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 uniform \(L_{loc}^{p}, p \ge 1, \) bound of the approximation solutions \( \rho _{i}^{ \varepsilon ,\delta } \) and other estimates of \( (u_{i}^{ \varepsilon ,\delta }, E^{ \varepsilon ,\delta })\) 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 \( \gamma _{i} > 1\).
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References
Degond, P., Markowich, P.A.: On a one-dimensional steady-state hydrodynamic model for semiconductors. Appl. Math. Lett. 3, 25–29 (1990)
DiPerna, R.J.: Convergence of the viscosity method for isentropic gas dynamics. Commun. Math. Phys. 91, 1–30 (1983)
Fang, W., Ito, K.: Weak solutions to a one-dimensional hydrodynamic model of two carrier types for semiconductors. Nonlinear Anal. TMA 28, 947–963 (1997)
Gasser, I., Natalini, R.: The energy transport and the drift diffusion equations as relaxation limits of the hydrodynamic model for semiconductors. Quart. Appl. Math. 57, 269–282 (1999)
Junca, S., Rascle, M.: Relaxation of the Isothermal Euler-Poisson System to the Drift-Diffusion Equations. Quart. Appl. Math. 58, 511–521 (2000)
Jungel, A., Peng, Y.J.: A hierarchy of hydrodynamic models for plasmas: zero-relaxation time-limits. Comm. Partial Differ. Equ. 58, 1007–1033 (1999)
Lax, P.D.: Hyperbolic systems of conservation laws II. Comm. Pure Appl. Math. 10, 537–566 (1957)
Lions, P.L., Perthame, B., Souganidis, P.E.: Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates. Comm. Pure Appl. Math. 49, 599–638 (1996)
Lions, P.L., Perthame, B., Tadmor, E.: Kinetic formulation of the isentropic gas dynamics and p-system. Commun. Math. Phys. 163, 415–431 (1994)
Lu, Y.-G.: Global existence of resonant isentropic gas dynamics. Nonlinear Anal. Real World Appl. 12, 2802–2810 (2011)
Lu, Y.-G.: Global solutions and Relaxation Limit to the Cauchy Problem of a Hydrodynamic Model for Semiconductors, preprint
Lu, Y.-G.: Hyperbolic Conservation Laws and the Compensated Compactness Method, vol. 128. Chapman and Hall, CRC Press, New York (2002)
Lu, Y.-G., Tsuge, N.: Uniformly time-independent \(L^{\infty }\) estimate for a one-dimensional hydrodynamic model of semiconductors, preprint
Marcati, P., Milani, A.: The one-dimensional Darcy’s law as the limit of a compressible Euler flow. J. Differ. Equ. 84, 129–147 (1990)
Marcati, P., Natalini, R.: Weak solutions to a hydrodynamic model for semconductors and relaxation to the drift-difusion equation. Arch. Ration. Mech. Anal. 129, 129–145 (1995)
Murat, F.: Compacité par compensation. Ann. Scuola Norm. Sup. Pisa 5, 489–507 (1978)
Natalini, R.: The bipolar hydrodynamic model for semiconductors and the drift-diffusion equations. J. Math. Anal. Appl. 198, 262–281 (1996)
Tadmor, E., Wei, D.: On the global regularity of sub-critical Euler–Poisson equations with pressure. J. Eur. Math. Soc. 10, 757–769 (2008)
Tartar, T.: Compensated compactness and applications to partial differential equations. In: Knops, R.J. (ed.) Research Notes in Mathematics, Nonlinear Analysis and Mechanics, Heriot-Watt symposium, vol. 4. Pitman Press, London (1979)
Wang, D.-H.: Global solutions to the Euler–Poisson equations of two-carrier types in one dimension. J. Appl. Math. Phys. (ZAMP) 48, 680–693 (1997)
Acknowledgements
The authors thank the anonymous reviewers for their valuable comments. The third author (Y. Lu) is supported by the NSFC grant No. LY20A010023, a Qianjiang professorship of Zhejiang Province of China and a Humboldt renewed research fellowship of Germany, who is very grateful to the colleagues in University of Wuerzburg for their warm hospitality.
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Communicated by Y. Giga.
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Hu, Yb., Klingenberg, C. & Lu, Yg. Zero relaxation time limits to a hydrodynamic model of two carrier types for semiconductors. Math. Ann. 382, 1031–1046 (2022). https://doi.org/10.1007/s00208-020-02071-9
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DOI: https://doi.org/10.1007/s00208-020-02071-9