Relaxation-time limit and initial layer in the isentropic hydrodynamic model for semiconductors

This work is concerned with the relaxation-time limit in the multidimensional isentropic hydrodynamic model for semiconductors in the critical Besov space. Firstly, we construct formal approximations of the initial layer solution to the nonlinear problem by the matched expansion method. Then, assuming some regularity of the solution to the reduced problem, and proves the existence of classical solutions in the uniform time interval where the reduced problem has a smooth solution and justify the validity of the formal approximations in any fixed compact subset of the uniform time interval.