March  2020, 40(3): 1775-1798. doi: 10.3934/dcds.2020093

Existence and a blow-up criterion of solution to the 3D compressible Navier-Stokes-Poisson equations with finite energy

Department of Mathematics and Information Technology, The Education University of Hong Kong, 10 Lo Ping Road, Tai Po, New Territories, Hong Kong, China

* Corresponding author: Anthony Suen

Received  July 2019 Published  December 2019

We study the low-energy solutions to the 3D compressible Navier-Stokes-Poisson equations. We first obtain the existence of smooth solutions with small $ L^2 $-norm and essentially bounded densities. No smallness assumption is imposed on the $ H^4 $-norm of the initial data. Using a compactness argument, we further obtain the existence of weak solutions which may have discontinuities across some hypersurfaces in $ \mathbb R^3 $. We also provide a blow-up criterion of solutions in terms of the $ L^\infty $-norm of density.

Citation: Anthony Suen. Existence and a blow-up criterion of solution to the 3D compressible Navier-Stokes-Poisson equations with finite energy. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1775-1798. doi: 10.3934/dcds.2020093
References:
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S. Cordier and E. Grenier, Quasineutral limit of an Euler-Poisson system arising from plasma physics, Comm. Partial Differential Equations, 25 (2000), 1099-1113.  doi: 10.1080/03605300008821542.  Google Scholar

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J. Lin, J. Zhang and J. Zhao, On the motion of three-dimensional compressible isentropic flows with large external potential forces and vacuum, arXiv: 1111.2114. Google Scholar

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P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2. Compressible Models, Oxford Lecture Series in Mathematics and its Applications, 10. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

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A. Matsumura and T. Nishida, The initial value problem for the equation of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A Math. Sci., 55 (1979), 337-342.  doi: 10.3792/pjaa.55.337.  Google Scholar

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E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970.  Google Scholar

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A. Suen, A blow-up criterion for the 3D compressible magnetohydrodynamics in terms of density, Discrete Contin. Dyn. Syst., 33 (2013), 3791-3805.  doi: 10.3934/dcds.2013.33.3791.  Google Scholar

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A. Suen, Global solutions of the Navier-Stokes equations for isentropic flow with large external potential force, Z. Angew. Math. Phys., 64 (2013), 767-784.  doi: 10.1007/s00033-012-0263-3.  Google Scholar

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A. Suen, Existence of global weak solution to Navier-Stokes equations with large external potential force and general pressure, Math. Methods Appl. Sci., 37 (2014), 2716-2727.  doi: 10.1002/mma.3012.  Google Scholar

[19]

A. Suen, Existence and uniqueness of low-energy weak solutions to the compressible 3D magnetohydrodynamics equations, J. Diff. Eqns., (2019). doi: 10.1016/j.jde.2019.09.037.  Google Scholar

[20]

A. Suen and D. Hoff, Global low-energy weak solutions of the equations of 3D compressible magnetohydrodynamics, Arch. Rational Mechanics Ana., 205 (2012), 27-58.  doi: 10.1007/s00205-012-0498-3.  Google Scholar

[21]

Y. Z. SunC. Wang and Z. F. Zhang, A Beale-Kato-Majda Blow-up criterion for the 3-D compressible Navier-Stokes equations, J. Math. Pures Appl., 95 (2011), 36-47.  doi: 10.1016/j.matpur.2010.08.001.  Google Scholar

[22]

Y. H. Zhang and Z. Tan, On the existence of solutions to the Navier-Stokes-Poisson equations of a two-dimensional compressible flow, Math. Methods Appl. Sci., 30 (2007), 305-329.  doi: 10.1002/mma.786.  Google Scholar

[23]

W. P. Ziemer, Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation, Graduate Texts in Mathematics, 120. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

show all references

References:
[1]

S. Cordier and E. Grenier, Quasineutral limit of an Euler-Poisson system arising from plasma physics, Comm. Partial Differential Equations, 25 (2000), 1099-1113.  doi: 10.1080/03605300008821542.  Google Scholar

[2]

P. Degond, Mathematical modelling of microelectronics semiconductor devices, Some Current Topics on Nonlinear Conservation Laws, AMS/IP Stud. Adv. Math., Amer. Math. Soc., Providence, RI, 15 (2000), 77-110.   Google Scholar

[3]

