Zero-electron-mass limit of Euler-Poisson equations

This paper is concerned with multidimensional Euler-Poisson equations for plasmas. The equations take the form of Euler equations for the conservation laws of the mass density and current density for charge-carriers (electrons and ions), coupled to a Poisson equation for the electrostatic potential. We study the limit to zero of some physical parameters which arise in the scaled Euler-Poisson equations, more precisely, which is the limit of vanishing ratio of the electron mass to the ion mass. When the initial data are small in critical Besov spaces, by virtue of the ``Shizuta-Kawashima"skew-symmetry condition, we establish the uniform global existence and uniqueness of classical solutions. Then we develop new frequency-localization Strichartz-type estimates for the equation of acoustics (a modified wave equation) with the aid of the detailed analysis of the semigroup formulation generated by this modified wave operator. Finally, it is shown that the uniform classical solutions converge towards that of the incompressible Euler equations (for \textit{ill-prepared} initial data) in a refined way as the scaled electron-mass tends to zero.


Introduction
The Euler-Poisson equations, sometimes called as the (macroscopic) hydrodynamic models, are to describe the transport of charge-carriers (electrons and ions) in semiconductor devices or plasmas. The system consists of the conservation laws for the mass density and current density for carriers, with a Poisson equation for the electrostatic potential. Using the classical moment method, it is derived from the semiclassical Boltzmann-Poisson equations, for details, see [14]. In recent three decades, the Euler-Poisson equations has attracted increasing attention both numerical simulation and theoretical analysis, since it can capture more physical phenomena that occur in submicron devices or in the presence of high electric fields than the traditional drift-diffusion models.
More specifically, we consider an un-magnetized plasmas consisting of electrons with charge q e = −1 and of a single species of ions with charge q i = +1. Denote by n e = n e (t, x), v e = v e (t, x) (resp., n i , v i ) the scaled density and mean velocity of the electrons (resp., ions) and by φ = φ(t, x) the scaled electrostatic potential. From [9], these unknowns satisfy the following scaled Euler-Poisson equations:            ∂ ∂t n a + div(n a v a ) = 0, δ a ∂ ∂t (n a v a ) + δ a div(n a v a ⊗ v a ) + ∇P a (n a ) = −q a n a ∇φ − δ a nava τa , λ 2 ∆φ = n e − n i + C(x), where a = e, i and (t, x) ∈ [0, +∞) × R N (N ≥ 2). The symbols div, ∇, ∆ and ⊗ are the x-divergence operator, gradient operator, Laplacian operator and the tensor product of two vectors respectively; the parameters λ, τ a > 0 are the scaled constants for the Debye length and the momentum relaxation time of electrons (if a = e; otherwise, ions if a = i) respectively; the pressure P a (n a ) is a smooth function satisfying P ′ a (n a ) > 0 for all n a > 0.
For the sake of (mathematical) simplicity, we assume that it satisfies the usual "γ-law" P a (n a ) = c a n γa a , where c a > 0 is a physical constant. In such case, the plasmas is called isothermal if γ a = 1 and isentropic if γ a > 1. The function C(x), which only depends on the space variable, represents the density of fixed charged background ions (doping profile).
The dimensionless parameter δ a in the Euler-Poisson equations (1.1) is given by where m e (resp. m i ) is the mass of a single electron (resp. ion), k B is the Boltzmann constant, v 0 and T 0 are typical velocity and temperature values for the plasmas respectively. The reader is refer to [9] for details about the scaling and the physical assumptions. Since m i is not smaller than the mass of a proton, we have m e m i ≤ m e the mass of a proton ≃ 5.45 × 10 −4 .
Namely, me mi is a small number. Thus, if v 2 0 is chosen to be k B T 0 /m i , then δ i = 1 and δ e = me mi ≪ 1. The main goal of this paper is to investigate the limit as δ e goes to zero.
