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 wellprepared initial data. It is demonstrated that the quantum effect do play important roles in the estimates and the norm introduced depends

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... on the Planck constant > 0. 2000 Mathematics Subject Classification. Primary: 76Y05, 35B40, 35C20; Secondary: 35Q35. Although have many applications in various fields of physics, the full quantum Navier-Stokes-Poisson equation (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 [5, 28, 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, 26] 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 [6]. 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 [4, 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, 27] 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) in R 3 . We show that as Debye length goes to zero, the smooth solutions converges to solutions of the incompressible Navier-Stokes equation (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), 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 (14) 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 (4) for the leading terms and then we derive the remainder equation (12) and state the main result in Theorem 1.3.