# A Sharp Estimate of Entropy Solution to Euler-Poisson System for Semiconductors in the Whole Domain

## Main Article Content

## Abstract

In this paper, we are concerned with the global existence, large time behavior, and timeincreasing-rate of entropy solutions to the one-dimensional unipolar hydrodynamic model for semiconductors in the form of Euler-Poisson equations. When the adiabatic index γ > 2, the L∞ estimates of artificial viscosity approximate solutions are obtained by using entropy inequality and maximum principle. Then the L∞ compensated compactness framework demonstrates the

convergence of approximate solutions. Finally, the global entropy solutions are proved to decay exponentially fast to the stationary solution, without any assumption on the smallness of initial data and doping profile.

Keywords:

Global entropy solutions, Large time behavior, Entropy inequality, Maximum principle, Compactness framework

## Article Details

How to Cite

*Journal of Advances in Mathematics and Computer Science*,

*32*(2), 1-12. https://doi.org/10.9734/jamcs/2019/v32i230140

Section

Original Research Article

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## References

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of semiconductors. Osaka J. Math. 2007;44:639-665.

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diffusion and damping relaxation. J. Differential Equations. 1998;144:44-65.

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cauchy problem. Proc. Roy. Soc. Edinburgh Sect. A. 1995;125:115-131.

Li T. Convergence of the Lax-Friedrichs scheme for isothermal gas dynamics with

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J. Differential Equations. 2009;247:3070-3099.

Huang F, Pan R, Yu H. Large time behavior of Euler-Poisson system for semiconductor. J.

Science in China. 2008;51:965-972.

Yu H. Large time behavior of entropy solutions to a unipolar hydrodynamic model of

semiconductors. Commun. Math. Sci. 2016;14:69-82.

Fang X, Yu H. Uniform boundedness in weak solutions to a specific dissipative system. J.

Math. Anal. Appl. 2018;461:1153-1164.

Huang F, Li T, Yu H, Yuan D. Large time behavior of entropy solutions to one-dimensional

unipolar hydrodynamic model for semiconductor devices. Z. Angew. Math. Phys. 2018;69.

Daniel Y, Aziz Z, Ismail Z, Salah F. Entropy analysis in electrical magnetohydrodynamic

(MHD) flow of nanofluid with effects of thermal radiation, viscous dissipation, and chemical

reaction. Theoretical and Applied Mechanics Letters. 2017;7:235-42.

Daniel Y, Aziz Z, Ismail Z, Salah F. Numerical study of Entropy analysis for electrical unsteady

natural magnetohydrodynamic flow of nanofluid and heat transfer. Chinese Journal of Physics.

;55:1821-1848.

Li H, Markowich P, Mei M. Asymptotic behavior of subsonic entropy solutions of isentropic

Euler-Poisson equations. Quart. Appl. Math. 2002;60:773-796.

Huang F, Pan R. Convergence rate for compressible Euler Equations with damping and

vacuum, Arch. Rational Mech. Anal. 2003;166:359-376.

Di Perna R. Convergence of the viscosity method for isentropic gas dynamics. Commun.Math.

Phys. 1983;91:1-30.

Cao W, Huang F, Li T, Yu H. Global entropy solutions to an inhomogenous isentropic

compressible Euler system. Acta. Math. Sci. 2016;36:1215-1224.

Ding X, Chen G, Luo P. Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics

(I)-(II). Acta Math. Sci. 1985;5:415-472.

Lions P, Perthame B, Souganidis P. Existence and stability of entropy solutions for the

hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates.

Comm.Pure Appl. Math. 1996;49:599-638. Press, 2002.

Degond P, Markowich P. On a one-dimensional steady-state hydrodynamic model. Appl. Math.

Lett. 1990;3:25-29.

Aseher U, Markowich P, Pietra R, Schmeiser C. A phase plane analysis of transonic solutions

for the hydrodynamic semiconductor model. Math. Models Appl. Sci. 1991;1:347-976.

Degond P, Markowich P. A steady-state potential flow model for semiconductors. Ann. Mat.

Pura. Appl. 1993;IV:87-98.

Gamba I. Stationary transonic solutions of a one-dimensional hydrodynamic model for

semiconductor. Comm. Partial Differential Equations. 1992;17:553-577.

Luo T, Rauch J, Xie C, Xin Z. Stability of transonic shock solutions for one-dimensional

Euler-Poisson equations. Arch. Rational Mech. Anal. 2011;202:787-827.

Markowich P. On steady state Euler-Poisson model for semiconductors. Z. Ang. Math. Phys.

;62:389-407.

Peng Y, Violet I. Example of supersonic solutions to a steady state Euler-Poisson system.

Appl.Math.Lett. 2006;19:1335-1340.