At the formal level, if ∇P (n) → 0, ∇φ → 0, when ǫ goes to zero. Hence n must be the positive constantn. Then passing to the limit in the mass conservation equation, we get divv → 0. Coming back to the momentum equation, we conclude that u (the limit of v) must satisfy the incompressible Euler equations with damping    u t + u · ∇u + u + ∇Π = 0, divu = 0, u| t=0 = u 0 , which also can be written    u t + P(u · ∇u) + u = 0, divu = 0, u| t=0 = u 0 = Pv 0 , (1.4) where P stands for the Leray projector on solenoidal vector fields. Moreover, if we assume that n =n + O(ǫ 2 ), divv = O(ǫ), then ∇P (n) = O(ǫ 2 ), ∇φ = O(ǫ 2 ). Indeed, this entails that n t , v t are uniformly bounded so that the dangerous time oscillations can not occur. Starting from this simple consideration, Alì, Chen, Jüngel and Peng [1] justified the zero-electron-mass limit (ǫ → 0) of (1.2) with data of the following type n 0 =n + ǫ 2 n 0,1 , v 0 =v(x, 0) + ǫv 0,1 with divv(x, 0) = 0, and (n 0,1 , v 0,1 ) uniformly bounded in the the periodic Sobolev space H σ (T N )(σ > 1 + N/2). Such data are referred as well-prepared data in the usual PDE's terminology. In the case of well-prepared data, the zero-electron-mass limit has some similarities with the classical low-Machnumber limit in the compressible Euler equations studied by Klainerman and Majda [11,12]. The unique difference is that there is an extra singularity from the electron-field term. Alì et al. [1] overcame this difficulty by a careful use of the mass conservation and the Poisson equation, and obtained the uniform a priori estimates with respect to the scaled zero-electron-mass. Then they justified the zero-electron-mass limit for smooth solutions from (1.2)-(1.3) to (1.4) (incompressible limit) by virtue of the Aubin-Lions compactness lemma. The precise limit behaviors of smooth solutions were shown by the recent work [17]. The zero-electron-mass limit for weak entropy solutions was studied by Goudon, Jüngel and Peng [6] under some restrictive assumptions, with the help of the kinetic formulation, the monotonicity and compactness arguments. It is worth noting that these results only provided local convergence or the case of well-prepared data.
In the present paper, we shall study the zero-electron-mass limit since few works can be found in mathematics literatures, except [1,6]. We focus on the case of ill-prepared data and hope to get the global convergence of classical solutions. We make a weaker assumption that the initial density only satisfies n 0 =n + ǫn 0,1 with (n 0,1 , v 0 ) uniformly bounded (in an appropriate functional space).

Symmetry
It is convenient to state the basic ideas and main results of this paper, we first introduce a function transform to reduce (1.2) to a symmetric hyperbolic-elliptic form.
For the isentropic case (γ > 1), by defining the sound speed and the sound speedψ = ψ(n) at a background densityn, we set For the isothermal case where γ = 1, the form (1.6) is still valid withψ = √ A. However, it depends on the following enthalpy variable change m = √ A(ln n − lnn), (1.8) which the details are referred to [5]. About the equivalence for classical solutions away from the vacuum between (1.2)-(1.3) and (1.6)-(1.7), see Section 3. Moreover we introduce the variables as follows: Then the new variables satisfy (1.9) with the initial data (1.10) One expects v ǫ to tend to u in (1.9) which u solves the incompressible Euler equations (1.4) as ǫ → 0. The expected convergence however is not easy to justify rigorously. Here, there are two main difficulties. The first is the singularity from the electron-field term, which can not be overcome by using the symmetrizer of main part of hyperbolic system as in [11,12]. We adopt a new (but small!) technique to deal with the singular electron-field term, which can help us to simplify the similar analysis as [1] heavily, see (4.6)-(4.7). The second is that one has to face the propagation of acoustic waves with the speed ǫ −1 , a phenomenon which does not occur in the case of well-prepared data. To solve this, we need to develop some new ideas. Inspired by [4], we split the velocity into a divergence-free part and a gradient part to obtain the equation of acoustics (for details, see Section 5). Then we establish some dispersive estimates according to the semigroup theory of this modified wave operator, and achieve a new frequency-localization Strichartz-type estimate which is used to pass to the zero-electron-mass limit in a refined way. Based on the recent work [5,8], we still choose the critical Besov space framework in space-variable x (a subalgebra of W 1,∞ ) to study the global well-posedness and the zero-electron-mass limit of classical solutions to the system (1.2)-(1.3). The main results are states as follows.
