Global existence and semiclassical limit for quantum hydrodynamic equations with viscosity and heat conduction

The hydrodynamic equations with quantum effects are studied in this paper. First we establish the global existence of smooth solutions with small initial data and then in the second part, we establish the convergence of the solutions of the quantum hydrodynamic equations to those of the classical hydrodynamic equations. The energy equation is considered in this paper, which added new difficulties to the energy estimates, especially to the selection of the appropriate Sobolev spaces.


Introduction
The hydrodynamic equations and related models with quantum effects are extensively studied in recent two decades. In these models, the quantum effects is included into the classical hydrodynamic equations by incorporating the first quantum corrections of O( 2 ), where is the Planck constant. One of the main applications of the quantum hydrodynamic equations is as a simplified but not a simplistic approach for quantum plasmas. In particular, the nonlinear aspects of quantum plasmas of quantum plasmas are much more accessible using a fluid description, in comparasion with kinetic theory. One may see the recent monograph of Haas [8] for many physics backgrounds and mathematical derivation of many interesting models. Many other applications of the quantum hydrodynamic equations consisting of analyzing the flow the electrons in quantum semiconductor devices in nano-size [7], where quantum effects like particle tunnelling through potential barriers and built-up in quantum wells, can not be simulated by classical hydrodynamic model. Similar macroscopic quantum models are also used in many other physical fields such as superfluid and superconductivity [6].
Let us first consider the following classical hydrodynamic equations in conservation form, describing the motion of the electrons in plasmas by omitting the electric potential where n is the density, m is the effective electron mass, u = (u 1 , u 2 , u 3 ) is the velocity, Π j is the momentum density, P ij is the stress tensor, W is the energy density and q is the heat flux. In this system, repeated indices are summed over under the Einstein convention.
2000 Mathematics Subject Classification. 35M20; 35Q35. This work is supported in part by NSFC (11471057) and Natural Science Foundation Project of CQ CSTC (cstc2014jcyjA50020).
This system also emerges from descriptions of the motion of the electrons in semiconductor devices, with the electrical potential and the relaxation omitted. As in the classical hydrodynamic equations, the quantum conservation laws have the same form as their classical counterparts. However, to close the moment expansion at the third order, we define the above quantities Π i , P ij and W in terms of the density n, the velocity u and the temperature T . As usual, the heat flux is assumed to obey the Fourier law q = −κ∇T and the momentum density is defined by Π i = mnu i , where m is the electron mass and u the velocity. The symmetric stress tensor P ij and the energy density W are defined, with quantum corrections, by respectively, where is the Planck constant, and is very small compared to macro quantities. As far as the quantum corrections are concerned, the quantum correction to the energy density was first derived by Wigner [24] for thermodynamic equilibrium, and the quantum correction to the stress tensor was proposed by Ancona and Tiersten [2] and Ancona and Iafrate [1] on the Wigner formalism. See also [7] for derivation of the system (1.1) by a moment expansion of the Wigner-Boltzmann equation and an expansion of the thermal equilibrium Wigner distribution function to O( 2 ), leading to the expression for Π and W above. We also remark the quantum correction term is closely related to the quantum Bohm potential [4] where n is the charge density. It relates to the quantum correction term in P ij with For the system (1.1), there is no dissipation in the second equation. Given n and T , the second equation if hyperbolic, and generally we can not expect global smooth solutions for this system. In this paper, we consider the following viscous system by taking into account the stress tensor S, Here, S is the stress tensor defined by where I is the d × d identity matrix, µ > 0 and λ are the primary coefficients of viscosity and the second coefficients of viscosity, respectively, satisfying 2µ + 3λ > 0. Without quantum corrections (i.e., setting = 0), this system is exactly the classical hydrodynamic equations studied in the seminal paper of Matsumura and Nishida [19]. Although important, there is little result on the system (1.2) to the best of our knowledge. But there does exist a large amount of work for system very similar to (1.2). These work comes from two main origins. The first one is from the quantum correction to various hydrodynamic equations, especially in semiconductors and in plasmas. Gardner [7] derived belong to L 2 with norm · H k = ( k j=0 D j · 2 ) 1/2 .Ḣ s,p denotes the homogeneous Sobolev spaces and [A, B] = AB − BA denotes the commutator of A and B.

