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The combined quasineutral and low Mach number limit of the Navier-Stokes-Poisson system. (English) Zbl 1407.35164

Summary: In this paper, the quasineutral limit of the compressible Navier-Stokes-Poisson system in the critical \(L^p\)-type Besov space is considered. More precisely, we will show that the solution of compressible Navier-Stokes-Poisson equations will converge to that of incompressible Navier-Stokes equations in the \(L^p\) framework when the Debye length is proportional to the Mach number and tends to zero. Moreover, the convergence rate will be obtained.

MSC:

35Q35 PDEs in connection with fluid mechanics
76W05 Magnetohydrodynamics and electrohydrodynamics
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[1] Degond, P.: Mathematical modelling of microelectronics semiconductor devices, some current topics on nonlinear conservation laws. In: AMS/IP Stud. Adv. Math., vol. 15, Am. Math. Sco., Providence, RI, pp. 77-110 (2007)
[2] Wang, S.: Quasineutral limit of Euler-Poisson system with and without viscosity. Commun. Partial Differ. Equ. 29(3-4), 419-456 (2004) · Zbl 1140.35551
[3] Wang, S., Jiang, S.: The convergence of the Navier-Stokes-Poisson system to the incompressible Euler equations. Commun. Partial Differ. Equ. 31(4-6), 571-591 (2006) · Zbl 1137.35416 · doi:10.1080/03605300500361487
[4] Gasser, I., Marcati, P.: The combined relaxation and vanishing Debye length limit in the hydrodynamic model for semiconductors. Math. Methods Appl. Sci. 24(2), 81-92 (2001) · Zbl 0974.35119 · doi:10.1002/1099-1476(20010125)24:2<81::AID-MMA198>3.0.CO;2-X
[5] Donatelli, D., Marcati, P.: A quasineutral type limit for the Navier-Stokes-Poisson system with large data. Nonlinearity 21(1), 135-148 (2008) · Zbl 1136.35068 · doi:10.1088/0951-7715/21/1/008
[6] Donatelli, D., Feireisl, E., Novotny, A.: On the vanishing electron-mass limit in plasma hydrodynamics in unbounded media. J. Nonlinear Sci. 22(6), 985-1012 (2012) · Zbl 1259.35157 · doi:10.1007/s00332-012-9134-5
[7] Donatelli, D., Feireisl, E., Novotny, A.: Scale analysis of a hydrodynamic model of plasma. M3AS Math. Models Methods Appl. Sci. 25(2), 371394 (2015) · Zbl 1308.76339
[8] Donatelli, D., Marcati, P.: Analysis of oscillations and defect measures for the quasineutral limit in plasma physics. Arch. Rat. Mech. Anal. 206(1), 159-188 (2012) · Zbl 1256.35052 · doi:10.1007/s00205-012-0531-6
[9] Donatelli, D., Marcati, P.: Quasineutral limit, dispersion and oscillations for Korteweg type fluids. SIAM J. Math. Anal. 47(3), 2265-2282 (2015) · Zbl 1320.35277 · doi:10.1137/140987651
[10] Ju, Q., Li, F., Li, H.: The quasineutral limit of compressible Navier-Stokes-Poisson system with heat conductivity and general initial data. J. Differ. Equ. 247(1), 203-224 (2009) · Zbl 1170.35075 · doi:10.1016/j.jde.2009.02.019
[11] Ju, Q., Li, F., Wang, S.: Convergence of the Navier-Stokes-Poisson system to the incompressible Navier-Stokes equations. J. Math. Phys. 49(7), 073515, 8 (2008) · Zbl 1152.76356 · doi:10.1063/1.2956495
[12] Ju, Q., Li, Y., Wang, S.: Rate of convergence from the Navier-Stokes-Poisson system to the incompressible Euler equations. J. Math. Phys. 50(1), 013533, 12 (2009) · Zbl 1200.76044 · doi:10.1063/1.3054866
[13] Danchin, R.: Zero Mach number limit in critical spaces for compressible Navier-Stokes equations. Ann. Sci. École Norm. Sup. (4) 35(1), 27-75 (2002) · Zbl 1048.35054 · doi:10.1016/S0012-9593(01)01085-0
[14] Danchin, R.: Zero Mach number limit for compressible flows with periodic boundary conditions. Am. J. Math. 124(6), 1153-1219 (2002) · Zbl 1048.35075 · doi:10.1353/ajm.2002.0036
[15] Desjardins, B., Grenier, E., Lions, P.L., Masmoudi, N.: Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions. J. Math. Pures Appl. 78, 461-71 (1999) · Zbl 0992.35067 · doi:10.1016/S0021-7824(99)00032-X
[16] Danchin, R., He, L.: The incompressible limit in \[L^p\] Lp type critical spaces. Math. Ann. 366(3-4), 1365-1402 (2016) · Zbl 1354.35096 · doi:10.1007/s00208-016-1361-x
[17] Lions, P.L., Masmoudi, N.: Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl. 77, 585-627 (1998) · Zbl 0909.35101 · doi:10.1016/S0021-7824(98)80139-6