Remark 1.1. The symbol ∇∆ −1 means where G(x, y) is a solution to ∆ x G(x, y) = δ(x − y) with x, y ∈ R N . In the periodic setting, the regularity assumption on e 0 can be removed.
Remark 1.2. In the proof of Theorem 1.1, different from that in [5], "Shizuta-Kawashima" skew-symmetry condition developed for general hyperbolic systems of balance laws [13,19] is used, which helps us avoid differentiating the system with respect to time-variable t and the proof is shortened. Thanks to the isentropic Euler-Poisson equations also including isentropic Euler equations [3], the concrete information of skew-symmetry matrix K(ξ) is well known (unknown for general systems) which is very effective to estimate the coupled electron-field ∇φ ǫ , see (4.21). Remark 1.4. Let us mention that the approach developed by this paper can also be applied to study the low-Mach-number limits of the compressible Euler equations with damping for the perfect gas flow. Therefore, this work can be regarded as a supplement to the theory of asymptotic limits for hyperbolic problems.
Remark 1.5. Prescribing ill-prepared initial data and periodic boundary conditions preclude from using dispersive properties in order to pass to the limit in (1.9). Indeed, there is no chance that the acoustic waves go at infinity. Hence, there could be resonances which may hinder the convergence to the incompressible Euler equations. This will be shown in a forthcoming paper.
Our paper is organized as follows. In Section 2, we introduce Besov spaces and their properties. In Section 3, we give some remarks on the hyperbolic symmetrization and recall a local existence result of classical solutions. In Section 4, we derive the a-priori estimate which is used to achieve the global existence of uniform classical solutions. For clarity, Section 5 is divided into two parts. Based on the detailed analysis of the equation of acoustics, we first establish a new frequency-localization Strichartztype estimate. Then using this estimate, we perform the zero-electron-mass limit in a refined way.
Finally, some efforts on other kinds of asymptotic limits (such as relaxation-time limit and quasineutral limit) of classical solutions to the Euler-Poisson equations (1.2)-(1.3) should be mentioned, the interested reader is referred to [15,16,18] and the literature quoted therein.

Preliminary
Throughout this paper, C is a uniform positive constant independent of ǫ. f ≈ g means that f ≤ Cg and g ≤ Cf . We denote by L ρ (0, T ; X), C([0, T ], X) (resp., C 1 ([0, T ], X)) the space of ρ-power integrable and continuation (resp., continuously differentiable) functions on [0, T ] with values in a Banach space X, respectively. In the case T = +∞, we sometimes label L ρ (R + ; X), C(R + , X), etc. For brevity, we use the notation (a, b, c) X := a X + b X + c X , where a, b, c ∈ X. All functional spaces of the present paper are considered in R N , so we may omit the space dependence for simplicity.
In this section, we review briefly the Littlewood-Paley decomposition theory and the characterization of Besov spaces; see also, e.g., [4] or [5].
Let S be the Schwarz class. (ϕ, χ) is a couple of smooth functions valued in [0,1] such that ϕ is supported in the shell C(0, 3 4 For f ∈ S ′ (denote the set of temperate distributions which is the dual of S), one can define the Fourier dyadic blocks as follows: where F f , F −1 f represent the Fourier transform and the inverse Fourier transform on f , respectively. The nonhomogeneous Littlewood-Paley decomposition is Define the low-frequency cut-off by The above Littlewood-Paley decomposition is almost orthogonal in L 2 .
