Advertisement

Some Formal Properties of the Hydrodynamical Model

  • Vito Dario Camiola
  • Giovanni Mascali
  • Vittorio Romano
Chapter
  • 50 Downloads
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 31)

Abstract

In this chapter we investigate the formal properties of the hydrodynamical model for semiconductors based on MEP. We will prove that it forms a hyperbolic system in the physically relevant region of the space of the dependent variables. Such a property is a consequence of the general theory developed in Chap.  3 when applied to the complete non linear model but since we have performed an expansion of the original non linear MEP distribution function, the hyperbolicity have to be checked.

References

  1. 1.
    Alì, G., Anile, A.M.: Moment equations for charged particles: global existence results. In: Modeling and Computational Methods for Kinetic Equations. Modeling and Simulation in Science, Engineering and Technology, pp. 59–80 (Birkhäuser, Boston, 2004)Google Scholar
  2. 2.
    Alì, G., Bini, D., Rionero, S.: Global existence and relaxation limit for smooth solutions to the Euler-Poisson model for semiconductors. SIAM J. Math. Anal. 32, 572–587 (2002)MathSciNetCrossRefGoogle Scholar
  3. 7.
    Anile, A.M., Romano, V.: Non parabolic band transport in semiconductors: closure of the moment equations. Contin. Mech. Thermodyn. 11, 307–325 (1999)MathSciNetCrossRefGoogle Scholar
  4. 9.
    Anile, A.M., Romano, V., Russo, G.: Hyperbolic hydrodynamical model of carrier transport in semiconductors. VLSI Des. 8, 521–525 (1998)CrossRefGoogle Scholar
  5. 12.
    Anile, A.M., Romano, V., Russo, G.: Extended hydrodynamical model of carrier transport in semiconductors. SIAM J. Appl. Math. 61, 74–101 (2000)MathSciNetCrossRefGoogle Scholar
  6. 29.
    Blokhin, A.M., Iohrdanidy, A.A.: Numerical investigation of a gas dynamic model for charge transport in semiconductors. COMPEL Int. J. Comput. Math. Electr. 18, 6–37 (1999)CrossRefGoogle Scholar
  7. 30.
    Blokhin, A.M., Tkachev, D.L.: Local-in-time well-posedness of a regularized mathematical model for silicon MESFET. Z. Angew. Math. Phys. 61, 849–864 (2010)MathSciNetCrossRefGoogle Scholar
  8. 31.
    Blokhin, A.M., Trakhinin, Y.L.: Symmetrization of the system of equations of radiation hydrodynamics and global solvability of the Cauchy problem. Sib. Math. J. 37, 1101–1109 (1996)CrossRefGoogle Scholar
  9. 32.
    Blokhin, A.M., Trakhinin, Y.L.: Stability of Strong Discontinuities in Magnetohydrodynamics and Electrodynamics. Nova Science Publishers, New York (2003)zbMATHGoogle Scholar
  10. 33.
    Blokhin, A.M., Bushmanova, A.S., Romano, V.: Stability of the equilibrium state for a hydrodynamical model of charge transport in semiconductors. Z. Angew. Math. Phys. 52, 476–499 (2001)MathSciNetCrossRefGoogle Scholar
  11. 34.
    Blokhin, A.M., Bushmanov, R.S., Romano, V.: Electron flow stability in bulk silicon in the limit of small electric field. In: 11th Conference on Waves and Stability in Continuous Media. Proceedings of WASCOM 2001, pp. 55–60. Word Scientific, Singapore (2002)Google Scholar
  12. 35.
    Blokhin, A.M., Bushmanov, R.S., Romano, V.: Asymptotic stability of the equilibrium state for the hydrodynamical model of charge transport in semiconductors based on the maximum entropy principle. Int. J. Eng. Sci. 42, 915–934 (2004)MathSciNetCrossRefGoogle Scholar
  13. 36.
    Blokhin, A.M., Bushmanov, R.S., Romano, V.: Nonlinear asymptotic stability of the equilibrium state for the MEP model of charge transport in semiconductors. Nonlinear Anal. 65, 2169–2191 (2006)MathSciNetCrossRefGoogle Scholar
  14. 37.
    Blokhin, A.M., Bushmanov, R.S., Rudometova, A.S., Romano, V.: Linear asymptotic stability of the equilibrium state for the 2-D MEP hydrodynamical model of charge transport in semiconductors. Nonlinear Anal. 65, 1018–1038 (2006)MathSciNetCrossRefGoogle Scholar
