## Weak solutions to a hydrodynamic model for semiconductors: The Cauchy problem.(English)Zbl 0831.35157

The paper is concerned with the following system of PDEs (hydrodynamic model for a one-dimensional semiconductor device): $(1)\;n_t + (nu)_x = 0, \quad (2)\;(nu)_t + \bigl( nu^2 + p(n) \bigr)_x = nE - nu, \quad (3)\;E_x = n - b$ in the region $$\mathbb{R} \times [0,T)$$, where $$n$$ denotes the electron density, $$u$$ the average particle velocity, $$E$$ the (negative) electric field and $$b$$ the doping profile; $$p = p(n)$$ is a given pressure function. System (1)–(3) is completed by initial conditions on $$n$$ and $$u$$.
The aim of the paper is to prove the existence of locally bounded weak solutions to (1)–(3), where $$n$$, $$Ju = nu$$ (= electron current density) and $$E$$ are the dependent variables. The authors construct approximate solutions by using two finite difference schemes: the fractional step Lax-Friedrichs scheme and the Godounov scheme. Then supremum-norm bounds and energy-type estimates on the approximate solutions are established. The passage to the limit is then carried out by the aid of compensated compactness techniques.
Reviewer: J.Naumann (Berlin)

### MSC:

 35Q60 PDEs in connection with optics and electromagnetic theory 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35Q35 PDEs in connection with fluid mechanics 35D05 Existence of generalized solutions of PDE (MSC2000) 78A35 Motion of charged particles
Full Text:

### References:

 [1] DOI: 10.1080/03605308808820544 · Zbl 0653.35057 [2] DOI: 10.1002/cpa.3160330502 · Zbl 0424.35057 [3] Lax, Conf. Board Math. Sci. 11 (1973) [4] DOI: 10.1007/BF01218624 · Zbl 0689.76022 [5] Ding, Acta Math. Sci. 7 pp 467– (1987) [6] Ding, Acta Math. Sci. 5 pp 483– (1985) [7] DOI: 10.1016/0893-9659(90)90130-4 · Zbl 0736.35129 [8] Chen, The theory of compensated compactness and the system of isentropic gas dynamics (1990) [9] Tartar, Systems of Nonlinear Partial Differential Equations (1983) [10] Chen, Acta Math. Sci. 6 pp 75– (1986) [11] Tartar, Research Notes in Mathematics, Nonlinear Analysis and Mechanics: Heriot–Watt Symposium 4 pp 136– (1979) [12] DOI: 10.1142/S0218202591000174 · Zbl 0800.76032 [13] Smoller, Shock Waves and Reaction-Diffusion Equations (1983) · Zbl 0508.35002 [14] Serre, J. Math. Pures Appl. 65 pp 432– (1986) [15] Lax, Contributions to Nonlinear Functional Analysis pp 603– (1971) [16] DOI: 10.1002/cpa.3160070112 · Zbl 0055.19404 [17] DOI: 10.1070/SM1970v010n02ABEH002156 · Zbl 0215.16203 [18] Hsiao, Ann. Scuola Norm. Sup. Pisa Cl. Sci. IV 15 pp 65– (1988) [19] Godounov, Math. Sb. 47 pp 271– (1959) [20] DOI: 10.1063/1.858321 · Zbl 0748.76066 [21] Gamba, Comm. Partial Differential Equations 17 pp 553– (1992) [22] DOI: 10.1109/43.68410 · Zbl 05447997 [23] DOI: 10.1007/BF01206047 · Zbl 0533.76071 [24] DOI: 10.1007/BF00251724 · Zbl 0519.35054 [25] Rubino, Ann. Inst. H. Poincaré Anal. Non Linéaire 10 pp 627– (1993) [26] Murat, J. Math. Pures Appl. 60 pp 309– (1981) [27] DOI: 10.1002/cpa.3160440818 · Zbl 0763.35056 [28] DOI: 10.1007/978-3-7091-6961-2 [29] DOI: 10.1002/mma.1670160603 · Zbl 0771.76075 [30] Markowich, The Steady-State Semiconductor Device Equations (1986) [31] DOI: 10.1016/0022-247X(80)90159-6 · Zbl 0457.35058 [32] DOI: 10.1512/iumj.1982.31.31039 · Zbl 0497.35058 [33] DOI: 10.2307/2154093 · Zbl 0761.35061
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.