Asymptotic stability of stationary solutions to the drift-diffusion model in the whole space

Ryo Kobayashi; Masakazu Yamamoto; Shuichi Kawashima

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 4, page 1097-1121
  • ISSN: 1292-8119

Abstract

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We study the initial value problem for the drift-diffusion model arising in semiconductor device simulation and plasma physics. We show that the corresponding stationary problem in the whole space ℝn admits a unique stationary solution in a general situation. Moreover, it is proved that when n ≥ 3, a unique solution to the initial value problem exists globally in time and converges to the corresponding stationary solution as time tends to infinity, provided that the amplitude of the stationary solution and the initial perturbation are suitably small. Also, we show the sharp decay estimate for the perturbation. The stability proof is based on the time weighted Lp energy method.

How to cite

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Kobayashi, Ryo, Yamamoto, Masakazu, and Kawashima, Shuichi. "Asymptotic stability of stationary solutions to the drift-diffusion model in the whole space." ESAIM: Control, Optimisation and Calculus of Variations 18.4 (2012): 1097-1121. <http://eudml.org/doc/272833>.

@article{Kobayashi2012,
abstract = {We study the initial value problem for the drift-diffusion model arising in semiconductor device simulation and plasma physics. We show that the corresponding stationary problem in the whole space ℝn admits a unique stationary solution in a general situation. Moreover, it is proved that when n ≥ 3, a unique solution to the initial value problem exists globally in time and converges to the corresponding stationary solution as time tends to infinity, provided that the amplitude of the stationary solution and the initial perturbation are suitably small. Also, we show the sharp decay estimate for the perturbation. The stability proof is based on the time weighted Lp energy method.},
author = {Kobayashi, Ryo, Yamamoto, Masakazu, Kawashima, Shuichi},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {drift-diffusion model; stability; decay estimates; weighted energy method; elliptic-parabolic system; semiconductor device simulation},
language = {eng},
number = {4},
pages = {1097-1121},
publisher = {EDP-Sciences},
title = {Asymptotic stability of stationary solutions to the drift-diffusion model in the whole space},
url = {http://eudml.org/doc/272833},
volume = {18},
year = {2012},
}

TY - JOUR
AU - Kobayashi, Ryo
AU - Yamamoto, Masakazu
AU - Kawashima, Shuichi
TI - Asymptotic stability of stationary solutions to the drift-diffusion model in the whole space
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2012
PB - EDP-Sciences
VL - 18
IS - 4
SP - 1097
EP - 1121
AB - We study the initial value problem for the drift-diffusion model arising in semiconductor device simulation and plasma physics. We show that the corresponding stationary problem in the whole space ℝn admits a unique stationary solution in a general situation. Moreover, it is proved that when n ≥ 3, a unique solution to the initial value problem exists globally in time and converges to the corresponding stationary solution as time tends to infinity, provided that the amplitude of the stationary solution and the initial perturbation are suitably small. Also, we show the sharp decay estimate for the perturbation. The stability proof is based on the time weighted Lp energy method.
LA - eng
KW - drift-diffusion model; stability; decay estimates; weighted energy method; elliptic-parabolic system; semiconductor device simulation
UR - http://eudml.org/doc/272833
ER -

References

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  1. [1] P. Biler and J. Dolbeault, Long time behavior of solutions to Nernst-Planck and Debye-Hückel drift-diffusion system. Ann. Henri Poincaré 1 (2000) 461−472. Zbl0976.82046MR1777308
  2. [2] P. Biler and N. Nadzieja, Existence and nonexistence of solutions for a model of gravitational interactions of particles I. Colloq. Math. 66 (1994) 319−334. Zbl0817.35041MR1268074
  3. [3] P. Biler and N. Nadzieja, A singular problem in electrolytes theory. Math. Methods Appl. Sci.20 (1997) 767–782. Zbl0885.35051MR1446210
  4. [4] P. Biler, G. Karch, P. Laurençot and T. Nadzieja, The 8π-problem for radially symmetric solutions of a chemotaxis model in the plane. Math. Methods Appl. Sci.29 (2006) 1563–1583. Zbl1105.35131MR2249579
  5. [5] S. Childress and J.K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci.56 (1981) 217–237. Zbl0481.92010MR632161
  6. [6] R. Farwig and H. Sohr, Weighted Lq-theory for the Stokes resolvent in exterior domains. J. Math. Soc. Jpn49 (1997) 251–288. Zbl0918.35106MR1601373
  7. [7] M. Escobedo and E. Zuazua, Large time behavior for convection-diffusion equations in ℝN. J. Funct. Anal.100 (1991) 119–161. Zbl0762.35011MR1124296
  8. [8] S. Kawashima, S. Nishibata and M. Nishikawa, Lp energy method for multi-dimensional viscous conservation laws and application to the stability of planar waves. J. Hyperbolic Differ. Equ.1 (2004) 581–603. Zbl1067.35019MR2094531
  9. [9] E.F. Keller and L.A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol.26 (1970) 399–415. Zbl1170.92306
  10. [10] R. Kobayashi and S. Kawashima, Decay estimates and large time behavior of solutions to the drift-diffusion system. Funkcial. Ekvac.51 (2008) 371–394. Zbl1173.35366MR2493875
  11. [11] R. Kobayashi, M. Kurokiba and S. Kawashima, Stationary solutions to the drift-diffusion model in the whole space. Math. Methods Appl. Sci.32 (2009) 640–652. Zbl1173.78301MR2504001
  12. [12] M. Kurokiba and T. Ogawa, Lp wellposedness for the drift-diffusion system arising from the semiconductor device simulation. J. Math. Anal. Appl.342 (2008) 1052–1067. Zbl1145.35067MR2445259
  13. [13] D.S. Kurtz and R.L. Wheeden, Results on weighted norm inequalities for multipliers, Trans. Amer. Math. Soc.255 (1979) 343–362. Zbl0427.42004MR542885
  14. [14] M.S. Mock, Asymptotic behavior of solutions of transport equations for semiconductor devices, J. Math. Anal. Appl.49 (1975) 215–225. Zbl0298.35033MR355383
  15. [15] T. Nagai, Global existence and decay estimates of solutions to a parabolic-elliptic system of drift-diffusion type in ℝ2. Differential Integral Equations24 (2011) 29–68. Zbl1240.35066MR2759351
  16. [16] L. Nirenberg, On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa13 (1959) 115–162. Zbl0088.07601MR109940
  17. [17] A. Raczyński, Weak-Lp solutions for a model of self-gravitating particles with an external potential. Stud. Math.179 (2007) 199–216. Zbl1113.35030
  18. [18] D.R. Smart, Fixed Point Theorems. Cambridge University Press, New York (1974). Zbl0427.47036MR467717
  19. [19] E.M. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970). Zbl0207.13501MR290095
  20. [20] G. Wolansky, On steady distributions of self-attracting clusters under friction and fluctuations, Arch. Rational Mech. Anal.119 (1992) 355–391. Zbl0774.76069MR1179691
  21. [21] W.P. Ziemer, Weakly Differentiable Functions. Springer-Verlag, New York (1989). Zbl0692.46022MR1014685

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