Relaxation-time limits of global solutions in full quantum hydrodynamic model for semiconductors

Sungjin Ra; Hakho Hong

Applications of Mathematics (2024)

  • Issue: 1, page 113-137
  • ISSN: 0862-7940

Abstract

top
This paper is concerned with the global well-posedness and relaxation-time limits for the solutions in the full quantum hydrodynamic model, which can be used to analyze the thermal and quantum influences on the transport of carriers in semiconductor devices. For the Cauchy problem in ℝ 3 , we prove the global existence, uniqueness and exponential decay estimate of smooth solutions, when the initial data are small perturbations of an equilibrium state. Moreover, we show that the solutions converge into that of the simplified quantum energy-transport model and the quantum drift-diffusion model for the moment relaxation limit, and the moment and energy relaxation limit, respectively.

How to cite

top

Ra, Sungjin, and Hong, Hakho. "Relaxation-time limits of global solutions in full quantum hydrodynamic model for semiconductors." Applications of Mathematics (2024): 113-137. <http://eudml.org/doc/299206>.

@article{Ra2024,
abstract = {This paper is concerned with the global well-posedness and relaxation-time limits for the solutions in the full quantum hydrodynamic model, which can be used to analyze the thermal and quantum influences on the transport of carriers in semiconductor devices. For the Cauchy problem in $\mathbb \{R\}^3$, we prove the global existence, uniqueness and exponential decay estimate of smooth solutions, when the initial data are small perturbations of an equilibrium state. Moreover, we show that the solutions converge into that of the simplified quantum energy-transport model and the quantum drift-diffusion model for the moment relaxation limit, and the moment and energy relaxation limit, respectively.},
author = {Ra, Sungjin, Hong, Hakho},
journal = {Applications of Mathematics},
keywords = {quantum hydrodynamic equation; quantum Euler-Poisson system; bipolar semiconductor model; relaxation-time limit},
language = {eng},
number = {1},
pages = {113-137},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Relaxation-time limits of global solutions in full quantum hydrodynamic model for semiconductors},
url = {http://eudml.org/doc/299206},
year = {2024},
}

TY - JOUR
AU - Ra, Sungjin
AU - Hong, Hakho
TI - Relaxation-time limits of global solutions in full quantum hydrodynamic model for semiconductors
JO - Applications of Mathematics
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
IS - 1
SP - 113
EP - 137
AB - This paper is concerned with the global well-posedness and relaxation-time limits for the solutions in the full quantum hydrodynamic model, which can be used to analyze the thermal and quantum influences on the transport of carriers in semiconductor devices. For the Cauchy problem in $\mathbb {R}^3$, we prove the global existence, uniqueness and exponential decay estimate of smooth solutions, when the initial data are small perturbations of an equilibrium state. Moreover, we show that the solutions converge into that of the simplified quantum energy-transport model and the quantum drift-diffusion model for the moment relaxation limit, and the moment and energy relaxation limit, respectively.
LA - eng
KW - quantum hydrodynamic equation; quantum Euler-Poisson system; bipolar semiconductor model; relaxation-time limit
UR - http://eudml.org/doc/299206
ER -