Preliminaries and Main Results
In this section, we reformulate the system (1.2) in convenient variables. First we take (n, u, T ) = (1, 0, 1) to be a constant solution to (1.2) and consider In these unknowns, with m = 1, the (1.2) transforms into We first state the local-in-time existence of smooth solutions to (2.1). To be precise, we first set |||(ρ, u, θ)||| 2 0 := (ρ, u, θ) 2 Theorem 2.1 (Local existence). For any initial data such that n 0 ≥ δ > 0 is satisfied and This theorem can be proved in a similar fashion as in [9] by the dual argument and iteration techniques, and hence omitted for brevity. Now, we consider the global existence of smooth solutions. Let T > 0, we set One of the main purpose is to show the following and set E 0 := |||(ρ, u, θ)(0)||| 3 < ∞. There exists some 0 > 0, ε 0 > 0, ν 0 > 0 and C 0 < ∞, such that if E 0 < ε 0 and < 0 , then there exists a unique global in time solution (ρ, u, θ) ∈ E 3 (0, T ) of the Cauchy problem (2.1) for any t ∈ (0, ∞), and the following estimates hold where the constants ν 0 > 0 and C 0 > 0 are independent of time t and .
The following three lemmas will be frequently used, and hence cited here for reader's convenience.

A priori estimates
In this section, we establish useful a priori estimates of the solutions to (2.1). First of all, we let the Planck constant < 1. To simplify the proof slightly, we assume that there exists a positive number ε ≪ 1 such that which together with Sobolev embedding, implies that and from (2.1) the following and In particular, we choose ε small enough such that 3.1. Basic estimates. Now, we consider the zeroth order estimates for the system (2.1). As in [19], we set s = (1 + θ)/(1 + ρ) 2/3 − 1, (3.6) and define a function E 0 (ρ, u, s) for ρ, u = (u 1 , u 2 , u 3 ) and s by The following lemma is proved in [19].
We first prove the zeroth order estimates in the following Proposition 3.1. There exists a constant ε 0 > 0 such that if E ≤ ε ≤ ε 0 , then the following a priori estimates holds for all t ∈ [0, T ], The proof if postponed to the end of Section 3.1.
for some constant C > 0 independent of t.
Proof. Under the transform of (3.6), the system (2.1) is transformed into the following system for (ρ, u, s) ∇ρ ∇ρ Recall that E 0 (ρ, u, s) is given in (3.7). We compute Now, we consider the integration in space of the last two terms I and II on the RHS of (3.11). For the first term I, by integration by parts, and using (3.10a), we obtain The last term on the RHS is easy to be bounded by 2 12 For the first term on the RHS, we use (2.1a) to obtain But from (2.1a), it is easy to know that and by integration by parts, Therefore it is easy to see from (3.12) that On the other hand, for the second term II in (3.11), we have In addition to (3.11), we compute ∂ ∂t After integration in space we obtain for the last term, For the term J 1 , we have For the term J 2 , we have Multiplying (3.13) with a constant β and integration in space, and then add the resultant to (3.11) integrated in space, we obtain (3.14) Note that as in [19], if we take β small such that where ρ 2 is given in Lemma 3.1, then and Integrating in time over [0, t] and taking δ 0 and E sufficiently small (say, δ 0 = 1/20), we obtain   for any positive constants δ 1 > 0 and t > 0.
Proof. We now take the inner product of (2.1b) with − 2 ∆u to obtain For L 3 , we have by integration by parts twice For R 1 , after integration by parts twice, we obtain which implies that For R 2 and R 3 , by Hölder inequality we obtain For R 4 , we have by integration by parts and (2.1a) It is easy to show the following estimates thanks to (3.3) and thanks to Lemma 2.6, and by integration by parts Therefore, we obtain For the term R 5 , it is easy to show that Hence, putting all the estimates together, we have from (3.17) that Take ε 0 and 0 small, then for any ε ≤ ε 0 and ≤ 0 , integration in time over [0, t] yields the result for any positive constant δ 1 > 0.
This proposition is proved as a direct sequence of the following lemmas.
Proof. Applying ∂ α to (2.1c) and then taking inner product of the resultant with ∂ α θ to obtain For the first term R 1 , we have For the term R 2 , we have thanks to Lemma 2.6. For the term R 3 , since |θ + 1| ≤ 3/2 by (3.3), we have For the term R 4 , by integration by parts, where s = |α|. Then making use of Lemma 2.4, 2.5 and 2.6 and (3.1)-(3.5), one obtains Similar to R 4 , we have for the term R 5 that Putting these estimates together, we obtain Integrating this inequality in time over [0, t] and noting 1/2 < 1 + ρ < 3/2, we know that there exists some constants δ 0 < 1 and ε 0 < 1 sufficiently small, such that (3.22) holds for all δ ≤ δ 0 and E ≤ ε ≤ ε 0 .
Lemma 3.5. Under the assumptions in Proposition 3.2, there exists some constant ε 0 < 1 sufficiently small and 0 < 1, such that for all E ≤ ε ≤ ε 0 and < 0 , where C is independent of t.
Proof. Applying ∂ α to (2.1b) and then taking inner product of the resultant with ∂ α u, we obtain Now, for the term L 2 , we have by integration by parts Invoking Lemma 2.5 and 2.6, we obtain Hence µ). Similarly, we have for L 3 that For the RHS term R 1 , we have by integration by parts that For the term R 2 , we have for any δ 0 > 0 that The term R 3 will be treated with much more effort, from which some good terms will appear. By integration by parts, It is easy to show that s−1 , thanks to Lemma 2.5. Differentiating the continuity equation (2.1a) with ∂ α yields which implies that

