Quasi-neutral limit to steady-state hydrodynamic model of semiconductors with degenerate boundary. (English) Zbl 1519.35319
Summary: This paper is concerned with the quasi-neutral limit to a one-dimensional steady hydrodynamic model of semiconductors in the form of Euler-Poisson equations with degenerate boundary, a difficult case caused by the boundary layers and degeneracy. We establish a so-called convexity structure of the sequence of subsonic-sonic solutions near the boundary domains in this limit process, which efficiently overcomes the degenerate effect. We first show the strong convergence in the \(L^2\) norm with the order \(O(\lambda^{\frac{1}{2}})\) for the Debye length \(\lambda\) when the doping profile is continuous. Then we derive the uniform error estimates in the \(L^\infty\) norm with the order \(O(\lambda)\) when the doping profile has higher regularity. The proof of \(L^{\infty}\) boundedness is based on a new bounded estimate method, which is used to replace the maximum principle utilized in the nondegenerate case. These newly proposed techniques in asymptotic limit analysis develop and improve the existing studies.
MSC:
| 35Q81 | PDEs in connection with semiconductor devices |
| 35Q35 | PDEs in connection with fluid mechanics |
| 35L60 | First-order nonlinear hyperbolic equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35C06 | Self-similar solutions to PDEs |
| 76G25 | General aerodynamics and subsonic flows |
| 76H05 | Transonic flows |
| 82D37 | Statistical mechanics of semiconductors |
Keywords:
quasi-neutral limit; hydrodynamic model of semiconductors; degenerate boundary; boundary layers; subsonic-sonic solutionsReferences:
| [1] | Ascher, U., Markowich, P., Pietra, P., and Schmeiser, C., 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] | Bløtekjær, K., Transport equations for electrons in two-valley semiconductors, IEEE Trans. Electron Devices, 17 (1970), pp. 38-47. |
| [3] | Chen, L., Mei, M., Zhang, G., and Zhang, K., Steady hydrodynamic model of semiconductors with sonic boundary and transonic doping profile, J. Differential Equations, 269 (2020), pp. 8173-8211. · Zbl 1509.35388 |
| [4] | Chen, L., Mei, M., Zhang, G., and Zhang, K., Radial solutions of the hydrodynamic model of semiconductors with sonic boundary, J. Math. Anal. Appl., 501 (2021), 125187. · Zbl 1467.76072 |
| [5] | Chen, L., Mei, M., Zhang, G., and Zhang, K., Transonic steady-states of Euler-Poisson equations for semiconductor models with sonic boundary, SIAM J. Math. Anal., 54 (2022), pp. 363-388, doi:10.1137/21M1416369. · Zbl 1487.35366 |
| [6] | Cordier, S. and Grenier, E., Quasi-neutral limit of an Euler-Poisson system arising from plasma physics, Comm. Partial Differential Equations, 251 (2000), pp. 99-113. · Zbl 0978.82086 |
| [7] | Degond, P. and Markowich, P., On a one-dimensional steady-state hydrodynamic model for semiconductors, Appl. Math. Lett., 3 (1990), pp. 25-29. · Zbl 0736.35129 |
| [8] | Degond, P. and Markowich, P., A steady state potential flow model for semiconductors, Ann. Mat. Pura Appl., 4 (1993), pp. 87-98. · Zbl 0808.35150 |
| [9] | Donatelli, D. and Mensah, P. R., The combined incompressible quasineutral limit of the stochastic Navier-Stokes-Poisson system, SIAM J. Math. Anal., 52 (2020), pp. 5090-5120, doi:10.1137/20M1338915. · Zbl 1450.35213 |
| [10] | Donatelli, D. and Spirito, S., Vanishing dielectric constant regime for the Navier Stokes Maxwell equations, NoDEA Nonlinear Differential Equations Appl., 23 (2016), 28. · Zbl 1358.35114 |
| [11] | Fife, P. C., Semilinear elliptic boundary value problems with small parameters, Arch. Ration. Mech. Anal., 52 (1973), pp. 205-232. · Zbl 0268.35007 |
| [12] | Gamba, I. M., Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductors, Comm. Partial Differential Equations, 17 (1992), pp. 553-577. · Zbl 0748.35049 |
| [13] | Giuseppe, A., Chen, L., and Jüngel, A., The zero-electron-mass limit in the hydrodynamic model for plasmas, Nonlinear Anal., 12 (2010), pp. 4415-4427. · Zbl 1189.35019 |
| [14] | Goudon, T., Jüngel, A., and Peng, Y. J., Zero-electron-mass limits in hydrodynamic models for plasmas, Appl. Math. Lett., 12 (1999), pp. 75-79. · Zbl 0959.76096 |
| [15] | Guo, Y. and Strauss, W., Stability of semiconductor states with insulating and contact boundary conditions, Arch. Ration. Mech. Anal., 179 (2005), pp. 1-30. · Zbl 1148.82030 |
| [16] | Hajjej, M. L. and Peng, Y. J., Initial layers and zero-relaxation limits of Euler-Maxwell equations, J. Differential Equations, 252 (2012), pp. 1441-1465. · Zbl 1233.35184 |
| [17] | Huang, F., Mei, M., Wang, Y., and Yu, H., Asymptotic convergence to stationary waves for unipolar hydrodynamic model of semiconductors, SIAM J. Math. Anal., 43 (2011), pp. 411-429, doi:10.1137/100793025. · Zbl 1227.35063 |