References

top
  1. Chen, X., 10.1007/s00033-008-7068-4, Z. Angew. Math. Phys. 60 (2009), 416-437. (2009) Zbl1173.35518MR2505412DOI10.1007/s00033-008-7068-4
  2. Chen, X., Chen, L., 10.1016/j.jmaa.2008.01.015, J. Math. Anal. Appl. 343 (2008), 64-80. (2008) Zbl1139.35010MR2409458DOI10.1016/j.jmaa.2008.01.015
  3. Chen, X., Chen, L., Jian, H., 10.1016/j.nonrwa.2008.01.008, Nonlinear Anal., Real World Appl. 10 (2009), 1321-1342. (2009) Zbl1171.35329MR2502947DOI10.1016/j.nonrwa.2008.01.008
  4. Chen, L., Ju, Q., 10.1007/s00033-005-0051-4, Z. Angew. Math. Phys. 58 (2007), 1-15. (2007) Zbl1107.35037MR2293100DOI10.1007/s00033-005-0051-4
  5. Dong, J., Mixed boundary-value problems for quantum hydrodynamic models with semiconductors in thermal equilibrium, Electron. J. Differ. Equ. 2005 (2005), Article ID 123, 8 pages. (2005) Zbl1245.35029MR2181267
  6. Gardner, C. L., 10.1137/S003613999224042, SIAM J. Appl. Math. 54 (1994), 409-427. (1994) Zbl0815.35111MR1265234DOI10.1137/S003613999224042
  7. Gianazza, U., Savaré, G., Toscani, G., 10.1007/s00205-008-0186-5, Arch. Ration. Mech. Anal. 194 (2009), 133-220. (2009) Zbl1223.35264MR2533926DOI10.1007/s00205-008-0186-5
  8. Gualdani, M. P., Jüngel, A., Toscani, G., 10.1137/S0036141004444615, SIAM. J. Math. Anal. 37 (2006), 1761-1779. (2006) Zbl1102.35045MR2213393DOI10.1137/S0036141004444615
  9. Huang, F., Li, H.-L., Matsumura, A., 10.1016/j.jde.2006.02.002, J. Differ. Equations 225 (2006), 1-25. (2006) Zbl1160.76444MR2228690DOI10.1016/j.jde.2006.02.002
  10. Jia, Y., Li, H., 10.1016/S0252-9602(06)60038-6, Acta Math. Sci., Ser. B, Engl. Ed. 26 (2006), 163-178. (2006) Zbl1152.76505MR2206278DOI10.1016/S0252-9602(06)60038-6
  11. Jüngel, A., 10.1007/s002200050364, Commun. Math. Phys. 194 (1998), 463-479. (1998) Zbl0916.76099MR1627673DOI10.1007/s002200050364
  12. Jüngel, A., Li, H., Quantum Euler-Poisson systems: Existence of stationary states, Arch. Math., Brno 40 (2004), 435-456. (2004) Zbl1122.35140MR2129964
  13. Jüngel, A., Li, H., 10.1090/qam/2086047, Q. Appl. Math. 62 (2004), 569-600. (2004) Zbl1069.35012MR2086047DOI10.1090/qam/2086047
  14. Jüngel, A., Li, H.-L., Matsumura, A., 10.1016/j.jde.2005.11.007, J. Differ. Equations 225 (2006), 440-464. (2006) Zbl1147.82364MR2225796DOI10.1016/j.jde.2005.11.007
  15. Jüngel, A., Matthes, D., Milišić, J. P., 10.1137/050644823, SIAM J. Appl. Math. 67 (2006), 46-68. (2006) Zbl1121.35117MR2272614DOI10.1137/050644823
  16. Jüngel, A., Milišić, J. P., 10.1016/j.nonrwa.2010.08.026, Nonlinear Anal., Real World Appl. 12 (2011), 1033-1046. (2011) Zbl1206.35152MR2736191DOI10.1016/j.nonrwa.2010.08.026
  17. Jüngel, A., Pinnau, R., 10.1137/S0036142900369362, SIAM J. Numer. Anal. 39 (2001), 385-406. (2001) Zbl0994.35047MR1860272DOI10.1137/S0036142900369362
  18. Jüngel, A., Violet, I., The quasineutral limit in the quantum drift-diffusion equations, Asymptotic Anal. 53 (2007), 139-157. (2007) Zbl1156.35077MR2349559
  19. Kim, Y.-H., Ra, S., Kim, S.-C., 10.1016/j.nonrwa.2020.103261, Nonlinear Anal., Real World Appl. 59 (2021), Article ID 103261, 18 pages. (2021) Zbl1468.35202MR4177987DOI10.1016/j.nonrwa.2020.103261
  20. Klainerman, S., Majda, A., 10.1002/cpa.3160340405, Commun. Pure Appl. Math. 34 (1981), 481-524. (1981) Zbl0476.76068MR0615627DOI10.1002/cpa.3160340405
  21. Li, H., Marcati, P., 10.1007/s00220-003-1001-7, Commun. Math. Phys. 245 (2004), 215-247. (2004) Zbl1075.82019MR2039696DOI10.1007/s00220-003-1001-7
  22. Li, H., Zhang, G., Zhang, M., Hao, C., 10.1063/1.2949082, J. Math. Phys. 49 (2008), Article ID 073503, 14 pages. (2008) Zbl1152.81528MR2432041DOI10.1063/1.2949082
  23. Mao, J., Zhou, F., Li, Y., 10.1016/j.jmaa.2009.11.039, J. Math. Anal. Appl. 364 (2010), 186-194. (2010) Zbl1186.35165MR2576062DOI10.1016/j.jmaa.2009.11.039
  24. Markowich, P. A., Ringhofer, C. A., Schmeiser, C., 10.1007/978-3-7091-6961-2, Springer, Vienna (1990). (1990) Zbl0765.35001MR1063852DOI10.1007/978-3-7091-6961-2
  25. Nirenberg, L., On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 13 (1959), 115-162. (1959) Zbl0088.07601MR0109940
  26. Nishibata, S., Shigeta, N., Suzuki, M., 10.1142/S0218202510004477, Math. Models Methods Appl. Sci. 20 (2010), 909-936. (2010) Zbl1193.82057MR2659742DOI10.1142/S0218202510004477
  27. Nishibata, S., Suzuki, M., 10.1016/j.jde.2007.10.035, J. Differ. Equations 244 (2008), 836-874. (2008) Zbl1139.82042MR2391346DOI10.1016/j.jde.2007.10.035
  28. Ra, S., Hong, H., 10.1007/s00033-021-01540-8, Z. Angew. Math. Phys. 72 (2021), Article ID 107, 32 pages. (2021) Zbl1467.76078MR4252285DOI10.1007/s00033-021-01540-8
  29. Ri, J., Ra, S., Solution to a multi-dimensional isentropic quantum drift-diffusion model for bipolar semiconductors, Electron. J. Differ. Equ. 2018 (2018), Article ID 200, 19 pages. (2018) Zbl07004591MR3907819
  30. Simon, J., 10.1007/BF01762360, Ann. Mat. Pura Appl. (4) 146 (1986), Article ID 146, 32 pages. (1986) MR0916688DOI10.1007/BF01762360
  31. Zhang, G., Li, H.-L., Zhang, K., 10.1016/j.jde.2008.06.019, J. Differ. Equations 245 (2008), 1433-1453. (2008) Zbl1154.35071MR2436449DOI10.1016/j.jde.2008.06.019

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.