Quasineutral limit for the quantum Navier-Stokes-Poisson equation

In this paper, we study the quasineutral limit and asymptotic behaviors for the quantum Navier-Stokes-Possion equation. We apply a formal expansion according to Debye length and derive the neutral incompressible Navier-Stokes equation. To establish this limit mathematically rigorously, we derive uniform (in Debye length) estimates for the remainders, for well-prepared initial data. It is demonstrated that the quantum effect do play important roles in the estimates and the norm introduced depends on the Planck constant $\hbar>0$.


Introduction
The purpose of this paper is to consider the following hydrodynamic Navier-Stokes-Possion equation, describing the motion of the electrons in plasmas with the electric potential [16], where V = −eφ is the electrostatic potential, n is the electron density, u = (u 1 , u 2 , u 3 ) is the velocity, nu is the momentum density, P = (P ij ) 3×3 is the stress tensor, W is the energy density and q is the heat flux. It is customary that the heat flux is assumed to obey the Fourier law q = −κ∇T . To close the moment expansion at the third order, we define the above stress tensor P and W in terms of the density n, u and T by P ij = −nT δ ij + 2 n 12 ∂ 2 ∂x i ∂x j log n, and W = 3 2 nT + 1 2 n|u| 2 − 2 n 24m ∆ log n, respectively, where > 0 is the Planck constant and is small compared to macro quantities. We have omitted the terms that are O( 4 ). The viscous stress tensor S is given by S = µ(∇u + (∇u) ⊤ ) + λ(divu)I, 2010 Mathematics Subject Classification. 76Y05; 35B40; 35C20. The second author is supported in part by NSFC (11471057) and the Fundamental Research Funds for the Central Universities (Project No. 106112016CDJZR105501).The third author is supported by NSFC (11371042) and the key fundation of Beijing Municipal Education Commission. 1 to include the viscosity, with µ > 0 and 2µ + 3λ ≥ 0. The quantum stress tensor is closely related to the quantum Bohm potential [1,37] By simple calculations, it gives that −n∇Q(n) = { (∆n∇n + ∇n · ∇ 2 n) n − (∇n · ∇n)∇n n 2 }.
In the above equation (1.1), ε is the scaled squared Debye length, which vary by many orders of magnitude. Typical values of the Debye length go from 10 −3 m to 10 −8 m. The Debye length is a fundamental and important parameter in plasmas, below which charge separation occurs. In practical applications, the Debye length is small compared to characteristic observation length and hence is interesting to study the limit as ε goes to zero. Formally, this will leads to an incompressible Navier-Stokes equation (See Sect. 1.1 below) and this paper aims to study this limit mathematically rigorously.
Before we proceed, we recall some derivations for the quantum fluid dynamics equation. The full compressible quantum Navier-Stokes-Poisson (FCQNSP) model (1.1) has many practical applications in fluid models of nucleus, superconductivity and superfluidity and semiconductor devices, and was derived by Gardner [16] from a moment expansion of the Wigner-Boltzmann equation. When there is no electrostatic potential V , it reduces to the full compressible quantum Navier-Stokes (FCQNS) equation, which was also derived by Jüngel and Miliŝić [25] from the collisional Wigner equation [37] by performing a Chapman-Enskog expansion. See also the references therein for the derivation of other quantum hydrodynamics equations from Wigner equation. In particular, Degond and Ringhofer [11] formally derived nonlocal quantum hydrodynamic models, where the quantum stress tensor may by nondiagonal and the quantum heat flux many not vanish, in contrast to the classical equations. For more models taking quantum effects into consideration, see also [7][8][9][10]. By performing the zero-mean-free-path limit in the collisional Wigner equation, Brull and Méhats derived the isothermal quantum Navier-Stokes equation [3].
Although have many applications in various fields of physics, the full quantum Navier-Stokes-Poisson equation (1.1) and the full quantum Navier-Stokes equation are less studied mathematically rigorously, to our best knowledge. Without quantum effects, they are all comprehensively studied in various aspects of mathematics, see for example [4,26,33,34]. For the full compressible quantum Navier-Stokes equation, the only result known to the authors is the global existence of small classical solutions in R 3 when the viscosity and heat conductivity are present, obtained recently in [30] following the seminal paper of Matsumura and Nishida [29]. For other results on the quantum hydrodynamic equations and related models, the readers may refer to [15, 19-21, 23, 24, 27] and the references therein.
Next, we recall some mathematical results on quasi-neutral limit for various hydrodynamic equation. Indeed, in the recent two decades, the quasi-neutral limit problem has attracted many attentions of physicists and applied mathematicians. To the author's best knowledge, the first quasi-neutral limit is on the Euler-Poisson equation for ions with positive ion temperature by Cordier and Grenier [5]. This result was recently generalized to the Euler-Poisson equation with zero ion temperature for cold plasmas in [31]. Since then, quasi-neutral limit results have been obtained for various hydrodynamic models in plasmas. See [35] for the Euler/Navier-Stokes-Poisson system with and without viscosity, [36] for the Navier-Stokes-Poisson equation, [32] for the Cauchy problem for the non-isentropic Euler-Poisson equation with prepared initial data, [22] for the non-isentropic compressible Navier-Stokes-Poisson, [13] for the analysis of oscillations and defect measures, to list only a few. See also [6,12,14,17,18] for the other related limits. For hydrodynamic models with capillarity or Korteweg effects, there are also some quasineutral limit results, see [2,28] for example. There is a vast literature concerning the quasi-neutral limit for various models, and we cannot give a complete list here. The interested readers may refer to these papers above and the references therein.
As pointed out above, we aim to study the quasineutral limit mathematically rigorously. To be precise, we confine ourselves to the Cauchy problem for (1.1) in R 3 . We show that as Debye length goes to zero, the smooth solutions converges to solutions of the incompressible Navier-Stokes equation (1.4), at least for well-prepared initial data (The case for ill-prepared initial data will be treated somewhere else). We also obtained convergence rate w.r.t. the Debye length parameter ε. The main result is stated in Theorem 1.3. Due to the special structure of (1.1), to get uniform in ε estimates for the remainder terms we need to carefully use the structure of the equation and construct suitable energy norms in estimates. The norm we finally adopt is the triple norm defined in (2.1) incorporating the quantum parameter > 0. Since higher order terms appear in this triple norm, much effort is needed to close the estimate in the proof.
In the rest of Introduction, we first give the formal expansions and derive the incompressible Navier-Stokes equation (1.4) for the leading terms and then we derive the remainder equation (1.12) and state the main result in Theorem 1.3.

