Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
quantum hydrodynamic equation; quantum Euler-Poisson system; bipolar semiconductor model; relaxation-time limit
quantum hydrodynamic equation; quantum Euler-Poisson system; bipolar semiconductor model; relaxation-time limit
Summary:
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.
References:
[1] Chen, X.: The isentropic quantum drift-diffusion model in two or three dimensions. Z. Angew. Math. Phys. 60 (2009), 416-437. DOI 10.1007/s00033-008-7068-4 | MR 2505412 | Zbl 1173.35518
[2] Chen, X., Chen, L.: Initial time layer problem for quantum drift-diffusion model. J. Math. Anal. Appl. 343 (2008), 64-80. DOI 10.1016/j.jmaa.2008.01.015 | MR 2409458 | Zbl 1139.35010
[3] Chen, X., Chen, L., Jian, H.: Existence, semiclassical limit and long-time behavior of weak solution to quantum drift-diffusion model. Nonlinear Anal., Real World Appl. 10 (2009), 1321-1342. DOI 10.1016/j.nonrwa.2008.01.008 | MR 2502947 | Zbl 1171.35329
[4] Chen, L., Ju, Q.: Existence of weak solution and semiclassical limit for quantum drift-diffusion model. Z. Angew. Math. Phys. 58 (2007), 1-15. DOI 10.1007/s00033-005-0051-4 | MR 2293100 | Zbl 1107.35037
[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. MR 2181267 | Zbl 1245.35029
[6] Gardner, C. L.: The quantum hydrodynamic model for semiconductor devices. SIAM J. Appl. Math. 54 (1994), 409-427. DOI 10.1137/S003613999224042 | MR 1265234 | Zbl 0815.35111
[7] Gianazza, U., Savaré, G., Toscani, G.: The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation. Arch. Ration. Mech. Anal. 194 (2009), 133-220. DOI 10.1007/s00205-008-0186-5 | MR 2533926 | Zbl 1223.35264
[8] Gualdani, M. P., Jüngel, A., Toscani, G.: A nonlinear fourth-order parabolic equation with nonhomogeneous boundary conditions. SIAM. J. Math. Anal. 37 (2006), 1761-1779. DOI 10.1137/S0036141004444615 | MR 2213393 | Zbl 1102.35045
[9] Huang, F., Li, H.-L., Matsumura, A.: Existence and stability of steady-state of one-dimensional quantum hydrodynamic system for semiconductors. J. Differ. Equations 225 (2006), 1-25. DOI 10.1016/j.jde.2006.02.002 | MR 2228690 | Zbl 1160.76444
[10] Jia, Y., Li, H.: Large-time behavior of solutions of quantum hydrodynamic model for semiconductors. Acta Math. Sci., Ser. B, Engl. Ed. 26 (2006), 163-178. DOI 10.1016/S0252-9602(06)60038-6 | MR 2206278 | Zbl 1152.76505
[11] Jüngel, A.: A steady-state quantum Euler-Poisson system for potential flows. Commun. Math. Phys. 194 (1998), 463-479. DOI 10.1007/s002200050364 | MR 1627673 | Zbl 0916.76099
[12] Jüngel, A., Li, H.: Quantum Euler-Poisson systems: Existence of stationary states. Arch. Math., Brno 40 (2004), 435-456. MR 2129964 | Zbl 1122.35140
[13] Jüngel, A., Li, H.: Quantum Euler-Poisson systems: Global existence and exponential decay. Q. Appl. Math. 62 (2004), 569-600. DOI 10.1090/qam/2086047 | MR 2086047 | Zbl 1069.35012
[14] Jüngel, A., Li, H.-L., Matsumura, A.: The relaxation-time limit in the quantum hydrodynamic equations for semiconductors. J. Differ. Equations 225 (2006), 440-464. DOI 10.1016/j.jde.2005.11.007 | MR 2225796 | Zbl 1147.82364
[15] Jüngel, A., Matthes, D., Milišić, J. P.: Derivation of new quantum hydrodynamic equations using entropy minimization. SIAM J. Appl. Math. 67 (2006), 46-68. DOI 10.1137/050644823 | MR 2272614 | Zbl 1121.35117
