dx. ε Td Td 1 d

∫

The terms like Ft can be easily controlled by

∫ Td

(|Ft |2 + |Gt |2 + |Ht |2 + |Lt |2 )dx ≤ c (M )

∫ Td

(|∂t h˜ i |2 + |∂t h˜ e |2 + |∂t u|2 + |∂t v|2 + 1)dx.

There are still two terms including ∇∂t φ˜ in the above energy estimate that need to be controlled. They are −

∂t udx and ε ∇∂t φ˜ · ∂t v dx. The first one can be easily controlled by Td ∫ ∫ − ∇∂t φ˜ · ∂t udx ≤ c (M ) (|∂t h˜ i |2 + |∂t h˜ e |2 + |∂t u|2 )dx.  −1

Td



Td

∇∂t φ˜ ·

Td

The second one involving 1/ε makes things much more complicated. In order to control it we use (3.21): 1

ε

∫ Td

∇∂t φ˜ · ∂t v dx = −

1

∫

ε ∫

˜ · ∂t v dx ∂t φ∇ ∫ ∫ ∂t φ˜ Ht dx + ∂t φ˜ Ae (εh˜ e )v · ∇∂t h˜ e dx + ∂t φ˜ Ae (εh˜ e )∂tt h˜ e dx

Td

=− Td

= P1 + P2 + P3 .

Td

Td

The above estimate for Ht gives P1 ≤ c ( M )

∫ Td

˜ 2 + |∂t h˜ e |2 + |∂t v|2 )dx. (|∂t φ|

In order to avoid the term ∇∂t h˜ e in P2 , we integrate by parts: P2 = −

∫  Td

≤ c (M )

 ˜ Ae (εh˜ e ) · v + ∂t φ˜ Ae (εh˜ e )∇ · v ∂t h˜ e dx ∇∂t φ˜ Ae (ε h˜ e ) · v + ∂t φ∇

∫ Td

˜ 2 + |∂t φ| ˜ 2 + |∂t h˜ e |2 )dx. (|∇∂t φ|

For the estimate of the remaining integral P3 , we take the time derivative of the Poisson equation in (3.21) and multiply the resulting equation by ∂t φ˜ Ae (ε h˜ e )/n′e (ε h˜ e + h0e ). This yields

˜ tt h˜ e = Ae (ε h˜ e )∂t φ∂

Ae (ε h˜ e ) n′ (ε h˜ e

+

∂t φ˜ 1∂tt φ˜ −

+ ) ˜ Ae (ε he )n′i (ε h˜ i + h0i ) e

h0e

n′e (ε h˜ e + h0e )

Ae (ε h˜ e ) n′ (ε h˜ e

e

˜ tt h˜ i + ∂t φ∂

Notice that

∇ · ∂t u Ft ∂tt h˜ i = −(εu + u0 ) · ∇∂t h˜ i − + . ˜ Ai (ε hi ) Ai (ε h˜ i )

+ h0e )

˜ t h˜ e ∂t (n′e (εh˜ e + h0e ))∂t φ∂

Ae (ε h˜ e ) n′e (ε h˜ e + h0e )

˜ t h˜ i . ∂t (n′i (ε h˜ i + h0i ))∂t φ∂

L. Chen et al. / Nonlinear Analysis: Real World Applications 12 (2011) 1002–1012

1011

We have by integration by parts to avoid terms like ∇∂t h˜ i and ∇ · ∂t u, Ae (ε h˜ e )n′i (ε h˜ i + h0i )

∫

n′e (ε h˜ e + h0e )

Td

˜ tt h˜ i dx = ∂t φ∂



∫ ∇·

Ae (ε h˜ e )n′i (ε h˜ i + h0i ) n′e (ε h˜ e + h0e )

Td



∫ +

∇

Ai (ε h˜ i )n′e (ε h˜ e + h0e )

Ae (ε h˜ e )n′i (ε h˜ i + h0i )

+

Ai (ε h˜ i )n′e (ε h˜ e + h0e )

Td

≤ c (M )

∫ Td

˜ u + u ) ∂t h˜ i dx ∂t φ(ε

Ae (ε h˜ e )n′i (ε h˜ i + h0i )