Proposition 2.1. For any f, g ∈ S ′ , the following properties hold: Based on the above Littlewood-Paley decomposition, we introduce the explicit definitions of nonhomogeneous Besov spaces.
Some conclusions as follows will be used in subsequent analysis. The first one is the classical Bernstein's inequality.
Lemma 2.2 (Bernstein's inequality). Let k ∈ N and 0 < R 1 < R 2 . There exists a constant C depending only on R 1 , R 2 , and N such that for all 1 ≤ a ≤ b ≤ ∞ and f ∈ L a , we have The second one is the embedding properties in Besov spaces.
B s p,r ֒→ Bs p,r whenevers < s ors = s and r ≤r; where C 0 is the space of continuous bounded functions which decay at infinity.
The three one is the compactness result for Besov spaces.
The last one is a continuity result for compositions.
Proposition 2.5. Let 1 ≤ p, r ≤ ∞, and I be an open interval of R. Let s > 0 and let n be the smallest integer such that n ≥ s. Let F : takes values in J ⊂⊂ I. Then F (v) ∈ B s p,r and there exists a constant C depending only on s, I, J, and N such that

Local existence
In order to obtain the effective a priori estimates by the low-and high-frequency decomposition methods, we formulate (1.2) into the symmetric hyperbolic-elliptic form (1.6) in the Introduction.
Here, we give some remarks.
It is easy to show that for classical solutions (n, v, ∇φ) away from the vacuum, the system (1.2) is equivalent to (1.6).
For the isothermal case (γ = 1), we also have the similar equivalence. Note that √ A ln n is the enthalpy, instead of the sound speed.
It is easy to show that for classical solutions (n, v, ∇φ) away from the vacuum, the system (1.2) is equivalent to (1.6). The symmetric hyperbolic system (1.9) can also be rewrite as the vector form coupled with the dynamic electron-potential equation Now, we recall a local existence result on classical solutions to (1.9)-(1.10), which has been obtained in [5].

A uniform a priori estimate
In this section, we establish a crucial a priori estimate by using the low-and high-frequency decomposition methods, which is used to derive the global existence and exponential stability of classical solutions to (1.9)-(1.10).
Proposition 4.1. There exist three positive constants δ 1 , C 1 and µ 1 independent of ǫ, such that for any Having this Proposition 4.1, we can extend the local-in-time solutions in Proposition 3.1 by virtue of the standard continuation argument and obtain the global existence of uniform classical solutions to the system (1.9)-(1.10). Using the imbedding property in Besov space Furthermore, we arrive at Theorem 1.1.
The main ingredients in the proof of Proposition 4.1 are the high-frequency (q ≥ 0) estimates and low-frequency (q = −1) estimates on (m ǫ , v ǫ , ∇φ ǫ ). We divide it into several lemmas, since the proof is a bit longer.
is a solution of (1.9)-(1.10) for any given T > 0, then the following estimate holds (q ≥ −1): where the commutator [f, g] := f g − gf and C is a uniform positive constant independent of ǫ.
Proof. Applying the localization operator ∆ q to (1.9) gives Then, by multiplying the first equation of (4.4) by ∆ q m ǫ , the second one by ∆ q v ǫ respectively, and adding the two resulting equations together, then integrating the resulting equations over R N , we get In above equality (4.5), we may use the spectral localization mass equation and Poisson equation in (4.4) in order to eliminate the singularity from the electron-field term similar to the idea in [1], but this will cause very tedious calculations. Here, we observe an equality following from the first equation and the third one in (1.2) under the symmetrization. Hence, we deduce that Together with (4.5) and (4.7), we arrive at (4.3) immediately with the aid of Cauchy-Schwartz inequality.
In this position, we formulate an important skew-symmetry lemma which has been well developed in [3,13,19], which is sometimes referred to as the "Kawashima condition".
where A ǫ j is the matrix appearing in the system (3.1). Due to the skew-symmetry structure of the system (3.1), we can develop some new frequencylocalization estimates and avoid performing the t-derivative to (3.1) as in [5].
is a solution of (1.9)-(1.10) for any given T > 0, then the following estimates hold: where the function G is given by (4.13), is a smooth function on {m|ςm +h > 0, ς ∈ [0, 1]} satisfying H(0) = 0 and C is a uniform positive constant independent of ǫ.