[16] Jüngel, A., Milišić, J. P.: A simplified quantum energy-transport model for semiconductors. Nonlinear Anal., Real World Appl. 12 (2011), 1033-1046. DOI 10.1016/j.nonrwa.2010.08.026 | MR 2736191 | Zbl 1206.35152
[17] Jüngel, A., Pinnau, R.: A positivity-preserving numerical scheme for a nonlinear fourth order parabolic system. SIAM J. Numer. Anal. 39 (2001), 385-406. DOI 10.1137/S0036142900369362 | MR 1860272 | Zbl 0994.35047
[18] Jüngel, A., Violet, I.: The quasineutral limit in the quantum drift-diffusion equations. Asymptotic Anal. 53 (2007), 139-157. MR 2349559 | Zbl 1156.35077
[19] Kim, Y.-H., Ra, S., Kim, S.-C.: Asymptotic behavior of strong solutions of a simplified energy-transport model with general conductivity. Nonlinear Anal., Real World Appl. 59 (2021), Article ID 103261, 18 pages. DOI 10.1016/j.nonrwa.2020.103261 | MR 4177987 | Zbl 1468.35202
[20] Klainerman, S., Majda, A.: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math. 34 (1981), 481-524. DOI 10.1002/cpa.3160340405 | MR 0615627 | Zbl 0476.76068
[21] Li, H., Marcati, P.: Existence and asymptotic behavior of multi-dimensional quantum hydrodynamic model for semiconductors. Commun. Math. Phys. 245 (2004), 215-247. DOI 10.1007/s00220-003-1001-7 | MR 2039696 | Zbl 1075.82019
[22] Li, H., Zhang, G., Zhang, M., Hao, C.: Long-time self-similar asymptotic of the macroscopic quantum models. J. Math. Phys. 49 (2008), Article ID 073503, 14 pages. DOI 10.1063/1.2949082 | MR 2432041 | Zbl 1152.81528
[23] Mao, J., Zhou, F., Li, Y.: Some limit analysis in a one-dimensional stationary quantum hydrodynamic model for semiconductors. J. Math. Anal. Appl. 364 (2010), 186-194. DOI 10.1016/j.jmaa.2009.11.039 | MR 2576062 | Zbl 1186.35165
[24] Markowich, P. A., Ringhofer, C. A., Schmeiser, C.: Semiconductor Equations. Springer, Vienna (1990). DOI 10.1007/978-3-7091-6961-2 | MR 1063852 | Zbl 0765.35001
[25] Nirenberg, L.: On elliptic partial differential equations. Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 13 (1959), 115-162. MR 0109940 | Zbl 0088.07601
[26] Nishibata, S., Shigeta, N., Suzuki, M.: Asymptotic behaviors and classical limits of solutions to a quantum drift-diffusion model for semiconductors. Math. Models Methods Appl. Sci. 20 (2010), 909-936. DOI 10.1142/S0218202510004477 | MR 2659742 | Zbl 1193.82057
[27] Nishibata, S., Suzuki, M.: Initial boundary value problems for a quantum hydrodynamic model of semiconductors: Asymptotic behaviors and classical limits. J. Differ. Equations 244 (2008), 836-874. DOI 10.1016/j.jde.2007.10.035 | MR 2391346 | Zbl 1139.82042
[28] Ra, S., Hong, H.: The existence, uniqueness and exponential decay of global solutions in the full quantum hydrodynamic equations for semiconductors. Z. Angew. Math. Phys. 72 (2021), Article ID 107, 32 pages. DOI 10.1007/s00033-021-01540-8 | MR 4252285 | Zbl 1467.76078
[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. MR 3907819 | Zbl 07004591
[30] Simon, J.: Compact sets in the space $L^p(0,T;B)$. Ann. Mat. Pura Appl. (4) 146 (1986), Article ID 146, 32 pages. DOI 10.1007/BF01762360 | MR 0916688
[31] Zhang, G., Li, H.-L., Zhang, K.: Semiclassical and relaxation limits of bipolar quantum hydrodynamic model for semiconductors. J. Differ. Equations 245 (2008), 1433-1453. DOI 10.1016/j.jde.2008.06.019 | MR 2436449 | Zbl 1154.35071
Search
Partner of