  15. 38.
    Bobylev, A.V.: The Chapman-Enskog and Grad methods for solving the Boltzmann equation. Sov. Phys. Dokl. 27, 29–31 (1982)Google Scholar
  16. 59.
    Colangeli, M., Karlin, I.V., Kröger, M.: From hyperbolic regularization to exact hydrodynamics for linearized Grad’s equations. Phys. Rev. E 75, 051204 (2007)MathSciNetCrossRefGoogle Scholar
  17. 83.
    Gardner, C.L., Jerome, J.W., Rose, D.J.: Numerical methods for the hydrodynamic device model: subsonic flow. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 8, 501–507 (1989)CrossRefGoogle Scholar
  18. 86.
    Gorban, A.N., Karlin, I.V., Ilg, P., Öttinger, H.C.: Corrections and enhancements of quasi-equilibrium states. J. Non-Newtonian Fluid. Mech. 96, 203–219 (2001)CrossRefGoogle Scholar
  19. 107.
    Karlin, I.V., Colangeli, M., Kröger, M.: Exact linear hydrodynamics from the Boltzmann equation. Phys. Rev. Lett. 100, 214503 (2008)CrossRefGoogle Scholar
  20. 113.
    Ladyzhenskaja, O.A.: The mathematical theory of viscous incompressible flow. Translated from the Russian by R. S. Silvermann and J. Chu. Gordon and Breach, New York (1969)Google Scholar
  21. 120.
    Li, H., Markowich, P., Mei, M.: Asymptotic behaviour of solutions of the hydrodynamic model of semiconductors. Proc. R. Soc. Edinburgh A Math. 132, 359–378 (2002)MathSciNetCrossRefGoogle Scholar
  22. 122.
    Ling, H., Shu, W.: The asymptotic behavior of global smooth solutions to the macroscopic models of semiconductors. Chin. Ann. Math. 22B, 195–210 (2001)MathSciNetzbMATHGoogle Scholar
  23. 123.
    Ling, H., Shu, W.: Asymptotic behavior of global smooth solutions to the full 1D hydrodynamic model for semiconductors. Math. Models Methods Appl. Sci. 12, 777–796 (2002)MathSciNetCrossRefGoogle Scholar
  24. 127.
    Luo, T., Natalini, R., Xin, Z.P.: Large time behavior of the solutions to a hydrodynamic model for semiconductors. SIAM J. Appl. Math. 59, 810–830 (1998)MathSciNetCrossRefGoogle Scholar
  25. 135.
    Marcati, P., Natalini, R.: Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation. Arch. Rational Mech. Anal. 129, 129–145 (1995)MathSciNetCrossRefGoogle Scholar
  26. 148.
    Mizohata, S.: The Theory of Partial Differential Equations. Cambridge University Press, Cambridge (1979)zbMATHGoogle Scholar
  27. 173.
    Romano, V.: Non parabolic band transport in semiconductors: closure of the production terms in the moment equations. Contin. Mech. Thermodyn. 12, 31–51 (2000)MathSciNetCrossRefGoogle Scholar
  28. 174.
    Romano, V.: Non-parabolic band hydrodynamical model of silicon semiconductors and simulation of electron devices. Math. Methods Appl. Sci. 24, 439–471 (2001)MathSciNetCrossRefGoogle Scholar
  29. 175.
    Romano, V.: 2D simulation of a silicon MESFET with a nonparabolic hydrodynamical model based on the maximum entropy principle. J. Comput. Phys. 176, 70–92 (2002)CrossRefGoogle Scholar
  30. 179.
    Romano, V., Rusakov, A.: 2D numerical simulations of an electron phonon hydrodynamical model based on the maximum entropy principle. Comput. Methods Appl. Mech. Eng. 199, 2741–2751 (2010)MathSciNetCrossRefGoogle Scholar
  31. 180.
    Romano, V., Russo, G.: Numerical solution for hydrodymamical models of semiconductors. Math. Models Methods Appl. Sci. 10, 1099–1120 (2000)MathSciNetCrossRefGoogle Scholar
  32. 187.
    Sobolev, S.L.: Applications of Functional Analysis in Mathematical Physics (Translation of Mathemtical Monograph). AMS, Providence (2000)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Vito Dario Camiola
    • 1
  • Giovanni Mascali
    • 2
  • Vittorio Romano
    • 1
  1. 1.Department of Mathematics and Computer ScienceUniversity of CataniaCataniaItaly
  2. 2.Department of Mathematics and Computer ScienceUniversity of CalabriaArcavacata di RendeItaly

Personalised recommendations