Zero relaxation time limits to a hydrodynamic model of two carrier types for semiconductors
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Mathematische Annalen
Zero relaxation time limits to a hydrodynamic model of two carrier types for semiconductors Yan-bo Hu1 · C. Klingenberg2 · Yun-guang Lu1 Received: 15 January 2020 / Revised: 6 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we study the zero relaxation time limits to a one dimensional hydrodynamic model of two carrier types for semiconductors. First, we introduce the flux approximation coupled with the classical viscosity method to obtain the unip form L loc , p ≥ 1, bound of the approximation solutions ρiε,δ and other estimates ε,δ of (u i , E ε,δ ) with the help of the high energy estimates (Jungel and Peng Comm Partial Differ Equ 58:1007–1033, 1999). Then, we apply the compensated compactness method coupled with the scaled variables technique (Marcati and Natalini Arch Ration Mech Anal 129:129–145, 1995) to prove the zero-relaxation-time limits with arbitrarily large initial data, and arbitrary adiabatic exponents γi > 1. Mathematics Subject Classification 35L65 · 76N10 · 65M12 · 78A35
1 Introduction In this paper, we study the zero relaxation time limits to the following one-dimensional hydrodynamic model of two carrier types for semiconductors ⎧ ρit + (ρi u i )x = 0, ⎪ ⎪ ⎪ ⎪ ⎨ (ρi u i )t + (ρi (u i )2 + Pi (ρi ))x = ρi E − ⎪ ⎪ ⎪ ⎪ ⎩ E x = ρ1 + ρ2 − b(x),
ai (x)ρi u i τi
, i = 1, 2,
(1.1)
Communicated by Y. Giga.
B
Yun-guang Lu [email protected]
1
K.K.Chen Inst. for Advanced Studies, Hangzhou Normal University, Hangzhou, China
2
Deptartment of Mathematics, Wuerzburg University, Würzburg, Germany
123
Y.-b. Hu et al.
in the region (−∞, +∞) × [0, ∞], with bounded initial data (ρi , u i )|t=0 = (ρi0 (x), u i0 (x)),
lim (ρi0 (x), u i0 (x)) = (0, 0), ρi0 (x) ≥ 0
|x|→∞
(1.2) and a condition at −∞ for the electric potential lim E(x, t) = E 0 ,
x→−∞
for a.e. t ∈ (0, ∞),
(1.3)
where E 0 is a fixed constant, (ρ1 , u 1 ) and (ρ2 , u 2 ) are the (density, velocity) pairs for electrons (i = 1) and holes (i = 2) respectively, E is the electric potential and the given function b(x) represents the impurity doping profile, and ai (x) ≥ 0 are damping coefficients (cf. [1,3,6,17,20] and the references cited therein). The pressure-density relations are Pi (ρi ) = γ1i (ρi )γi , where γi > 1 correspond to the adiabatic exponents, τi > 0 are the momentum relaxation times. When damping coefficients ai (x) = 1, the global existence of entropy solutions of the initial-boundary value problem of (1.1) was first studied by using the viscosity method [3] and the Godunov scheme method [20]), respectively, where the adiabatic exponents γi are limited in the region (1, 53 ] to ensure the uniform L ∞ estimates of the approximation solutions. The global solutions of the Cauchy problem of (1.1) was obtained in [6,17], where the approximation solutions were constructed by the Lax-Friedrichs scheme and the Godunov scheme. Due to the lack of a technique to obtain the a-priori L ∞ estimate, it is a long-standing open problem to study the Cauchy problem of
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