Acta mathematica scientia, Series B >
COMPRESSIBLE NON-ISENTROPIC BIPOLAR NAVIER-STOKES-POISSON SYSTEM IN R3
Received date: 2011-07-26
Online published: 2011-11-20
Supported by
The research of L. Hsiao was partially supported by the NSFC (10871134). The research of H. Li was partially supported by the NSFC (10871134, 10910401059), and the funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality (PHR201006107). The research of T. Yang is supported by the General Research Fund of Hong Kong, City Univ. 103108.
The compressible non-isentropic bipolar Navier–Stokes–Poisson (BNSP) sys-tem is investigated in R3 in the present paper, and the optimal time decay rates of global strong solution are shown. For initial data being a perturbation of equilibrium state in Hl(R3)∩˙B −s
1,1(R3) for l ≥4 and s ∈ (0, 1], it is shown that the density and temperature for each charged particle (like electron or ion) decay at the same optimal rate (1 + t)−3/4 , but the momentum for each particle decays at the optimal rate (1 + t)−1/4−s/2 which is slower than the rate (1+t)−3/4−s/2 for the compressible Navier–Stokes (NS) equations [19] for same initial data. However, the total momentum tends to the constant state at the rate (1+t)−3/4 as well, due to the interplay interaction of charge particles which counteracts the influence of electric field.
Hsiao Ling , Li Hailiang , Yang Tong , Zou Chen . COMPRESSIBLE NON-ISENTROPIC BIPOLAR NAVIER-STOKES-POISSON SYSTEM IN R3[J]. Acta mathematica scientia, Series B, 2011 , 31(6) : 2169 -2194 . DOI: 10.1016/S0252-9602(11)60392-5
[1] Deckelnick K. L2-decay for the compressible Navier-Stokes equations in unbounded domains. Comm Partial Diff Eqns, 1993, 18: 1445–1476
[2] Donatelli D. Local and global existence for the coupled Navier-Stokes-Poisson problem. Quart Appl Math, 2003, 61: 345–361
[3] Donatelli D, Marcati P. A quasineutral type limit for the Navier-Stokes-Poisson system with large data. Nonlinearity, 2008, 21(1): 135–148
[4] Duan R -J, Liu H, Ukai S, Yang T. Optimal Lp-Lq convergence rates for the compressible Navier-Stokes equations with potential force. J Diff Eqns, 2007, 238(1): 220–233
[5] Ducomet B, Feireisl E, Petzeltova H, Skraba I. S. Global in time weak solution for compressible barotropic self-gravitating fluids. Discrete Continous Dynamical System, 2004, 11(1): 113–130
[6] Ducomet B, Zlotnik A. Stabilization and stability for the spherically symmetric Navier-Stokes-Poisson system. Appl Math Lett, 2005, 18(10): 1190–1198
[7] Guo Y. Smooth irrotational fows in the large to the Euler-Poisson system. Comm Math Phys, 1998, 195: 249–265
[8] Hao C, Li H. Global Existence for compressible Navier-Stokes-Poisson equations in three and higher dimensions. J Diff Eqns, 2009, 246: 4791–4812
[9] Hoff D, Zumbrun K. Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow. Indiana Univ Math J, 1995, 44: 603–676
[10] Ju Q, Li F, Li H -L. The quasineutral limit of Navier-Stokes-Poisson system with heat conductivity and general initial data. J Diff Eqns, 2009, 247: 203–224
[11] Ju Q, Li H -L, Li Y. The quasineutral limit of full two fluid Euler-Poisson system. Comm Pure Appl Anal, 2010, 9(6): 1577–1590
[12] Kagei Y, Kobayashi T. Asymptotic behavior of solutions of the compressible Navier-Stokes equations on the half space. Arch Rat Mech Anal, 2005, 177: 231–330
[13] Kobayashi T. Some estimates of solutions for the equations of motion of compressible viscous fluid in an exterior domain in R3. J Diff Eqns, 2002, 184: 587–619
[14] Kobayashi T, Shibata Y. Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in R3. Comm Math Phys, 1999, 200: 621–659
[15] Kobayashi T, Suzuki T. Weak solutions to the Navier-Stokes-Poisson equations, 2004, preprint.
[16] Li D L. The Green’s function of the Navier-Stokes equations for gas dynamics in R3. Comm Math Phys, 2005, 257: 579–619
[17] Li H -L, Matsumura A, Zhang G. Optimal decay rate of the compressible Navier-Stokes-Poisson system in R3. Arch Rat Mech Anal, 2010, 196: 681–713
[18] Li H -L, Yang T, Zou C. Time asymptotic behavior of the bipolar Navier-Stokes-Poisson system. Acta Math Sci, 2009, 29B: 1721–1736
[19] Li H -L, Zhang T. Large time behavior of isentropic compressible Navier–Stokes system in R3. preprint.
[20] Li H -L, Zhang T. Large time behavior of solutions to 3D compressible Navier-Stokes-Poisson system. Science China Math, doi: 10.1007/s, 1425-011-4280-z
[21] Lin Y -Q. Global well-posedness of compressible Navier-Stokes-Poisson system in multi-dimensions. preprint.
[22] Liu T -P, Wang W -K. The pointwise estimates of diffusion waves for the Navier-Stokes equations in odd multi-dimensions. Comm Math Phys, 1998, 196: 145–173
[23] Markowich P A, Ringhofer C A, Schmeiser C. Semiconductor Equations. Springer, 1990
[24] Matsumura A, Nishida T. The initial value problem for the equations of motion of viscous and heat-conductive gases. J Math Kyoto Univ, 1980, 20: 67–104
[25] Ponce G. Global existence of small solution to a class of nonlinear evolution equations. Nonlinear Anal, 1985, 9: 339–418
[26] Wang S, Jiang S. The convergence of the Navier-Stokes-Poisson system to the incompressible Euler equa-tions. Comm Partial Diff Eqns, 2006, 31: 571–591
[27] Wang W, Wu Z. Pointwise estimates of solution for the non-isentropic Navier-Stokes-Poisson equations in multi-dimensions. J Diff Eqns, 2010, 248: 1617–1636
[28] Zhang G, Li H -L, Zhu C. Optimal decay rate of the non-isentropic compressible Navier-Stokes-Poisson system in R3. J Diff Eqns, 2011, 250(2): 866–891
[29] Zhang Y, Tan Z. On the existence of solutions to the Navier-Stokes-Poisoon equations of a two-dimensional compressible flow. Math Methods Appl Sci, 2007, 30(3): 305–329
[30] Zou C. Asymptotical behaviors of bipolar non-isentropic Navier-Stokes-Poisson system. Acta Math Appl Sin Engl Ser, (accepted)
[31] Zou C. Large time behaviors of the isentropic bipolar compressible Navier-Stokes-Poisson system. Acta Math Sci, 2011, 31B(5): 1725–1740
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