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Asymptotical behavior of bipolar non-isentropic compressible Navier-Stokes-Poisson system

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Abstract

The bipolar non-isentropic compressible Navier-Stokes-Poisson (BNSP) system is investigated in R3 in the present paper, and the optimal L 2 time decay rate for the global classical solution is established. It is shown that the total densities, total momenta and total temperatures of two carriers converge to the equilibrium states at the rate

$${\left( {1 + t} \right)^{ - \frac{3}{4} + \varepsilon }}$$

in L 2-norm for any small and fix ε > 0. But, both the difference of densities and the difference of temperatures of two carriers decay at the optimal rate

$${\left( {1 + t} \right)^{ - \frac{3}{4}}}$$

, and the difference of momenta decays at the optimal rate

$${\left( {1 + t} \right)^{ - \frac{1}{4}}}$$

. This phenomenon on the charge transport shows the essential difference between the non-isentropic unipolar NSP and the bipolar NSP system.

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References

  1. Donatelli, D. Local and global existence for the coupled Navier-Stokes-Poisson problem. Quart. Appl. Math., 61: 345–361 (2003)

    MathSciNet  MATH  Google Scholar 

  2. Donatelli, D., Marcati, P. A quasineutral type limit for the Navier-Stokes-Poisson system with large data. Nonlinearity, 21(1): 135–148 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Duan, R.J., Liu, H., Ukai, S., Yang, T., Zhao, H.J. Opitmal convergence rates for the compressible Navier- Stokes equations with potential forces. Math. Models Methods App. Sci., 17(5): 737–758 (2007)

    Article  MATH  Google Scholar 

  4. Ducomet, B., Feireisl, E., Petzeltova, H., Skraba, I.S. Global in time weak solution for compressible barotropic self-gravitating fluids. Discrete Contin. Dyn. Syst., 11(1): 113–130 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ducomet, B., Zlotnik, A. Stabilization and stability for the spherically symmetric Navier-Stokes-Poisson system. Appl. Math. Lett., 18(10): 1190–1198 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hao, C., Li, H.L. Global Existence for compressible Navier-Stokes-Poisson equations in three and higher dimensions. J. Diff. Eqns., 246: 4791–4812 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hoff, D., Zumbrun, K. Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow. Indiana Univ. Math. J., 44: 603–676 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hsiao, L., Li, H.L., Yang, T., Zou, C. Compressible Non-isentropic Bipolar Navier-Stokes-Poisson System in R3. preprint ???

  9. 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., 247: 203–224 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kobayashi, T., Suzuki, T. Weak solutions to the Navier-Stokes-Poisson equations. ???preprint, 2004

  11. Li, D.L. The Green’s function of the Navier-Stokes equations for gas dynamics in R3. Comm. Math. Phys., 257: 579–619 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Li, H.L., Matsumura, A., Zhang, G. Optimal decay rate of the compressible Navier-Stokes-Poisson system in R3. Arch. Ration. Mech. Anal., 196: 681–713 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Li, H.-L., Yang, T., Zou, C. Time asymptotic behavior of the bipolar Navier-Stokes-Poisson system. Acta. Math. Sci. Ser. B, 29: 1721–1736 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, H.L., Zhang, T. Large time behavior of isentropic compressible Navier-Stokes system in R3. preprint ???

  15. Liu T.P., Wang, W.K. The pointwise estimates of diffusion waves for the Navier-Stokes equations in odd multi-dimensions. Comm. Math. Phys., 196: 145–173 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Markowich, P.A., Ringhofer, C.A., Schmeiser, C. Semiconductor Equations. Springer-Verlag, New York, 1990

  17. Matsumura, A., Nishida, T. The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids. Proc. Japan. Acad. Ser. A, 55: 337–342 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  18. Matsumura, A., Nishida. The initial value problem for the equations of motion of viscous and heatconductive gases. J. Math. Kyoto. Univ., 20: 67–104 (1980)

    MathSciNet  MATH  Google Scholar 

  19. Ponce, G. Global existence of small solution to a class of nonlinear evolution equations. Nonlinear Anal., 9: 339–418 (1985)

    Article  MathSciNet  Google Scholar 

  20. Wang, S., Jiang, S. The convergence of the Navier-Stokes-Poisson system to the incompressible Euler equations. Comm. Partial Diff. Eqns., 31: 571–591 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wang, W., Wu, Z. Pointwise estimates of solution for the Navier-Stokes-Poisson equations in multidimensions. J. Diff. Eqns., 248: 1617–1636 (2010)

    Article  MATH  Google Scholar 

  22. Zhang, G., Li, H.L., Zhu, C. Optimal decay rate of the non-isentropic compressible Navier-Stokes-Poisson system in R3. J. Diff. Eqns., accepted ???

  23. 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., 30(3): 305–329 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanics and Aerospace Engineering, COE, Peking University, Beijing, 100871, China

    Chen Zou

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  1. Chen Zou
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Correspondence to Chen Zou.

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Supported by the National Natural Science Foundation of China (No. 10872004).

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Zou, C. Asymptotical behavior of bipolar non-isentropic compressible Navier-Stokes-Poisson system. Acta Math. Appl. Sin. Engl. Ser. 32, 813–832 (2016). https://doi.org/10.1007/s10255-016-0596-3

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  • DOI: https://doi.org/10.1007/s10255-016-0596-3

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