Proof. The system (3.1) can be written as the linearized form where Applying the operator ∆ q to the system (4.12) gives By performing the Fourier transform with respect to the space variable x for (4.14) and multiplying the resulting equation by −iǫ( ∆ q W ǫ ) * K(ξ)( * represents transpose and conjugate), then taking the real part of each term in the equality, we can obtain Using the skew-symmetry of K(ξ), we have Substituting (4.9) into the second term on the left-hand side of (4.15), it is not difficult to get With the help of Young inequality, the right-hand side of (4.15) can be estimated as where we have used the uniform boundedness of the matrix K(ξ)(ξ = 0). Combining the equality (4.15) and the inequality (4.17)-(4.18), we deduce Multiplying (4.19) by |ξ| and integrating it over R N , using Plancherel's theorem, we obtain Furthermore, the last term on the right-hand side of (4.20) can be estimated as where H(m) = we get the estimate (4.10) and (4.11) immediately.
On the electron field ∇φ, we have the following a priori estimates.
is a solution of (1.9)-(1.10) for any given T > 0, then where C is a uniform positive constant independent of ǫ.
Proof. By applying the localization operator ∆ q (q ≥ 0) to both sides of div∇φ ǫ = ǫ −1 h(ǫm ǫ ) , integrating it over R N after multiplying ∆ q div∇φ ǫ , and noticing the irrotationality of ∇φ ǫ , we can obtain (4.23) in virtue of Hölder's inequality. From (1.2) and (1.5), we get where the non-local term ∇∆ −1 ∇ · f is the product of Riesz transforms on f . From (1.9) and (4.25), we have where I can be estimated as Then using the L 2 -boundedness of Riesz transform and Hölder's inequality, we derive (4.24) immediately.
For the estimates of the commutators in (4.3) and (4.10)-(4.11) and (4.24), we have the following conclusion.
Lemma 4.6 (see [5]). Let s > 0 and 1 < p < ∞; then the following inequalities are true: In particular, if f = g, then where the operator A = div or ∇, C is a harmless constant, and c q denotes a sequence such that (c q ) l 1 ≤ 1.
With these lemmas for ready, now, we complete the proof of the unform a priori estimate (4.2). Proof of Proposition 4.1. Note that the a priori assumption (4.1), we deduce from the embedding inequality in Besov spaces that (4.28) To ensure the smoothness of functions h(ǫm ǫ ) and H(ǫm ǫ ), together with the smallness of ǫ, it suffices to choose 0 < δ 1 ≤ψ (γ−1)C such that For the proof of Proposition 4.1, it can be divided into the following high-frequency part and lowfrequency part.
Lemma 4.7 (q ≥ 0). There exist some positive constants K 1 , K 2 , µ 2 independent of ǫ such that the following estimate holds: where K 1 , K 2 are given by (4.31) and C is a uniform positive constant independent of ǫ.
Proof. Combining (4.3), (4.10) and (4.23), we have d dt where the uniform positive constants K 1 , K 2 and K 3 (independent of ǫ) satisfy (4.31) Due to so we introduce these uniform constants in order to ensure and eliminate the quadratic terms in the right-hand side of (4.30) with the aid of Young's inequality, for similar details, see [5]. Dividing the resulting inequality by after eliminating the quadratic terms, then multiplying the factor 2 q(σ−1) on the both sides of inequality, we arrive at (4.29) immediately with the help of Lemma 4.6 and the smallness of ǫ(0 < ǫ ≤ 1).
Similarly, we also have the a priori estimate for the case of low frequency.
Lemma 4.8. (q = −1) There exist some positive constantsK 1 ,K 2 ,K 3 and µ 3 independent of ǫ, such that the following estimate holds: where C is a uniform positive constant independent of ǫ.
