##
**Transonic steady-states of Euler-Poisson equations for semiconductor models with sonic boundary.**
*(English)*
Zbl 07465239

In this paper the authors consider radial transonic solutions for the steady hydrodynamic model of semiconductors represented by Euler-Poisson (E-P) equations with sonic boundary in \(n\)-dimensions. It is considered a system of three partial differential equations (PDE) written in the form:

\(\mathrm{div}{(\rho\mathbf{u})}=0\), \((\mathbf{u}\cdot \nabla )\mathbf{u}+ \nabla P/\rho =\nabla\Phi - \mathbf{u}/\tau \), \(\bigtriangleup\Phi =\rho - b(x)\), where \(x\in \mathbb{R}^n\) (\(n=2,3\)), \(\rho (x)\) is the density of electrons, \(\mathbf{u}\) presents the average particle velocity at location \(x\), and \(\Phi (x)\) denotes the electrostatic potential of electrons. The pressure \(P = P (\rho )\) depends on the density for the isothermal flow with the constant temperature \(T > 0\). The constant \(\tau > 0\) represents the momentum relaxation time, and the known function \(b(x) > 0\) is the doping profile standing for a background density of changed ions.

The authors investigate the structure of radial transonic solutions to E-P equations in an annulus domain which is defined by: \(\mathcal{A}= \{x\in\mathbb{R}^n|r_0<|x|<r_1\}\) with fixed constants \(r_0\in (0,r_1)\), where the inner boundary is given by \(\Gamma_0 = \{x\in\mathbb{R}^n:|x|=r_0\}\), and the outer boundary \(\Gamma_1 = \{x\in\mathbb{R}^n:|x|=r_1\}\). The closure of \(\mathcal{A}\) is \(\overline{\mathcal{A}}= \Gamma_0\cup\mathcal{A} \cup\Gamma_1\). The goal of this paper is to show the existence results about radial transonic solutions to the E-P system with sonic boundary. It is given a sonic density \(\rho_0\) and prescribed constant current \(j_0\) at the inner boundary \(\Gamma_0\), and a sonic density \(\rho_1\) at the outer boundary \(\Gamma_1\). It is shown that there exist infinitely many radial transonic shock steady-states of E-P system with a large relaxation time and infinitely many radial differentiable transonic steady-states of E-P system with a small relaxation time. It is investigated a two-dimensional system. It is shown that there exists a transonic shock solution over \([r_0,r_1]\) of this system satisfying both the entropy condition and Rankine-Hugoniot condition at a point \(x_0\in (r_0, r_1)\). Because of the arbitrary choices of \(x_0\), the transonic shock solutions are infinitely many.

\(\mathrm{div}{(\rho\mathbf{u})}=0\), \((\mathbf{u}\cdot \nabla )\mathbf{u}+ \nabla P/\rho =\nabla\Phi - \mathbf{u}/\tau \), \(\bigtriangleup\Phi =\rho - b(x)\), where \(x\in \mathbb{R}^n\) (\(n=2,3\)), \(\rho (x)\) is the density of electrons, \(\mathbf{u}\) presents the average particle velocity at location \(x\), and \(\Phi (x)\) denotes the electrostatic potential of electrons. The pressure \(P = P (\rho )\) depends on the density for the isothermal flow with the constant temperature \(T > 0\). The constant \(\tau > 0\) represents the momentum relaxation time, and the known function \(b(x) > 0\) is the doping profile standing for a background density of changed ions.

The authors investigate the structure of radial transonic solutions to E-P equations in an annulus domain which is defined by: \(\mathcal{A}= \{x\in\mathbb{R}^n|r_0<|x|<r_1\}\) with fixed constants \(r_0\in (0,r_1)\), where the inner boundary is given by \(\Gamma_0 = \{x\in\mathbb{R}^n:|x|=r_0\}\), and the outer boundary \(\Gamma_1 = \{x\in\mathbb{R}^n:|x|=r_1\}\). The closure of \(\mathcal{A}\) is \(\overline{\mathcal{A}}= \Gamma_0\cup\mathcal{A} \cup\Gamma_1\). The goal of this paper is to show the existence results about radial transonic solutions to the E-P system with sonic boundary. It is given a sonic density \(\rho_0\) and prescribed constant current \(j_0\) at the inner boundary \(\Gamma_0\), and a sonic density \(\rho_1\) at the outer boundary \(\Gamma_1\). It is shown that there exist infinitely many radial transonic shock steady-states of E-P system with a large relaxation time and infinitely many radial differentiable transonic steady-states of E-P system with a small relaxation time. It is investigated a two-dimensional system. It is shown that there exists a transonic shock solution over \([r_0,r_1]\) of this system satisfying both the entropy condition and Rankine-Hugoniot condition at a point \(x_0\in (r_0, r_1)\). Because of the arbitrary choices of \(x_0\), the transonic shock solutions are infinitely many.

