Subsonic Solutions for Steady Euler-Poisson System in Two-Dimensional Nozzles

In this paper, we prove the existence and stability of subsonic flows for a steady full Euler--Poisson system in a two-dimensional nozzle of finite length when imposing the electric potential difference on a noninsulated boundary from a fixed point at the entrance, and prescribing pressure at the exit of the nozzle. The Euler--Poisson system for subsonic flow is a hyperbolic-elliptic coupled nonlinear system. One of the crucial ingredients of this work is the combination of Helmholtz decomposition for the velocity field and stream function formulation. In terms of the Helmholtz decomposition, the Euler--Poisson system is rewritten as a second order nonlinear elliptic system of three equations and transport equations for entropy and pseudo-Bernoulli's invariant. The associated elliptic system in a Lipschitz domain with nonlinear boundary conditions is solved with the help of the estimates developed in [M. Bae, B. Duan, and C. J. Xie, Existence and Stability of Multidimensional Steady Potential Flows for Eu...

[1]  M. Rosini.  Stability of transonic strong shock waves for the one-dimensional hydrodynamic model for semiconductors , 2004 .

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

[3]  G. M. Lieberman.  Hölder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions , 1987 .

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

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

[6]  Mikhail Feldman,et al.  Transonic Shocks in Multidimensional Divergent Nozzles , 2010, 1003.4335.

[7]  R. Eisenberg,et al.  Hydrodynamic model of temperature change in open ionic channels. , 1995, Biophysical journal.

[8]  Ming Mei,et al.  Asymptotic behavior of subsonic entropy solutions of the isentropic Euler-Poisson equations , 2002 .

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

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

[11]  S. Weng.  A new formulation for the 3-D Euler equations with an application to subsonic flows in a cylinder , 2012, 1212.1635.

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

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

[14]  On a steady state Euler-Poisson model for semiconductors , 1996 .

[15]  Yan Guo,et al.  Smooth Irrotational Flows in the Large to the Euler–Poisson System in R3+1 , 1998 .

[16]  Z. Xin,et al.  Subsonic Irrotational Flows in a Finitely Long Nozzle with Variable end Pressure , 2014 .

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

[18]  Chunjing Xie,et al.  Existence of Global Steady Subsonic Euler Flows Through Infinitely Long Nozzles , 2009, SIAM J. Math. Anal..

[19]  Shangkun Weng,et al.  On Steady Subsonic Flows for Euler-Poisson Models , 2012, SIAM J. Math. Anal..

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

[21]  Gui-Qiang G. Chen,et al.  Formation of Singularities in Compressible Euler-Poisson Fluids with Heat Diffusion and Damping Relaxation , 1998 .

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

[23]  Bo Zhang,et al.  Convergence of the Godunov scheme for a simplified one-dimensional hydrodynamic model for semiconductor devices , 1993 .

[24]  Chunjing Xie,et al.  Three dimensional steady subsonic Euler flows in bounded nozzles , 2013, 1305.2242.

[25]  Zhouping Xin,et al.  Large Time Behavior of the Solutions to a Hydrodynamic Model for Semiconductors , 1998, SIAM J. Appl. Math..

[26]  Huimin Yu,et al.  Large time behavior of Euler-Poisson system for semiconductor , 2008 .

[27]  Jun Li.  ON TRANSONIC SHOCKS IN A NOZZLE WITH VARIABLE END PRESSURES , 2011 .

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

[29]  Asymptotic Behavior of the Solution to a Nonisentropic Hydrodynamic Model of Semiconductors , 1998 .

[30]  Gui-Qiang G. Chen,et al.  Convergence of shock capturing schemes for the compressible Euler-Poisson equations , 1996 .

[31]  Peter A. Markowich,et al.  On steady state Euler-Poisson models for semiconductors , 1991 .

[32]  A. Ionescu,et al.  The Euler–Poisson System in 2D: Global Stability of the Constant Equilibrium Solution , 2011, 1110.0798.

[33]  L. Nirenberg,et al.  On elliptic partial differential equations , 1959 .

[34]  Mikhail Feldman,et al.  MULTIDIMENSIONAL TRANSONIC SHOCKS AND FREE BOUNDARY PROBLEMS FOR NONLINEAR EQUATIONS OF MIXED TYPE , 2003 .

[35]  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 .

[36]  Hairong Yuan,et al.  Transonic Shocks in Compressible Flow Passing a Duct for Three-Dimensional Euler Systems , 2008 .