| [18] | Ju, Q. and Wang, S., Quasi-neutral limit of the multidimensional drift-diffusion models for semiconductors, Math. Models Methods Appl. Sci., 20 (2010), pp. 1649-1679. · Zbl 1206.35025 |
| [19] | Junca, S. and Rascle, M., Relaxation of the isothermal Euler-Poisson system to the drift-diffusion equations, Quart. Appl. Math., 58 (2000), pp. 511-521. · Zbl 1127.35354 |
| [20] | Jüngel, A. and Peng, Y. J., A hierarchy of hydrodynamic models for plasmas: Zero-relaxation-time limits, Comm. Partial Differential Equations, 24 (1999), pp. 1007-1033. · Zbl 0946.35074 |
| [21] | Jüngel, A. and Peng, Y. J., A hierarchy of hydrodynamic models for plasmas: Zero-electron-mass limits in the drift-diffusion equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 17 (2000), pp. 83-118. · Zbl 0956.35010 |
| [22] | Jüngel, A. and Peng, Y. J., A hierarchy of hydrodynamic models for plasmas: Quasi-neutral limits in the drift-diffusion equations, Asymptotic Anal., 28 (2001), pp. 49-73. · Zbl 1045.76058 |
| [23] | Li, H. L., Markowich, P., and Mei, M., Asymptotic behavior of solutions of the hydrodynamic model of semiconductors, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), pp. 359-378. · Zbl 1119.35310 |
| [24] | Li, J., Mei, M., Zhang, G., and Zhang, K., Steady hydrodynamic model of semiconductors with sonic boundary: (I) Subsonic doping profile, SIAM J. Math. Anal., 49 (2017), pp. 4767-4811, doi:10.1137/17M1127235. · Zbl 1379.35350 |
| [25] | Li, J., Mei, M., Zhang, G., and Zhang, K., Steady hydrodynamic model of semiconductors with sonic boundary: (II) Supersonic doping profile, SIAM J. Math. Anal., 50 (2018), pp. 718-734, doi:10.1137/17M1129477. · Zbl 1380.35168 |
| [26] | Mai, L.-S., Li, H. L., and Marcati, P., Non-relativistic limit analysis of the Chandrasekhar-Thorne relativistic Euler equations with physical vacuum, Math. Models Methods Appl. Sci., 29 (2019), pp. 531-579. · Zbl 1428.35331 |
| [27] | Mai, L.-S. and Mei, M., Newtonian limit for the relativistic Euler-Poisson equations with vacuum, J. Differential Equations, 313 (2022), pp. 336-381. · Zbl 1511.35344 |
| [28] | Markowich, P., The Steady-State Semiconductor Device Equations, Springer-Verlag, Vienna, 1986. |
| [29] | Markowich, P., Ringhofer, C. A., and Schmeiser, C., Semiconductor Equations, Springer-Verlag, Vienna, 1990. · Zbl 0765.35001 |
| [30] | Marcati, P. and Natalini, R., Weak solutions to a hydrodynamic model for semiconductors: The Cauchy problem, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), pp. 115-131. · Zbl 0831.35157 |
| [31] | Mei, M., Wu, X., and Zhang, Y., Stability of steady-states for 3-D hydrodynamic model of unipolar semiconductor with Ohmic contact boundary in hollow ball, J. Differential Equations, 277 (2021), pp. 57-113. · Zbl 1456.82936 |
| [32] | Meng, R., Mai, L.-S., and Mei, M., Free boundary value problem for damped Euler equations and related models with vacuum, J. Differential Equations, 321 (2022), pp. 349-380. · Zbl 1507.35156 |
| [33] | Nishibata, S. and Suzuki, M., Asymptotic stability of a stationary solution to a hydrodynamic model of semiconductors, Osaka J. Math., 44 (2007), pp. 639-665. · Zbl 1138.82033 |
| [34] | Peng, Y. J., Asymptotic limits of one-dimensional hydrodynamic models for plasmas and semiconductors, Chinese Ann. Math. Ser. B, 23 (2002), pp. 25-36. · Zbl 0994.35112 |
| [35] | Peng, Y. J., Some asymptotic analysis in steady-state Euler-Poisson equations for potential flow, Asymptotic Anal., 36 (2003), pp. 75-92. · Zbl 1051.82026 |
| [36] | Peng, Y. J. and Wang, Y. G., Boundary layers and quasi-neutral limit in steady state Euler-Poisson equations for potential flows, Nonlinearity, 17 (2004), pp. 835-849. · Zbl 1073.35183 |
| [37] | Peng, Y.-J., Wang, S., and Gu, Q., Relaxation limit and global existence of smooth solutions of compressible Euler-Maxwell equations, SIAM J. Math. Anal., 43 (2011), pp. 944-970, doi:10.1137/100786927. · Zbl 1231.35039 |
| [38] | Peng, Y. J. and Violet, I., Example of supersonic solutions to a steady state Euler-Poisson system, Appl. Math. Lett., 19 (2006), pp. 1335-1340. · Zbl 1139.35374 |
| [39] | Qiu, Y. and Zhang, K., On the relaxation limits of the hydrodynamic model for semiconductor devices, Math. Models Methods Appl. Sci., 12 (2002), pp. 333-363. · Zbl 1174.82351 |
| [40] | Slemrod, M. and Sternberg, N., Quasi-neutral limit for the Euler-Poisson system, J. Nonlinear Sci., 11 (2001), pp. 193-209. · Zbl 0997.34033 |
| [41] | Zhang, B., Convergence of the Godunov scheme for a simplified one-dimensional hydrodynamic model for semiconductor devices, Comm. Math. Phys., 157 (1993), pp. 1-22. · Zbl 0785.76053 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