Reviewer: Dimitar A. Kolev (Sofia)

### MSC:

35Q81 | PDEs in connection with semiconductor devices |

35G50 | Systems of nonlinear higher-order PDEs |

35B40 | Asymptotic behavior of solutions to PDEs |

35B65 | Smoothness and regularity of solutions to PDEs |

35C06 | Self-similar solutions to PDEs |

35L67 | Shocks and singularities for hyperbolic equations |

76H05 | Transonic flows |

76L05 | Shock waves and blast waves in fluid mechanics |

76J20 | Supersonic flows |

78A35 | Motion of charged particles |

35A01 | Existence problems for PDEs: global existence, local existence, non-existence |

35Q35 | PDEs in connection with fluid mechanics |

35Q60 | PDEs in connection with optics and electromagnetic theory |

### Keywords:

hydrodynamic model of semiconductors; Euler-Poisson equations; transonic shock solutions; smooth transonic solutions; local analysis
PDF
BibTeX
XML
Cite

\textit{L. Chen} et al., SIAM J. Math. Anal. 54, No. 1, 363--388 (2022; Zbl 07465239)

Full Text:
DOI

### References:

[1] | U. Ascher, P. Markowich, P. Pietra, and C. Schmeiser, A phase plane analysis of transonic solutions for the hydrodynamic semiconductor model, Math. Models Methods Appl. Sci., 1 (1991), pp. 347-376. · Zbl 0800.76032 |

[2] | P. Amster, M. P. B. Varela, A. Jüngel, and M. C. Mariani, Subsonic solutions to a one-dimensional non-isentropic hydrodynamic model for semiconductors, J. Math. Anal. Appl., 258 (2001), pp. 52-62. · Zbl 0982.35112 |

[3] | M. Bae, B. Duan, and C. Xie, Subsonic solutions for steady Euler-Poisson system in two-dimensional nozzles, SIAM J. Math. Anal., 46 (2014), pp. 3455-3480. · Zbl 1316.35221 |

[4] | M. Bae, B. Duan, and C. Xie, Subsonic flow for the multidimensional Euler-Poisson system, Arch. Ration. Mech. Anal., 220 (2016), pp. 155-191. · Zbl 1339.35222 |

[5] | M. Bae, B. Duan, J. J. Xiao, and C. Xie, Structural stability of supersonic solutions to the Euler-Poisson system, Arch. Ration. Mech. Anal., 239 (2021), pp. 679-731. · Zbl 1456.35136 |

[6] | M. Bae and Y. Park, Radial transonic shock solutions of Euler-Poisson system in convergent nozzles, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), pp. 773-791. · Zbl 07006532 |

[7] | K. Bløtekj\aer, Transport equations for electrons in two-valley semiconductors, IEEE Trans. Electron. Devices, 17 (1970), pp. 38-47. |

[8] | L. Chen, M. Mei, G. Zhang, and K. Zhang, Steady hydrodynamic model of semiconductors with sonic boundary and transonic doping profile, J. Differential Equations, 269 (2020), pp. 8173-8211. · Zbl 07216749 |

[9] | L. Chen, M. Mei, G. Zhang, and K. Zhang, Radial solutions of the hydrodynamic model of semiconductors with sonic boundary, J. Math. Anal. Appl., 501 (2021), 125187. · Zbl 1467.76072 |

[10] | P. Degond and P. Markowich, On a one-dimensional steady-state hydrodynamic model for semiconductors, Appl. Math. Lett., 3 (1990), pp. 25-29. · Zbl 0736.35129 |

[11] | P. Degond and P. Markowich, A steady state potential flow model for semiconductors, Ann. Mat. Pura Appl., 4 (1993), pp. 87-98. · Zbl 0808.35150 |

[12] | B. Duan and Y. Zhou, Non-isentropic multi-transonic solutions of Euler-Poisson system, J. Differential Equations, 268 (2020), pp. 7029-7046. · Zbl 1437.35480 |

