Czechoslovak Mathematical Journal, first online, pp. 1-11


Blow-up for the compressible isentropic Navier-Stokes-Poisson equations

Jianwei Dong, Junhui Zhu, Yanping Wang

Received March 3, 2018.   Published online September 5, 2019.

Abstract:  We will show the blow-up of smooth solutions to the Cauchy problems for compressible unipolar isentropic Navier-Stokes-Poisson equations with attractive forcing and compressible bipolar isentropic Navier-Stokes-Poisson equations in arbitrary dimensions under some restrictions on the initial data. The key of the proof is finding the relations between the physical quantities and establishing some differential inequalities.
Keywords:  compressible isentropic Navier-Stokes-Poisson equations; unipolar; bipolar; smooth solution; blow-up
Classification MSC:  35Q35; 35B44
DOI:  10.21136/CMJ.2019.0156-18

PDF available at:  Springer   Institute of Mathematics CAS

References:
[1] H. Cai, Z. Tan: Existence and stability of stationary solutions to the compressible Navier-Stokes-Poisson equations. Nonlinear Anal., Real World Appl. 32 (2016), 260-293. DOI 10.1016/j.nonrwa.2016.04.010 | MR 3514925 | Zbl 1348.35183
[2] H. Cai, Z. Tan: Asymptotic stability of stationary solutions to the compressible bipolar Navier-Stokes-Poisson equations. Math. Methods Appl. Sci. 40 (2017), 4493-4513. DOI 10.1002/mma.4320 | MR 3672880 | Zbl 1373.35233
[3] Y. Cho, B. Jin: Blow up of viscous heat-conducting compressible flows. J. Math. Anal. Appl. 320 (2006), 819-826. DOI 10.1016/j.jmaa.2005.08.005 | MR 2225997 | Zbl 1121.35110
[4] D. P. Du, J. Y. Li, K. J. Zhang: Blow-up of smooth solutions to the Navier-Stokes equations for compressible isothermal fluids. Commun. Math. Sci. 11 (2013), 541-546. DOI 10.4310/CMS.2013.v11.n2.a11 | MR 3002564 | Zbl 1305.76089
[5] L. Hsiao, H. L. Li: Compressible Navier-Stokes-Poisson equations. Acta Math. Sci., Ser. B 30 (2010), 1937-1948. DOI 10.1016/S0252-9602(10)60184-1 | MR 2778703 | Zbl 1240.35406
[6] L. Hsiao, H. L. Li, T. Yang, C. Zou: Compressible non-isentropic bipolar Navier-Stokes-Poisson system in $\mathbb{R}^3$. Acta Math. Sci., Ser. B 31 (2011), 2169-2194. DOI 10.1016/S0252-9602(11)60392-5 | MR 2931498 | Zbl 1265.35265
[7] F. Jiang, Z. Tan: Blow-up of viscous compressible reactive self-gravitating gas. Acta Math. Appl. Sin., Engl. Ser. 28 (2012), 401-408. DOI 10.1007/s10255-012-0152-8 | MR 2914383 | Zbl 1359.35131
[8] Q. S. Jiu, Y. X. Wang, Z. P. Xin: Remarks on blow-up of smooth solutions to the compressible fluid with constant and degenerate viscosities. J. Differ. Equations 259 (2015), 2981-3003. DOI 10.1016/j.jde.2015.04.007 | MR 3360663 | Zbl 1319.35194
[9] N. A. Lai: Blow up of classical solutions to the isentropic compressible Navier-Stokes equations. Nonlinear Anal., Real World Appl. 25 (2015), 112-117. DOI 10.1016/j.nonrwa.2015.03.005 | MR 3351014 | Zbl 1327.35299
[10] O. Rozanova: Blow-up of smooth highly decreasing at infinity solutions to the compressible Navier-Stokes equations. J. Differ. Equations 245 (2008), 1762-1774. DOI 10.1016/j.jde.2008.07.007 | MR 2433485 | Zbl 1154.35070
[11] Z. Tan, Y. J. Wang: Blow-up of smooth solutions to the Navier-Stokes equations of compressible viscous heat-conducting fluids. J. Aust. Math. Soc. 88 (2010), 239-246. DOI 10.1017/S144678871000008X | MR 2629933 | Zbl 1191.35210
[12] T. Tang, Z. J. Zhang: Blow-up of smooth solution to the compressible Navier-Stokes-Poisson equations. Bull. Malays. Math. Sci. Soc. 39 (2016), 1487-1497. DOI 10.1007/s40840-015-0256-4 | MR 3549976 | Zbl 1358.35133
[13] G. W. Wang, B. L. Guo: Blow-up of the smooth solutions to the compressible Navier-Stokes equations. Math. Methods Appl. Sci. 40 (2017), 5262-5272. DOI 10.1002/mma.4384 | MR 3689262 | Zbl 1383.35034
[14] Y. Z. Wang, K. Y. Wang: Asymptotic behavior of classical solutions to the compressible Navier-Stokes-Poisson equations in three and higher dimensions. J. Differ. Equations 259 (2015), 25-47. DOI 10.1016/j.jde.2015.01.042 | MR 3335919 | Zbl 1317.35211
[15] H. Z. Xie: Blow-up of smooth solutions to the Navier-Stokes-Poisson equations. Math. Methods Appl. Sci. 34 (2011), 242-248. DOI 10.1002/mma.1353 | MR 2779329 | Zbl 1206.35201
[16] Z. P. Xin: Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density. Commun. Pure Appl. Math. 51 (1998), 229-240. DOI 10.1002/(SICI)1097-0312(199803)51:3%3C229::AID-CPA1%3E3.0.CO;2-C | MR 1488513 | Zbl 0937.35134
[17] Z. P. Xin, W. Yan: On blowup of classical solutions to the compressible Navier-Stokes equations. Commun. Math. Phys. 321 (2013), 529-541. DOI 10.1007/s00220-012-1610-0 | MR 3063918 | Zbl 1287.35059
[18] Z. Y. Zhao, Y. P. Li: Global existence and optimal decay rate of the compressible bipolar Navier-Stokes-Poisson equations with external force. Nonlinear Anal., Real World Appl. 16 (2014), 146-162. DOI 10.1016/j.nonrwa.2013.09.014 | MR 3123807 | Zbl 1297.35195
[19] C. Zou: Asymptotical behavior of bipolar non-isentropic compressible Navier-Stokes-Poisson system. Acta Math. Appl. Sin., Engl. Ser. 32 (2016), 813-832. DOI 10.1007/s10255-016-0596-3 | MR 3552850 | Zbl 1364.35291

Affiliations:   Jianwei Dong (corresponding author), Junhui Zhu, Yanping Wang, School of Mathematics, Zhengzhou University of Aeronautics, 100 Kexue Ave, Zhongyuan Qu, Zhengzhou 450015, P. R. China, e-mail: dongjianweiccm@163.com


 
PDF available at: