# American Institute of Mathematical Sciences

May  2019, 18(3): 1281-1302. doi: 10.3934/cpaa.2019062

## Stability and $L^{p}$ convergence rates of planar diffusion waves for three-dimensional bipolar Euler-Poisson systems

 Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China

* Corresponding author: Jie Liao

Received  May 2018 Revised  August 2018 Published  November 2018

Fund Project: The research is partially supported by NSFC grants 11671134 and 11871335

In the paper, we consider a three-dimensional bipolar hydrodynamic model from semiconductor devices and plasmas. This system takes the form of Euler-Poisson with electric field and relaxation term added to the momentum equations. We first construct the planar diffusion waves. Next we show the global existence of smooth solutions for the initial value problem of three-dimensional bipolar Euler-Poisson systems when the initial data are near the planar diffusive waves. Finally, we also establish the $L^p(p∈[2,+∞])$ convergence rates of the solutions toward the planar diffusion waves. A frequency decomposition, approximate Green function and delicate energy method are used to prove our results.

Citation: Yeping Li, Jie Liao. Stability and $L^{p}$ convergence rates of planar diffusion waves for three-dimensional bipolar Euler-Poisson systems. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1281-1302. doi: 10.3934/cpaa.2019062
##### References:
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Marcati, The combined relaxation and vanishing Debye length limit in the hydrodynamic model for semiconductors, Math. Meth. Appl. Sci., 24 (2001), 81-92. doi: 10.1002/1099-1476(20010125)24:2<81::AID-MMA198>3.3.CO;2-O. Google Scholar [6] F.-M. Huang and Y.-P. Li, Large time behavior and quasineutral limit of solutions to a bipolar hydrodynamic model with large data and vacuum, Dis. Contin. Dyn. Sys., A24 (2009), 455-470. doi: 10.3934/dcds.2009.24.455. Google Scholar [7] F.-M. Huang, M. Mei and Y. Wang, Large time behavior of solution to n-dimensional bipolar hydrodynamic model for semiconductors, SIAM J. Math. Anal., 43 (2011), 1595-1630. doi: 10.1137/100810228. Google Scholar [8] F.-M. Huang, M. Mei, Y. Wang and T. Yang, Large-time behavior of solution to the bipolar hydrodynamic model of semiconductors with boundary effects, SIAM J. Math. Anal., 44 (2012), 1134-1164. doi: 10.1137/110831647. Google Scholar [9] L. Hsiao and K.-J. Zhang, The global weak solution and relaxation limits of the initial boundary value problem to the bipolar hydrodynamic model for semiconductors, Math. Models Methods Appl. Sci., 10 (2000), 1333-1361. doi: 10.1142/S0218202500000653. Google Scholar [10] L. Hsiao and K.-J. Zhang, The relaxation of the hydrodynamic model for semiconductors to the drift-diffusion equations, J. Differential Equations, 165 (2000), 315-354. doi: 10.1006/jdeq.2000.3780. Google Scholar [11] L. Hsiao and T.-P. Liu, Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Comm. Math. Phys., 143 (1992), 599-605. Google Scholar [12] A. Jüngel, Quasi-hydrodynamic semiconductor equations, in Progress in Nonlinear Differential Equations, Birkhäuser, 2001. doi: 10.1007/978-3-0348-8334-4. Google Scholar [13] Q.-C. Ju, Global smooth solutions to the multidimensional hydrodynamic model for plasmas with insulating boundary conditions, J. Math. Anal. Appl., 336 (2007), 888-904. doi: 10.1016/j.jmaa.2007.03.038. Google Scholar [14] Q.-C. Ju, H.-L. Li, Y. Li and S. Jiang, Quasi-neutral limit of the two-fluid Euler-Poisson system, Comm. Pure Appl. Anal., 9 (2010), 1577-1590. doi: 10.3934/cpaa.2010.9.1577. Google Scholar [15] C. Lattanzio, On the 3-D bipolar isentropic Euler-Poisson model for semiconductors and the drift-diffusion limit, Math. Models Methods Appl. Sci., 10 (2000), 351-360. doi: 10.1142/S0218202500000215. Google Scholar [16] Y.-P. Li and T. Zhang, Relaxation-time limit of the multidimensional bipolar hydrodynamic model in Besov space, J. Differential Equations, 251 (2011), 3143-3162. doi: 10.1016/j.jde.2011.07.018. Google Scholar [17] Y.-P. Li, Diffusion relaxation limit of a bipolar isentropic hydrodynamic model for semiconductors, J. Math. Anal. Appl., 336 (2007), 1341-1356. doi: 10.1016/j.jmaa.2007.03.068. Google Scholar [18] Y.-P. Li, Existence and some limit analysis of stationary solutions for a multi-dimensional bipolar Euler-Poisson system, Dis. Contin. Dyn. Sys., B16 (2011), 345-360. doi: 10.3934/dcdsb.2011.16.345. Google Scholar [19] Y.-P. Li and X.-F. Yang, Global existence and asymptotic behavior of the solutions to the three dimensional bipolar Euler-Poisson systems, J. Differential Equations, 252 (2012), 768-791. doi: 10.1016/j.jde.2011.08.008. Google Scholar [20] J. Liao, W. K. Wang and T. Yang, $L^{p}$ convergence rates of planar waves for multi-dimensional Euler equations with damping, J. Differential Equations, 247 (2009), 303-329. doi: 10.1016/j.jde.2009.03.011. Google Scholar [21] P. A. Markowich, C. A. Ringhofev and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Wien, New York, 1990. doi: 10.1007/978-3-7091-6961-2. Google Scholar [22] R. Natalini, The bipolar hydrodynamic model for semiconductors and the drift-diffusion equation, J. Math. Anal. Appl., 198 (1996), 262-281. doi: 10.1006/jmaa.1996.0081. Google Scholar [23] Y.-J. Peng and J. Xu, Global well-posedness of the hydrodynamic model for two-carrier plasmas, J. Differential Equations, 255 (2013), 3447-3471. doi: 10.1016/j.jde.2013.07.045. Google Scholar [24] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. Google Scholar [25] A. Sitnko and V. Malnev, Plasma Physics Theory, London: Chapman & Hall, 1995. Google Scholar [26] N. Tsuge, Existence and uniqueness of stationary solutions to a one-dimensional bipolar hydrodynamic models of semiconductors, Nonlinear Anal. TMA, 73 (2010), 779-787. doi: 10.1016/j.na.2010.04.015. Google Scholar [27] Z.-G. Wu and W.-K. Wang, Decay of the solution to the bipolar Euler-Poisson system with damping in $R^3$, Commun. Math. Sci., 12 (2014), 1257-1276. doi: 10.4310/CMS.2014.v12.n7.a5. Google Scholar [28] W. Wang and T. Yang, Existence and stability of planar diffusion waves for $2-D$ Euler equations with damping, J. Differential Equations, 242 (2007), 40-71. doi: 10.1016/j.jde.2007.07.002. Google Scholar [29] C. Zhu and H. Hattori, Stability of steady state solutions for an isentropic hydrodynamic model of semiconductors of two species, J. Differential Equations, 166 (2000), 1-32. doi: 10.1006/jdeq.2000.3799. Google Scholar [30] F. Zhou and Y.-P. Li, Existence and some limits of stationary solutions to a one-dimensional bipolar Euler-Poisson system, J. Math. Anal. Appl., 351 (2009), 480-490. doi: 10.1016/j.jmaa.2008.10.032. Google Scholar

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##### References:
 [1] G. Ali and A. Jüngel, Global smooth solutions to the multi-dimensional hydrodynamic model for two-carrier plasma, J. Differential Equations, 190 (2003), 663-685. doi: 10.1016/S0022-0396(02)00157-2. Google Scholar [2] G. Ali and L. Chen, The zero-electron-mass limit in the Euler-Poisson system for both well- and ill-prepared initial data, Nonlinearity, 24 (2011), 2745-2761. doi: 10.1088/0951-7715/24/10/005. Google Scholar [3] D. Donatelli, M. Mei, B. Rubino and R. Sampalmieri, Asymptotic behavior of solutions to Euler-Poisson equations for bipolar hydrodynamic model of semiconductors, J. Differential Equations, 255 (2013), 3150-3184. doi: 10.1016/j.jde.2013.07.027. Google Scholar [4] I. Gasser, L. Hsiao and H.-L. Li, Large time behavior of solutions of the bipolar hydrodynamical model for semiconductors, J. Differential Equations, 192 (2003), 326-359. doi: 10.1016/S0022-0396(03)00122-0. Google Scholar [5] I. Gasser and P. Marcati, The combined relaxation and vanishing Debye length limit in the hydrodynamic model for semiconductors, Math. Meth. Appl. Sci., 24 (2001), 81-92. doi: 10.1002/1099-1476(20010125)24:2<81::AID-MMA198>3.3.CO;2-O. Google Scholar [6] F.-M. Huang and Y.-P. Li, Large time behavior and quasineutral limit of solutions to a bipolar hydrodynamic model with large data and vacuum, Dis. Contin. Dyn. Sys., A24 (2009), 455-470. doi: 10.3934/dcds.2009.24.455. Google Scholar [7] F.-M. Huang, M. Mei and Y. Wang, Large time behavior of solution to n-dimensional bipolar hydrodynamic model for semiconductors, SIAM J. Math. Anal., 43 (2011), 1595-1630. doi: 10.1137/100810228. Google Scholar [8] F.-M. Huang, M. Mei, Y. Wang and T. Yang, Large-time behavior of solution to the bipolar hydrodynamic model of semiconductors with boundary effects, SIAM J. Math. Anal., 44 (2012), 1134-1164. doi: 10.1137/110831647. Google Scholar [9] L. Hsiao and K.-J. Zhang, The global weak solution and relaxation limits of the initial boundary value problem to the bipolar hydrodynamic model for semiconductors, Math. Models Methods Appl. Sci., 10 (2000), 1333-1361. doi: 10.1142/S0218202500000653. Google Scholar [10] L. Hsiao and K.-J. Zhang, The relaxation of the hydrodynamic model for semiconductors to the drift-diffusion equations, J. Differential Equations, 165 (2000), 315-354. doi: 10.1006/jdeq.2000.3780. Google Scholar [11] L. Hsiao and T.-P. Liu, Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Comm. Math. Phys., 143 (1992), 599-605. Google Scholar [12] A. Jüngel, Quasi-hydrodynamic semiconductor equations, in Progress in Nonlinear Differential Equations, Birkhäuser, 2001. doi: 10.1007/978-3-0348-8334-4. Google Scholar [13] Q.-C. Ju, Global smooth solutions to the multidimensional hydrodynamic model for plasmas with insulating boundary conditions, J. Math. Anal. Appl., 336 (2007), 888-904. doi: 10.1016/j.jmaa.2007.03.038. Google Scholar [14] Q.-C. Ju, H.-L. Li, Y. Li and S. Jiang, Quasi-neutral limit of the two-fluid Euler-Poisson system, Comm. Pure Appl. Anal., 9 (2010), 1577-1590. doi: 10.3934/cpaa.2010.9.1577. Google Scholar [15] C. Lattanzio, On the 3-D bipolar isentropic Euler-Poisson model for semiconductors and the drift-diffusion limit, Math. Models Methods Appl. Sci., 10 (2000), 351-360. doi: 10.1142/S0218202500000215. Google Scholar [16] Y.-P. Li and T. Zhang, Relaxation-time limit of the multidimensional bipolar hydrodynamic model in Besov space, J. Differential Equations, 251 (2011), 3143-3162. doi: 10.1016/j.jde.2011.07.018. Google Scholar [17] Y.-P. Li, Diffusion relaxation limit of a bipolar isentropic hydrodynamic model for semiconductors, J. Math. Anal. Appl., 336 (2007), 1341-1356. doi: 10.1016/j.jmaa.2007.03.068. Google Scholar [18] Y.-P. Li, Existence and some limit analysis of stationary solutions for a multi-dimensional bipolar Euler-Poisson system, Dis. Contin. Dyn. Sys., B16 (2011), 345-360. doi: 10.3934/dcdsb.2011.16.345. Google Scholar [19] Y.-P. Li and X.-F. Yang, Global existence and asymptotic behavior of the solutions to the three dimensional bipolar Euler-Poisson systems, J. Differential Equations, 252 (2012), 768-791. doi: 10.1016/j.jde.2011.08.008. Google Scholar [20] J. Liao, W. K. Wang and T. Yang, $L^{p}$ convergence rates of planar waves for multi-dimensional Euler equations with damping, J. Differential Equations, 247 (2009), 303-329. doi: 10.1016/j.jde.2009.03.011. Google Scholar [21] P. A. Markowich, C. A. Ringhofev and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Wien, New York, 1990. doi: 10.1007/978-3-7091-6961-2. Google Scholar [22] R. Natalini, The bipolar hydrodynamic model for semiconductors and the drift-diffusion equation, J. Math. Anal. Appl., 198 (1996), 262-281. doi: 10.1006/jmaa.1996.0081. Google Scholar [23] Y.-J. Peng and J. Xu, Global well-posedness of the hydrodynamic model for two-carrier plasmas, J. Differential Equations, 255 (2013), 3447-3471. doi: 10.1016/j.jde.2013.07.045. Google Scholar [24] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. Google Scholar [25] A. Sitnko and V. Malnev, Plasma Physics Theory, London: Chapman & Hall, 1995. Google Scholar [26] N. Tsuge, Existence and uniqueness of stationary solutions to a one-dimensional bipolar hydrodynamic models of semiconductors, Nonlinear Anal. TMA, 73 (2010), 779-787. doi: 10.1016/j.na.2010.04.015. Google Scholar [27] Z.-G. Wu and W.-K. Wang, Decay of the solution to the bipolar Euler-Poisson system with damping in $R^3$, Commun. Math. Sci., 12 (2014), 1257-1276. doi: 10.4310/CMS.2014.v12.n7.a5. Google Scholar [28] W. Wang and T. Yang, Existence and stability of planar diffusion waves for $2-D$ Euler equations with damping, J. Differential Equations, 242 (2007), 40-71. doi: 10.1016/j.jde.2007.07.002. Google Scholar [29] C. Zhu and H. Hattori, Stability of steady state solutions for an isentropic hydrodynamic model of semiconductors of two species, J. Differential Equations, 166 (2000), 1-32. doi: 10.1006/jdeq.2000.3799. Google Scholar [30] F. Zhou and Y.-P. Li, Existence and some limits of stationary solutions to a one-dimensional bipolar Euler-Poisson system, J. Math. Anal. Appl., 351 (2009), 480-490. doi: 10.1016/j.jmaa.2008.10.032. Google Scholar
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