[18] Bahouri, H., Chemin, J.Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343, p. xvi+523. Springer, Heidelberg (2011) · Zbl 1227.35004
[19] Chemin, J.Y.: Théorèmes d’unicité pour le système de Navier-Stokes tridimensionnel, (French) [Uniqueness theorems for the three-dimensional Navier-Stokes system]. J. Anal. Math. 77, 27-50 (1999) · Zbl 0938.35125 · doi:10.1007/BF02791256
[20] Kozono, H., Yamazaki, M.: Semilinear heat equations and the Navier-Stokes equations with distributions in new function spaces as initial data. Commun. Partial Differ. Equ. 19, 959-1014 (1994) · Zbl 0803.35068 · doi:10.1080/03605309408821042
[21] Cannone, M., Meyer, Y., Planchon, F.: Solutions auto-similaires des équations de Navier-Stokes. (French) [Self-similar solutions of Navier-Stokes equations] Séminaire sur les Équations aux Dérivées Partielles, 1993-1994, Exp. No. VIII, p 12. École Polytech, Palaiseau (1994) · Zbl 0882.35090
[22] Donatelli, D.: Local and global existence for the coupled Navier-Stokes-Poisson problem. Quart. Appl. Math. 61, 345-361 (2003) · Zbl 1039.35075 · doi:10.1090/qam/1976375
[23] Zhang, Y., Tan, Z.: On the existence of solutions to the Navier-Stokes-Poisson equations of a two-dimensional compressible flow. Math. Methods Appl. Sci. 30(3), 305-329 (2007) · Zbl 1107.76065 · doi:10.1002/mma.786
[24] Hao, C., Li, H.: Global existence for compressible Navier-Stokes-Poisson equations in three and higher dimensions. J. Differ. Equ. 246(12), 4791-4812 (2009) · Zbl 1173.35098 · doi:10.1016/j.jde.2008.11.019
[25] Chikami, N., Danchin, R.: On the global existence and time decay estimates in critical spaces for the Navier-Stokes-Poisson system. Math. Nachr. 290(13), 1939-1970 (2017) · Zbl 1375.35367 · doi:10.1002/mana.201600238
[26] Chikami, N., Ogawa, T.: Well-posedness of the compressible Navier-Stokes-Poisson system in the critical Besov spaces. J. Evol. Equ. 17(2), 717-747 (2017) · Zbl 1378.35213 · doi:10.1007/s00028-016-0334-6
[27] Zheng, X.: Global well-posedness for the compressible Navier-Stokes-Poisson system in the \[L^p\] Lp framework. Nonlinear Anal. 75(10), 4156-4175 (2012) · Zbl 1239.35119 · doi:10.1016/j.na.2012.03.006
[28] Wu, Z., Wang, W.: Pointwise estimates for bipolar compressible Navier-Stokes-Poisson system in dimension three. Arch. Ration. Mech. Anal. 226(2), 587-638 (2017) · Zbl 1373.35258 · doi:10.1007/s00205-017-1140-1
[29] Tan, Z., Yang, T., Zhao, H., Zou, Q.: Global solutions to the one-dimensional compressible Navier-Stokes-Poisson equations with large data. SIAM J. Math. Anal. 45(2), 547-571 (2013) · Zbl 1291.35237 · doi:10.1137/120876174
[30] Li, H.L., Matsumura, A., Zhang, G.: Optimal decay rate of the compressible Navier-Stokes-Poisson system in \[R^3\] R3. Arch. Ration. Mech. Anal. 196(2), 681-713 (2010) · Zbl 1205.35201 · doi:10.1007/s00205-009-0255-4
[31] Wang, Y.: Decay of the Navier-Stokes-Poisson equations. J. Differ. Equ. 253(1), 273-297 (2012) · Zbl 1239.35117 · doi:10.1016/j.jde.2012.03.006
[32] Bie, Q., Wang, Q., Yao, Z.: Optimal decay rate for the compressible Navier-Stokes-Poisson system in the critical \[L^p\] Lp framework. J. Differ. Equ. 263(12), 8391-8417 (2017) · Zbl 1375.35360 · doi:10.1016/j.jde.2017.08.041
[33] Danchin, R.: Global existence in critical spaces for compressible Navier-Stokes equations. Invent. Math. 141(3), 579-614 (2000) · Zbl 0958.35100 · doi:10.1007/s002220000078
[34] Charve, F., Danchin, R.: A global existence result for the compressible Navier-Stokes equations in the critical \[L^p\] Lp framework. Arch. Ration. Mech. Anal. 198(1), 233-271 (2010) · Zbl 1229.35167 · doi:10.1007/s00205-010-0306-x
[35] Chen, Q., Miao, C., Zhang, Z.: Global well-posedness for compressible Navier-Stokes equations with highly oscillating initial velocity. Commun. Pure Appl. Math. 63(9), 1173-1224 (2010) · Zbl 1202.35002
[36] Haspot, B.: Existence of global strong solutions in critical spaces for barotropic viscous fluids. Arch. Ration. Mech. Anal. 202(2), 427-460 (2011) · Zbl 1427.76230 · doi:10.1007/s00205-011-0430-2
[37] Chemin, J.Y.: Localization in Fourier space and Navier-Stokes system. Phase space analysis of partial differential equations, Vol. I, 53-135, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa (2004) · Zbl 1081.35074
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