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

Abstract

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.the full 3D quantum hydrodynamic model by a moment expansion of the Wigner-Boltzmann equation. Hsiao and Li [13] reviewed the recent progress on well-posedness, stability analysis, and small scaling limits for the (bi-polar) quantum hydrodynamic models, where the interested readers may find many useful references therein. Jungel [11] proved global existence of weak solutions for the isentropic case. See also [16][17][18].The other one, being equally important, emerges from the study of the compressible fluid models of Korteweg type, which are usually used to describe the motion of compressible fluids with capillarity effect of materials. See Korteweg [15] and the pioneering work of Dunn and Serrin [5]. The reference list can be very long, and we only mention a few of them. Hattori and Li [9,10] considered the local and global existence of smooth solutions for for the fluid model of Korteweg type for small initial data. Wang and Tan [23] studied the optimal decay for the compressible fluid model of Korteweg type. Recently, Bian, Yao and Zhu [3] studied the global existence of small smooth solutions and the vanishing capillarity limit of this model. Jungel et al [12] showed a combined incompressible and vanishing capillarity limit in the barotropic compressible Navier-Stokes equations for smooth solutions.Almost all of the above mentioned results considered the isothermal case, studying only the continuity equation and the momentum equation, or with electric potential described by a Poisson equation. To the best of our knowledge, there is no mathematical studies for the full quantum hydrodynamic system (1.1). The system (1.2) is itself interesting, since the energy equation also includes the quantum effects through the energy density W , which brings new features into this system. This makes it different from the previous known results, to be precisely stated in the following.The aim of this paper is two fold. On one hand, we show the global existence of smooth solutions for (1.2) with fixed constant > 0 when the initial data is small near the constant stationary solution (n, u, T ) = (1, 0, 1). To be precise, we denote the perturbation by (ρ, u, θ) = (n − 1, u, T − 1) and transform the (1.2) into (2.1). The result is then stated in Theorem 2.2 for (2.1), where the estimates is stated in terms of the planck constant , and we can see clearly how the quantum corrections affect the estimates. On the other hand, since (1.2) modifies the classical hydrodynamic equations to a macro-micro level in the sense that it incorporates the (micro) qu...

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