D. Donatelli, Local and global existence for the coupled Navier-Stokes-Poisson problem, Quart. Appl. Math., 61 (2003), 345-361.  doi: 10.1090/qam/1976375.  Google Scholar

[4]

D. Donatelli and K. Trivisa, From the dynamics of gaseous stars to the incompressible euler equations, J. Differential Equations, 245 (2008), 1356-1385.  doi: 10.1016/j.jde.2008.05.018.  Google Scholar

[5]

E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford Lecture Series in Mathematics and its Applications, 26. Oxford University Press, Oxford, 2004.  Google Scholar

[6]

D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional, compressible flow with discontinuous initial data, J. Diff. Eqns., 120 (1995), 215-254.  doi: 10.1006/jdeq.1995.1111.  Google Scholar

[7]

D. Hoff, Compressible flow in a half-space with navier boundary conditions, J. Math. Fluid Mech., 7 (2005), 315-338.  doi: 10.1007/s00021-004-0123-9.  Google Scholar

[8]

D. Hoff, Uniqueness of weak solutions of the Navier-Stokes equations of multidimensional compressible flow, SIAM J. Math. Anal., 37 (2006), 1742-1760.  doi: 10.1137/040618059.  Google Scholar

[9]

H.-L. LiA. Matsumura and G. J. Zhang, Optimal decay rate of the compressible Navier-Stokes-Poisson system in $ \mathbb R^3$, Arch. Ration. Mech. Anal., 196 (2010), 681-713.  doi: 10.1007/s00205-009-0255-4.  Google Scholar

[10]

J. Li and A. Matsumura, On the Navier-Stokes equations for three-dimensional compressible barotropic flow subject to large external potential forces with discontinuous initial data, J. Math. Pures Appl. (9), 95 (2011), 495–512. doi: 10.1016/j.matpur.2010.12.002.  Google Scholar

[11]

J. Lin, J. Zhang and J. Zhao, On the motion of three-dimensional compressible isentropic flows with large external potential forces and vacuum, arXiv: 1111.2114. Google Scholar

[12]

P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2. Compressible Models, Oxford Lecture Series in Mathematics and its Applications, 10. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[13]

A. Matsumura and T. Nishida, The initial value problem for the equation of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A Math. Sci., 55 (1979), 337-342.  doi: 10.3792/pjaa.55.337.  Google Scholar

[14]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Kyoto Univ., 20 (1980), 67-104.  doi: 10.1215/kjm/1250522322.  Google Scholar

[15]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970.  Google Scholar

[16]

A. Suen, A blow-up criterion for the 3D compressible magnetohydrodynamics in terms of density, Discrete Contin. Dyn. Syst., 33 (2013), 3791-3805.  doi: 10.3934/dcds.2013.33.3791.  Google Scholar

[17]

A. Suen, Global solutions of the Navier-Stokes equations for isentropic flow with large external potential force, Z. Angew. Math. Phys., 64 (2013), 767-784.  doi: 10.1007/s00033-012-0263-3.  Google Scholar

[18]

A. Suen, Existence of global weak solution to Navier-Stokes equations with large external potential force and general pressure, Math. Methods Appl. Sci., 37 (2014), 2716-2727.  doi: 10.1002/mma.3012.  Google Scholar

[19]

A. Suen, Existence and uniqueness of low-energy weak solutions to the compressible 3D magnetohydrodynamics equations, J. Diff. Eqns., (2019). doi: 10.1016/j.jde.2019.09.037.  Google Scholar

[20]

A. Suen and D. Hoff, Global low-energy weak solutions of the equations of 3D compressible magnetohydrodynamics, Arch. Rational Mechanics Ana., 205 (2012), 27-58.  doi: 10.1007/s00205-012-0498-3.  Google Scholar

[21]

Y. Z. SunC. Wang and Z. F. Zhang, A Beale-Kato-Majda Blow-up criterion for the 3-D compressible Navier-Stokes equations, J. Math. Pures Appl., 95 (2011), 36-47.  doi: 10.1016/j.matpur.2010.08.001.  Google Scholar

[22]

Y. H. Zhang and Z. Tan, On the existence of solutions to the Navier-Stokes-Poisson equations of a two-dimensional compressible flow, Math. Methods Appl. Sci., 30 (2007), 305-329.  doi: 10.1002/mma.786.  Google Scholar

[23]

W. P. Ziemer, Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation, Graduate Texts in Mathematics, 120. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

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