Subsonic Solutions to a One-Dimensional Non-isentropic Hydrodynamic Model for Semiconductors

Abstract

The one-dimensional stationary full hydrodynamic model for semiconductor devices with non-isentropic pressure is studied. This model consists of the equations for the electron density, electron temperature, and electric field in a bounded domain supplemented with boundary conditions. The existence of a classical subsonic solution with positive particle density and positive temperature is shown in two situations: non-constant and constant heat conductivities. Moreover, we prove uniqueness of a classical solution in the latter case. The existence proofs are based on elliptic estimates, Stampacchia truncation methods, and fixed-point arguments.

Asymptotic profile in a one-dimensional steady-state nonisentropic hydrodynamic model for semiconductors

“…ε ) and (n (2) ε , T (2) ε , φ (2) ε ) be two strong solutions for (2.1) and (2.2)-(2.3), satisfying (2.4)-(2.5). Then taking the difference of Eq.…”

“…Then taking the difference of Eq. (2.1) 3 satisfying by (n (1) ε , T (1) ε ) and (n (2) ε , T (2) ε ), respectively, and using T (1) ε − T (2) ε as the test function in the weak formulation of the difference equation, we obtain…”

“…Let (n (1) ε , T (1) ε , φ (1) ε ) and (n (2) ε , T (2) ε , φ (2) ε ) be two solutions in H 2 (0, 1) × H 2 (0, 1) × H 2 (0, 1) to the problem (2.1)-(2.3) and satisfy the uniform (2.4)-(2.5) with respect to ε. Then (n (1) ε , T (1) ε , φ (1) ε ) = (n (2) ε , T (2) ε , φ (2) ε ) provided that 0 < j 1 holds.…”

“…To derive an estimate for n (1) ε − n (2) ε in L 2 -norm, we take the differences of equations (2.1) 2,3 for (n (1) ε , T (1) ε , φ (1) ε ) and (n (2) ε , T (2) ε , φ (2) ε ), respectively, further, using (n (1) ε − n (2) ε )(x) as the test function in the weak formulations of the corresponding difference equations, similar to (2.27), we have…”

“…Since a mathematical analysis for the simplified steady-state hydrodynamic model was introduced by Degond and Markowich [7], the steady-state solution for the one-and multi-dimensional simplified hydrodynamic semiconductor model was proved in the subsonic case [8,9,25], while the corresponding transonic steady-state solution only for the one-dimensional case was shown by means of the phase plane analysis [3] and the vanishing viscosity method [11], and an approximation transonic steady-state solution for the two-dimensional hydrodynamic semiconductor model was discussed in [12]. Regarding the nonisentropic hydrodynamic model with constant lattice temperature and the general energy transport equations, Amster [2] discussed the onedimensional subsonic solutions for the general pressure p(n, T ) satisfying ∂ n p(n, T ) large enough and the boundary value of T closed to the ambient temperature, and Yeh [31] showed the existence of a unique strong solution in several space dimensions if the flow is subsonic, the ambient temperature T L is large enough, and the vorticity on the inflow boundary and the variation of the electron density on the boundary are sufficiently small. In particular, we just studied the steady-state solution for the one-dimensional nonisentropic hydrodynamic model for semiconductors under the general assumptions in [21].…”