Td

∫

 0

 ∂t φ˜ · ∂t udx

∂t φ˜ Ft dx

˜ 2 + |∂t φ| ˜ 2 + |∂t u|2 + |∂t h˜ i |2 + |Ft |2 )dx. (|∇∂t φ|

˜ tt h˜ e in P3 by the above expression, we obtain, Substituting the product Ae (ε h˜ e )∂t φ∂ P3 ≤ −

1 d 2 dt

≤−

n′ (ε h˜

Td

+ c (M ) 1 d

Ae (ε h˜ e )

∫ e

∫ Td

+

)

Ae (ε h˜ e ) Td

1 2

∫ Td

∂t

Ae (ε h˜ e ) n′ (ε h˜ e

e

+ h0e )

˜ 2 dx |∇∂t φ|

˜ 2 + |∂t φ| ˜ 2 + |∂t u|2 + |∂t h˜ e |2 + |∂t h˜ i |2 + |Ft |2 + |ε∇∂tt φ| ˜ 2 )dx (|∇∂t φ|

∫

2 dt

e

h0e

˜ 2 dx + |∇∂t φ|

n′e (ε h˜ e + h0e )

˜ 2 dx + c (M ) |∇∂t φ|

∫ Td

˜ 2 + |∂t φ| ˜ 2 + |∂t h˜ e |2 + |∂t h˜ i |2 (|∇∂t φ|

˜ 2 )dx. + |∂t u|2 + |∂t v|2 + |ε∇∂tt φ| We employ again the Poisson equation to deal with the term ε∇∂tt φ˜ ,

1∂tt φ˜ = ∂t (n′e (ε h˜ e + h0e ))∂t h˜ e − ∂t (n′i (ε h˜ i + h0i ))∂t h˜ i + n′e (ε h˜ e + h0e )∂tt h˜ e − n′i (ε h˜ i + h0i )∂tt h˜ i 1 n′e (ε h˜ e + h0e ) = ∂t (n′e (ε h˜ e + h0e ))∂t h˜ e − ∂t (n′i (εh˜ i + h0i ))∂t h˜ i − n′e (εh˜ e + h0e )v · ∇∂t h˜ e − ∇ · ∂t v ε Ae (ε h˜ e ) n′ (ε h˜ e + h0e ) n′ (ε h˜ i + h0i ) n′ (ε h˜ i + h0i ) + e Ht + (ε u + u0 )n′i (ε h˜ i + h0i ) · ∇∂t h˜ i + i ∇ · ∂t u − i Ft . Ae (ε h˜ e ) Ai (ε h˜ i ) Ai (ε h˜ i ) Multiplying this equation by −ε 2 ∂tt φ˜ , integrating over Td and integrating by parts yields

∫ Td

˜ 2 dx ≤ |ε∇∂tt φ|

1 2

∫ Td

˜ 2 dx + c (M ) |ε∇∂tt φ|

∫ Td

(|∂t h˜ i |2 + |∂t h˜ e |2 + |∂t u|2 + |∂t v|2 )dx.

Thus, the estimate of P3 becomes P3 ≤ −

Ae (ε h˜ e )

∫

1 d 2 dt

Td

n′e (ε h˜ e + h0e )

˜ 2 dx + c (M ) |∇∂t φ|

We end up with an estimate of ε −1 1 d 2 dt

∫ Td



Td

∫ Td

˜ 2 + |∂t h˜ i |2 + |∂t h˜ e |2 + |∂t u|2 + |∂t v|2 )dx. (|∇∂t φ|

∇∂t φ˜ · vt dx, and thus, we obtain

(Ae (ε h˜ e )|∂t h˜ e |2 + |∂t u|2 + Ai (εh˜ i )|∂t h˜ i |2 + |∂t v|2 +

≤ c (M )

∫ Td

Ae (ε h˜ e ) n′e (ε h˜ e + h0e )

˜ 2 )dx + |∇∂t φ|

∫ Td

(|∂t u|2 + |∂t v|2 )dx

(|∂t h˜ i |2 + |∂t h˜ e |2 + |∂t u|2 + |∂t v|2 )dx + c (M ).

