On the classical solutions to the hydrodynamic model for semiconductors

This paper mainly deals with the multidimensional hydrodynamic model for semiconductors. Inspired by the previous papers [Y. Shizuta, S. Kawashima, Systems of equations of hyperbolic–parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J. 14 (1985) 249–275; S. Kawashima, W.-A. Yong, Dissipative structure and entropy for hyperbolic systems of balance laws, Arch. Ration. Mech. Anal. 174 (2004) 345–364; W.-A. Yong, Entropy and global existence for hyperbolic balance laws, Arch. Ration. Mech. Anal. 172 (2004) 247–266], we develop some new frequency-localization estimates to establish the global existence and exponential stability of (small) classical solutions in a class of critical Besov spaces, which are different from estimates in our recent paper [D.Y. Fang, J. Xu, T. Zhang, Global exponential stability of classical solutions to the hydrodynamic model for semiconductors, Math. Models Methods Appl. Sci. 17 (2007) 1507–1530]. Furthermore, this new method can also be applied to the multidimensional Euler equations with damping. The analytic tool used is the Littlewood–Paley decomposition.