Remark 4.1. Similar to the proof of Lemma 4.7, the constantsK 1 ,K 2 ,K 3 are introduced to ensure that and eliminate some quadratic terms appearing in the right-hand side of inequality (4.34).

Zero-electron-mass limit
In this section, our first aim at deducing a frequency-localization Strichartz-type estimate which is the crucial ingredient of the zero-electron-mass limit. For this end, we need to give a detailed analysis of the equation of acoustics (5.3) with the aid of the semigroup formulation, and obtain some dispersive estimates. Then, by using the classical T T * argument, we obtain the desired Strichartz-type estimate.

The linearized system
. Following from the idea in [4], we split the velocity into a divergence-free part Pv ǫ and a gradient part Qv, where P = I − ∇∆ −1 div and Q = I − P = ∇∆ −1 div. Then, the system (5.1) reads where d ǫ 0 := Λ −1 divQv ǫ 0 . According to the semigroup theory for evolutional equation, the solutions (m ǫ , d ǫ ) to the linear initial value problem (5.3) can be expressed for U ǫ = (m ǫ , d ǫ ) ⊤ as which gives rise to where S(t)U ǫ 0 := e tB U ǫ 0 . Then, we analyze the differential operator B by means of its Fourier expression A(ξ) and show the long time properties of the semigroup S(t). Applying the Fourier transform to the system (5.4) with F = 0 and G = 0, we get The eigenvalues of the matrix A(ξ) are computed from the determinant which implies Hence, the semigroup e tA exhibits the expression After a direct computation, we can verify the exact expression about the Fourier transform e tB of the Green's function e tB as e tB := e tA = e λ+t P + + e λ−t P − Lemma 5.1 (Dispersive estimate). With the above notations, when ǫ is small enough, we get the following estimate: and smooth function ψ satisfies suppψ(x) ∈ {x ∈ R| 1 6 ≤ x ≤ 3} and ψ(x)| 5 6 ≤|x|≤ 12 5 = 1, C > 0 denotes a uniform constant independent of k and ǫ.
Similarly, we can obtain the following lemma and omit the proofs for brevity.
Lemma 5.2 (Dispersive estimates). With the above notations, when ǫ is small enough, we get the following estimates: and C > 0 denotes a uniform constant independent of k and ǫ.
Proposition 5.3 (Strichartz-type estimate). Suppose U ǫ is the solution of the system (5.4) with When ǫ is small enough, we get the following estimate where p ∈ [2, +∞].
Proof. Duhamel's formula enable us to restrict our attentions to the case (F, G) = 0. if we can obtain by the same computations as that in [2], we can get the estimate on by Fubini theorem and interpolation. Finally, we can get (5.14).
In the following, we mainly prove the estimate (5.15). For simplicity, we only estimate the term τ = t ǫ , and ψ satisfies suppψ(x) ∈ {x ∈ R| 1 6 ≤ x ≤ 3} and ψ(x)| 5 6 ≤|x|≤ 12 5 = 1. Now we shall use the T T * argument. Define By the Fourier-Planchered theorem, we have Note that the following identity holdŝ Then we have Similarly, from Lemma 5.2, we can estimate the other terms in S(t)U ǫ 0 and obtain Similarly, we can obtain that the L 1 (R + ; L 2 (R N )) norm of U ǫ is bounded, uniformly in ǫ, by interpolation, we get (5.15).
Mf ), then we obtain where p ∈ [2, ∞]. From (5.14), we have where C is independent of ǫ and c k ≤ 1. Let P A≤·≤B f = F −1 (χ A≤|ξ|≤Bf ). Then, we have  4). Following from the standard frequency-localization method, we can easily obtain the following result.  where H = −P(w · ∇u) − P(v ǫ · ∇w) − P(Qv ǫ · ∇u) − P(v ǫ · ∇Qv ǫ ). Then, by multiplying by w and integrating the resulting equations over R N , we have for all T > 0. Using the Gronwall's inequality, we have By interpolation, we can easily obtain (5.28).