[13] | I. M. Gamba, Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductors, Comm. Partial Differential Equations, 17 (1992), pp. 553-577. · Zbl 0748.35049 |

[14] | I. M. Gamba and C. S. Morawetz, A viscous approximation for a 2-D steady semiconductor or transonic gas dynamic flow: Existence for potential flow, Comm. Pure Appl. Math., 49 (1996), pp. 999-1049. · Zbl 0863.76029 |

[15] | F. Huang, M. Mei, Y. Wang, and H. Yu, Asymptotic convergence to stationary waves for unipolar hydrodynamic model of semiconductors, SIAM J. Math. Anal., 43 (2011), pp. 411-429. · Zbl 1227.35063 |

[16] | F. Huang, M. Mei, Y. Wang, and H. Yu, Asymptotic convergence to planar stationary waves for multi-dimensional unipolar hydrodynamic model of semiconductors, J. Differential Equations., 251 (2011), pp. 1305-1331. · Zbl 1228.35177 |

[17] | A. Jüngel, Quasi-Hydrodynamic Semiconductor Equations, Progr. Nonlinear Differential Equations Appl. 41, Birkhäuser, Basel, 2001. · Zbl 0969.35001 |

[18] | H. L. Li, P. Markowich, and M. Mei, Asymptotic behavior of solutions of the hydrodynamic model of semiconductors, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), pp. 359-378. · Zbl 1119.35310 |

[19] | J. Li, M. Mei, G. Zhang, and K. Zhang, Steady hydrodynamic model of semiconductors with sonic boundary: (I) Subsonic doping profile, SIAM J. Math. Anal., 49 (2017), pp. 4767-4811. · Zbl 1379.35350 |

[20] | J. Li, M. Mei, G. Zhang, and K. Zhang, Steady hydrodynamic model of semiconductors with sonic boundary: (II) Supersonic doping profile, SIAM J. Math. Anal., 50 (2018), pp. 718-734. · Zbl 1380.35168 |

[21] | T. Luo, J. Rauch, C. Xie, and Z. Xin, Stability of transonic shock solutions for one-dimensional Euler-Poisson equations, Arch. Ration. Mech. Anal., 202 (2011), pp. 787-827. · Zbl 1261.76055 |

[22] | T. Luo and Z. Xin, Transonic shock solutions for a system of Euler-Poisson equations, Commun. Math. Sci., 10 (2012), pp. 419-462. · Zbl 1286.35165 |

[23] | P. Markowich, The Stationary Semiconductor Device Equations, Springer, New York, 1986. |

[24] | P. Markowich, C. Ringhofer, and C. Schmeiser, Semiconductor Equations, Springer, Vienna, 1990. · Zbl 0765.35001 |

[25] | P. A. Markowich, On steady state Euler-Poisson models for semiconductors, Z. Angew. Math. Phys., 42 (1991), pp. 389-407. · Zbl 0755.35138 |

[26] | S. Nishibata and M. Suzuki, Asymptotic stability of a stationary solution to a hydrodynamic model of semiconductors, Arch. Osaka J. Math., 44 (2007), pp. 639-665. · Zbl 1138.82033 |

[27] | S. Nishibata and M. Suzuki, Asymptotic stability of a stationary solution to a thermal hydrodynamic model for semiconductors, Arch. Ration. Mech. Anal., 192 (2009), pp. 187-215. · Zbl 1166.82020 |

[28] | Y. Peng and I. Violet, Example of supersonic solutions to a steady state Euler-Poisson system, Appl. Math. Lett., 19 (2006), pp. 1335-1340. · Zbl 1139.35374 |

[29] | M. D. Rosini, A phase analysis of transonic solutions for the hydrodynamic semiconductor model, Quart. Appl. Math., 63 (2005), pp. 251-268. · Zbl 1319.82024 |

[30] | S. Selberherr, Analysis and Simulation of Semiconductor Devices, Springer, New York, 1989. |

[31] | A. Sitenko and V. Malnev, Plasma Physics Theory, Appl. Math. Math. Comput. 10, Chapman and Hall, London, 1995. · Zbl 0845.76099 |

[32] | M. Wei, M. Mei, G. Zhang, and K. Zhang, Smooth transonic steady-states of hydrodynamic model for semiconductors, SIAM J. Math. Anal., 53 (2021), pp. 4908-4932. · Zbl 07391578 |

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.