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.
Highlights
The hydrodynamic equations and related models with quantum effects are extensively studied in recent two decades
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 may see the recent monograph of Haas [8] for many physics backgrounds and mathematical derivation of many interesting models
Summary
The hydrodynamic equations and related models with quantum effects are extensively studied in recent two decades. Since (1.2) modifies the classical hydrodynamic equations to a macro-micro level in the sense that it incorporates the (micro) quantum corrections, it is expected that as the Planck constant → 0, the solution of the system (1.2) converges to that of the classical hydrodynamic equations. This limit is rigorously studied in this paper and stated in Theorem 2.3.
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