Large time behavior of entropy solutions to a unipolar hydrodynamic model of semiconductors

In this paper, we study the large time behavior of entropy solutions to the onedimensional unipolar hydrodynamic model for semiconductors in the form of Euler–Poisson equations. First of all, a large time behavior framework for the time-increasing entropy solutions is given. In this framework, the global entropy solutions (which increase slowly with time) are proved to decay exponentially fast to the corresponding stationary solutions. Then, for an application purpose, the existence and time-increasing-rate of the global entropy solutions with large initial data is considered by using a modified fractional step Lax–Friedrichs scheme and the theory of compensated compactness. By using the large time behavior framework, the global entropy solutions are proved to decay exponentially fast to the stationary solutions when the adiabatic index $\gamma \gt 3$, without any assumption on smallness or regularity for the initial data.

Keywords

compressible Euler equation, entropy solution, large time behavior

2010 Mathematics Subject Classification

35Mxx, 35Q35, 76W05

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Published 16 September 2015