Zhang B. On a local existence theroem for a simplified one-dimensional hydrodynamic model

for semiconductor devices. SIAM J. Math. Anal. 1994;25:941-947.

Guo Y, Strauss W. Stability of semiconductor states with insulating and contact boundary

conditions. Arch. Ration. Mech. Anal. 2006;179:1-30.

Hsiao L, Yang T. Asymptotics of initial boundary value problems for hydrodynamic and drift

diffusion models for semiconductors. J. Differential Equations. 2001;170:472-493.

Huang F, Mei M, Wang Y, Yu H. Asymptotic convergence to stationary waves for unipolar

hydrodynamic model of semiconductors. SIAM J. Math. Anal. 2011;43:411-429.

Huang F, Mei M, Wang Y, Yu H. Asymptotic convergence to planar stationary waves for

multi-dimensional unipolar hydrodynamic model of semiconductors. J. Differential Equations.

;251:1305-1331.

J¨ ungel A, Peng Y. A hierarchy of hydrodynamic models for plasmas. Zero-relaxation-time

limits. Comm. P.D.E. 1999;4:1007-1033.

Li H, Markowich P, Mei M. Asymptotic behavior of solutions of the hydrodynamic model of

semiconductors. Proc. Roy. Soc. Edinburgh. 2002;132A:359-378.

Luo T, Natalini R, Xin ZP. Large time behavior of the solutions to a hydrodynamic model for

semiconsuctors. SIAM J. Appl. Math. 1998;59:810-830.

Nishibata S, Suzuki M. Asymptotic stability of a stationary solution to a hydrodynamic model

of semiconductors. Osaka J. Math. 2007;44:639-665.

Wang D, Chen G. Formation of singularities in compressible Euler-Poisson fluids with heat

diffusion and damping relaxation. J. Differential Equations. 1998;144:44-65.

Zhang B. Convergence of the Godunov scheme for a simplified one-dimensional hydrodynamic

model for semiconductor devices, Comm. Math. Phys. 1993;157:1-22.

Marcati P, Natalini R. Weak solutions to a hydrodynamic model for semiconductors: The

cauchy problem. Proc. Roy. Soc. Edinburgh Sect. A. 1995;125:115-131.

Li T. Convergence of the Lax-Friedrichs scheme for isothermal gas dynamics with

semiconductor devices. Z. Angew. Math. Phys. 2006;57:1-20.

Huang F, Li T, Yu H. Weak solutions to isothermal hydrodynamic model for semiconductors.

J. Differential Equations. 2009;247:3070-3099.

Huang F, Pan R, Yu H. Large time behavior of Euler-Poisson system for semiconductor. J.

Science in China. 2008;51:965-972.

Yu H. Large time behavior of entropy solutions to a unipolar hydrodynamic model of

semiconductors. Commun. Math. Sci. 2016;14:69-82.

Fang X, Yu H. Uniform boundedness in weak solutions to a specific dissipative system. J.

Math. Anal. Appl. 2018;461:1153-1164.

Huang F, Li T, Yu H, Yuan D. Large time behavior of entropy solutions to one-dimensional

unipolar hydrodynamic model for semiconductor devices. Z. Angew. Math. Phys. 2018;69.

Daniel Y, Aziz Z, Ismail Z, Salah F. Entropy analysis in electrical magnetohydrodynamic

(MHD) flow of nanofluid with effects of thermal radiation, viscous dissipation, and chemical

reaction. Theoretical and Applied Mechanics Letters. 2017;7:235-42.

Daniel Y, Aziz Z, Ismail Z, Salah F. Numerical study of Entropy analysis for electrical unsteady

natural magnetohydrodynamic flow of nanofluid and heat transfer. Chinese Journal of Physics.

;55:1821-1848.

Li H, Markowich P, Mei M. Asymptotic behavior of subsonic entropy solutions of isentropic

Euler-Poisson equations. Quart. Appl. Math. 2002;60:773-796.

Huang F, Pan R. Convergence rate for compressible Euler Equations with damping and

vacuum, Arch. Rational Mech. Anal. 2003;166:359-376.

Di Perna R. Convergence of the viscosity method for isentropic gas dynamics. Commun.Math.

Phys. 1983;91:1-30.

Cao W, Huang F, Li T, Yu H. Global entropy solutions to an inhomogenous isentropic

compressible Euler system. Acta. Math. Sci. 2016;36:1215-1224.

Ding X, Chen G, Luo P. Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics

(I)-(II). Acta Math. Sci. 1985;5:415-472.

Lions P, Perthame B, Souganidis P. Existence and stability of entropy solutions for the

hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates.

Comm.Pure Appl. Math. 1996;49:599-638. Press, 2002.