Steady Hydrodynamic Model of Semiconductors with Sonic Boundary: (II) Supersonic Doping Profile

In this paper, we study the well-posedness/ill-posedness and regularity of stationary solutions to the hydrodynamic model of semiconductors represented by Euler-Poisson equations with sonic boundary. When the doping profile is subsonic, we prove that, the steady-state equations with sonic boundary possess a unique interior subsonic solution, and at least one interior supersonic solution, and if the relaxation time is large and the doping profile is a small perturbation of constant, then the equations admit infinitely many transonic shock solutions, while, if the relaxation time is small enough and the doping profile is a subsonic constant, then the equations admits infinitely many $C^1$ smooth transonic solutions, and no transonic shock solution exists. When the doping profile is supersonic, we show that the system does not hold any subsonic solution, furthermore, the system doesn't admit any supersonic solution or any transonic solution if such a supersonic doping profile is small or the relaxation time is small, but it has at least one supersonic solution and infinitely many transonic solutions if the supersonic doping profile is close to the sonic line and the relaxation time is large. The interior subsonic/supersonic solutions all are global $C^{\frac{1}{2}}$ H\"older-continuous, and the exponent $\frac{1}{2}$ is optimal. The non-existence of any type solutions in the case of small doping profile or small relaxation time indicates that the semiconductor effect for the system is remarkable and cannot be ignored. The proof for the existence of subsonic/supersonic solutions is the technical compactness analysis combining the energy method and the phase-plane analysis, while the approach for the existence of multiple transonic solutions is artfully constructed. The results obtained significantly improve and develop the existing studies.

[1]  Yue-Jun Peng,et al.  Example of supersonic solutions to a steady state Euler-Poisson system , 2006, Appl. Math. Lett..

[2]  Lipman Bers,et al.  Existence and uniqueness of a subsonic flow past a given profile , 1954 .

[3]  K. Blotekjaer.  Transport equations for electrons in two-valley semiconductors , 1970 .

[4]  Pierre Degond,et al.  On a one-dimensional steady-state hydrodynamic model , 1990 .

[5]  Tian-Yi Wang,et al.  Subsonic-Sonic Limit of Approximate Solutions to Multidimensional Steady Euler Equations , 2013, 1311.3985.

[6]  Yong Wang,et al.  Asymptotic Convergence to Stationary Waves for Unipolar Hydrodynamic Model of Semiconductors , 2011, SIAM J. Math. Anal..

[7]  Constantine M. Dafermos,et al.  On Two-Dimensional Sonic-Subsonic Flow , 2007 .

[8]  Chunjing Xie,et al.  Subsonic Flow for the Multidimensional Euler–Poisson System , 2012, 1211.5234.

[9]  Zhouping Xin,et al.  On Sonic Curves of Smooth Subsonic-Sonic and Transonic Flows , 2016, SIAM J. Math. Anal..

[10]  Shinya Nishibata,et al.  Asymptotic stability of a stationary solution to a hydrodynamic model of semiconductors , 2007 .

[11]  C. Schmeiser,et al.  Semiconductor equations , 1990 .

[12]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[13]  Chunpeng Wang,et al.  Continuous subsonic-sonic flows in a general nozzle , 2015 .

[14]  Pierre Degond,et al.  A steady state potential flow model for semiconductors , 1993 .

[15]  Kazufumi Ito,et al.  Steady-State Solutions of a One-Dimensional Hydrodynamic Model for Semiconductors , 1997 .

[16]  Chunjing Xie,et al.  Subsonic Solutions for Steady Euler-Poisson System in Two-Dimensional Nozzles , 2013, SIAM J. Math. Anal..

[17]  Chunjing Xie,et al.  Global Subsonic and Subsonic-Sonic Flows through Infinitely Long Axially Symmetric Nozzles , 2009, 0907.3274.

[18]  Zhouping Xin,et al.  On a Degenerate Free Boundary Problem and Continuous Subsonic–Sonic Flows in a Convergent Nozzle , 2013 .

[19]  Yan Guo,et al.  Stability of Semiconductor States with Insulating and Contact Boundary Conditions , 2006 .

[20]  Chunjing Xie,et al.  Stability of Transonic Shock Solutions for One-Dimensional Euler–Poisson Equations , 2010, 1008.0378.

[21]  Kung-Ching Chang.  Methods in nonlinear analysis , 2005 .

[22]  Yong Wang,et al.  Asymptotic convergence to planar stationary waves for multi-dimensional unipolar hydrodynamic model of semiconductors , 2011 .

[23]  Irene M. Gamba.  Stationary transonic solutions of a one—dimensional hydrodynamic model for semiconductors , 1992 .

[24]  Massimiliano Daniele Rosini,et al.  A phase analysis of transonic solutions for the hydrodynamic semiconductor model , 2005 .

[25]  Uri M. Ascher,et al.  A PHASE PLANE ANALYSIS OF TRANSONIC SOLUTIONS FOR THE HYDRODYNAMIC SEMICONDUCTOR MODEL , 1991 .

[26]  Ansgar Jüngel,et al.  Quasi-hydrodynamic Semiconductor Equations , 2001 .

[27]  Irene M. Gamba,et al.  A viscous approximation for a 2‐D steady semiconductor or transonic gas dynamic flow: Existence theorem for potential flow , 1996 .

[28]  Zhouping Xin,et al.  Smooth Transonic Flows in De Laval Nozzles , 2013, 1304.2473.

[29]  Ming Mei,et al.  Asymptotic behaviour of solutions of the hydrodynamic model of semiconductors , 2002, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[30]  Zhouping Xin,et al.  Transonic shock solutions for a system of Euler-Poisson equations , 2012 .

[31]  Joseph W. Jerome,et al.  Steady Euler–Poisson systems: A differential/integral equation formulation with general constitutive relations☆ , 2009 .