It is immediately from (3.3) and (3.4) with p = 3/2 that
For the term R 312 , we have by integration by parts that The same estimate hold for R 313 . Combining all the estimates for R 3 , we obtain Now, we consider the estimate of R 4 . By integration by parts, we obtain Using the continuity equation (2.1a) and similar to the term R 31 , it can be shown that For R 411 , we obtain By integration by parts, and hence by Lemma 2.6 and (3.2) Similarly, by Lemma 2.6 and (3.2), For the term R 42 , we have But by the commutator estimates, we have s . Similarly, for R 43 , we obtain s . Putting all the estimates for R 4 together, we obtain Finally, for R 5 , it is easy to show Now, putting all these estimates for (3.24) together, and taking δ 0 = 1/4, we obtain, Integrating in time over [0, t] completes the proof, thanks to < 1.
Lemma 3.6. Under the assumptions in Proposition 3.2, there exists some constant ε 0 < 1 sufficiently small and 0 < 1, such that for all E ≤ ε ≤ ε 0 and < 0 , where C is independent of t.
Proof. Applying ∂ α to (2.1b) and then taking inner product of the resultant with − 2 ∆∂ α u, we obtain Now, for the term L 2 , we have by integration by parts thanks to Lemma 2.5 and (2.6), we have Similarly, for the L 3 , we have For the RHS term R 1 , we have by integration by parts twice that By integration by parts, It is easy to show that and by Lemma 2.5 and 2.6 ≤C 2 E ∇(ρ, u, θ) 2Ḣ s . Differentiating the continuity equation (2.1a) with ∂ α and then inserting the resultant to R 31 , we obtain It is immediately that thanks to (3.3) and (3.4) again. For the term R 312 , we have by integration by parts that The same estimate hold for R 313 . Now, we consider the estimate of R 4 . By integration by parts, we obtain For the last two terms, it can be shown that Using the continuity equation (2.1a) and similar to the term R 31 , it can be shown that Finally, for R 5 , we have Now, putting all these estimates together, and taking δ 0 = 1/4, we obtain, Integrating in time over [0, t] completes the proof.
Lemma 3.7. Under the assumptions in Proposition 3.2, there exists some constant ε 0 < 1 sufficiently small and 0 < 1, such that for any β > 0, E ≤ ε ≤ ε 0 and ≤ 0 , where C depends only on µ, λ and some Sobolev constants. In particular, C does not depend on > 0 or t > 0.
Proof. Let γ be a multi-index such that |γ| = |α| − 1 and γ ≤ α. We apply ∂ γ to (2.1b) and then take the inner product of the resultant with −β∂ γ ∇∆ρ to obtain We first note that by Hölder inequality and by integration by parts and Hölder inequality For the term L 8 , we have For the term L 5 , we have by integration by parts For the term L 4 , we have by integration by parts For the term L 3 , we have by integration by parts, ]∇divu∂ γ ∆ρ =: L 31 + L 32 + L 33 .