The proof of the lemma is completed by application of Gronwall’s lemma if a bound for the initial data is available, i.e., ˜ ‖(∂t h˜ i,e , ∂t u, ∂t v, ∇∂t φ)(·, 0)‖0 ≤ c (M0 , M1 ).  The proof of Theorem 1.2 can be done by similar arguments in [16] based on the estimates we obtained in Theorem 1.1 and Lemma 3.1. References [1] P. Markowich, C. Ringhofer, C. Schmeiser, Semiconductor Equations, Springer, Vienna, 1990. [2] A. Jüngel, Quasi-Hydrodynamic Semiconductor Equations, Birkhäuser, Basel, 2001. [3] P. Degond, P.A. Markowich, On a one-dimensional steady-state hydrodynamic model for semiconductors, Appl. Math. Lett. 3 (1990) 25–29.

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[4] I.M. Gamba, Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductor, Comm. Partial Differential Equations 17 (1992) 553–577. [5] B. Zhang, Convergence of the Godunov scheme for the simplifies one-dimensional hydrodynamic model for semiconductor devices, Comm. Math. Phys. 157 (1993) 1–22. [6] P. Marcati, R. Natalini, Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift diffusion equations, Arch. Ration. Mech. Anal. 129 (1995) 129–145. [7] L. Hsiao, K.J. Zhang, The relaxation of the hydrodynamic model for semiconductors to the drift-diffusion equations, J. Differential Equations 165 (2) (2000) 315–354. [8] T. Luo, R. Natalini, Z.P. Xin, Large-time behavior of the solutions to a hydrodynamic model for semiconductors, SIAM J. Appl. Math. 59 (1998) 810–830. [9] L. Hsiao, T. Yang, Asymptotics of initial boundary value problems for hydrodynamic and drift diffusion models for semiconductors, J. Differential Equations 170 (2) (2001) 472–493. [10] H.L. Li, P. Markowich, M. Mei, Asymptotic behavior of solutions of the hydrodynamic model of semiconductors, Proc. Roy. Soc. Edinburgh Sect. A 132 (2) (2002) 359–378. [11] G. Alì, A. Jüngel, Global smooth solutions to the multi-dimensional hydrodynamic model for two-carrier plasmas, J. Differential Equations 190 (2003) 663–685. [12] S. Cordier, E. Grenier, Quasineutral limit of an Euler–Poisson system arising from plasma physics, Comm. Partial Differential Equations 25 (2000) 1099–1113. [13] Y.-J. Peng, J.-G. Wang, Convergence of compressible Euler–Poisson equations to incompressible type Euler equations, Asymptot. Anal. 41 (2005) 141–160. [14] S. Wang, Quasineutral limit of the Euler–Poisson system with and without viscosity, Comm. Partial Differential Equations 29 (2004) 419–456. [15] A. Jüngel, Y.-J. Peng, A hierarchy of hydrodynamic models for plasmas: zero-relaxation-time limits, Comm. Partial Differential Equations 24 (1999) 1007–1033. [16] G. Alì, L. Chen, A. Jüngel, Y. Peng, The zero-electron-mass limit in the hydrodynamic model for plasmas, Nonlinear Anal. TMA 72 (2010) 4415–4427. [17] D. Hewett, Low-frequency electro-magnetic (Darwin) applications in plasma simulation, Comput. Phys. Comm. 84 (1994) 243–277. [18] F. Kazeminezhad, J. Dawson, J. Lebœuf, R. Sydora, D. Holland, A Vlasov ion zero mass electron model for plasma simulations, J. Comput. Phys. 102 (1992) 277–296. [19] S. Klainerman, A. Majda, Singular perturbations of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math. 34 (1981) 481–524. [20] S. Ukai, The incompressible limit and the initial layer of the compressible Euler equation, J. Math. Kyoto Univ. 26 (1986) 323–331. [21] E. Grenier, Oscillatory perturbations of the Navier Stokes equations, J. Math. Pures Appl. 76 (1997) 477–498. [22] G. Métivier, S. Schochet, The incompressible limit of the non-isentropic Euler equations, Arch. Ration. Mech. Anal. 158 (2001) 61–90. [23] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer, New York, 1984. [24] S. Klainerman, A. Majda, Compressible and incompressible fluids, Comm. Pure Appl. Math. 35 (1982) 629–651.