Formal Expansions.
In terms of (n, u, T, φ), the above quantum Navier-Stokes-Poisson system (1.1) with the electric potential can be rewritten as Plugging the following formal expansion into (1.2), we get a power series of ε. At O(1) order, we obtain n (0) = 1 and the system for (u (0) , T (0) , φ (0) ), Formally, as ε → 0, we known that the solutions of (1.2) should converge to those of (1.4).
Then for any given initial data (u 0 ) ∈ Hs +3 and divu (0) 0 = 0, there exists some τ * > 0 such that the initial value problem (1.4) has a unique solution such that for any τ < τ * , the following holds where C is a constant.
In the next two sections, we will give some uniform estimates for (N R , U R , T R , Φ R ). Finally in Section 4, we prove Theorem 1.3. Throughout this paper, we use to a b to stand for a ≤ Cb for some constant C > 0. Let [f, g] = f g − gf to denote the commutator of f and g.

Basic estimates
To give uniform estimates for (1.12), we first introduce the norm (2.1) LetC be a constant to be determined later, which is assumed to be independent of ε and much larger than the bound |||(U R , N R , T R , ∇Φ R )(0)||| 2 3 of the initial data. Next, it is classical that there exists τ ε > 0 such that on [0, τ ε ], Hence, we can assume that n is bounded from above and below 1/2 < n < 3/2, when ε is sufficiently small. We will show that for any given τ > 0, there is some ε 0 > 0 such that the existence time τ ε > τ for any 0 < ε < ε 0 . To prove the theorem 1.3, we need to derive the uniform estimate for the remainder system (1.12). To this end, we first give some estimates in Lemma 2.1, Lemma 2.2 and Lemma 2.3 in this section. a solution to (1.12), and α be a multi-index with |α| = k, then we obtain

4)
and Proof. Notice that applying (1.12a), (2.3), and Sobolev embedding, together with (5.1) and (5.2) in the Appendix, we can obtain Moreover, it is easy to have

Lemma 2.2.
Under the same conditions in Lemma 2.1, we have the following estimate,

7)
and (2.8) Proof. For |α| = 1, it is easy to prove, so we take |α| = 2 for example. By computation then taking L 2 -norm, using Hölder's inequality and Sobolev embedding H 1 ֒→ L 3 , L 6 , we can obtain Similarly, by computation, we have Due to Sobolev embedding H 2 ֒→ L ∞ , Young's inequality and the above results, we can verify Again taking L 2 -norm, using Hölder's inequality and Sobolev embedding H 1 ֒→ L 3 , L 6 , we know The case of |α| = 2 can be proved similarly, thus we omit them. Lemma 2.3. Let α be a multi-index with 0 ≤ |α| ≤ k and k be an integer, f ∈ S, the Schwartz class, then we obtain a prior bound Proof. Thanks to Plancherel's theorem, we derive where we use the Riesz operator R j , (R j f ) = iξj |ξ|f , and the conclusion in [33] that R i R j is bounded from L p to L p with 1 < p < ∞. Then summing all multi-index α with 0 ≤ |α| ≤ k, we complete the proof.

Uniform energy estimates
In this section, we will prove the following a solution to (1.12), then there holds

1)
where δ = min{ 1 32 , 1 32µ , 1 32(µ+λ) } and ≪ 1. This proposition is proved as a direct sequence of the following Lemmas. Moreover, combining Proposition 3.1 with the standard continuous induction method, we can obtain for any given τ > 0, there is some ε 0 > 0 such that the existence time τ ε > τ for any 0 < ε < ε 0 . a solution to (1.12), and α be a multi-index with |α| = k, then we can have Proof. Applying the operator ∂ α to (1.12a) and taking inner product with ∂ α N R , it holds 1 2 We shall estimate the first two terms of the right hand side. Combining the fact divu (0) = 0 with integration by parts, Sobolev embedding and the commutator estimates Remark 5.1 in the Appendix, we obtain for any sufficiently small positive constant δ. It is obvious that the last three terms can be bounded by ε + (N R , U R ) 2 H 3 . The proof is complete by taking δ = 1 32 .

Lemma 3.2.
Under the assumptions in Lemma 3.1, we obtain Proof. Applying the operator ∂ α to (1.12b) and taking inner product with ∂ α U R , we derive The first term R 2,1 can be bounded by Young's inequality, Lemma 2.2, Lemma 2.3 and (5.1)∼(5.4) in the Appendix that for any sufficiently small positive constant δ. Here, we only estimate the two terms in R 2,1 , For the second term, based on integration by parts and the commutator estimates Remark 5.1 in the Appendix, we deduce Now, we will estimate the term R 2,3 since there is ∇N R and we can not close our estimates with this. Owing to Lemma 2.2, integration by parts, Young's inequality, and the commutator estimates Remark 5.1 in the Appendix, we have The estimate for R 2,4 in (3.4) requires much efforts since it involves higher order terms. From (1.12a) and the commutator, it is easy to compute That is to say, we can insert the above results into R 2,4 to decompose where R 2,4,1 is denoted by Moreover, by the commutator estimates, integration by parts, Hölder inequality, Sobolev embedding H 2 ֒→ L ∞ and Lemma 2.1, Lemma 2.2, it follows that Plugging this above equality into R 2,7 , together with (1.12d), we obtain Finally, by Young's inequality and the commutator estimate Remark 5.1 in the Appendix, we have Putting the estimates together, and taking δ = 1 32 , we complete the proof of lemma 3.2. Lemma 3.3. Under the assumptions in Lemma 3.1, we obtain d dt (3.5) Proof. Applying the operator ∂ α to (1.12c) and taking inner product with ∂ α T R yield  Proof. Applying the operator ∂ α to (1.12b) and taking inner product with − 2 ∂ α ∆U R , we obtain Similar to R 2,1 , the first term can be treated by Young's inequality, Lemma 2.2, Lemma 2.3 and (5.1)∼(5.4) in the Appendix, Hence, it is obvious that, for any m ≥ 1, together with 1 2 ≤ n ≤ 2 3 , we can deduce the following inequality Similarly, we derive