# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 2020, 40 (3) : 1775-1798. doi: 10.3934/dcds.2020093
##### 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. [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. [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. [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. [5] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford Lecture Series in Mathematics and its Applications, 26. Oxford University Press, Oxford, 2004. [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. [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. [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. [9] H.-L. Li, A. 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. [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. [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. [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. [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. [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. [15] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970. [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. [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. [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. [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. [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. [21] Y. Z. Sun, C. 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. [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. [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.

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##### 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. [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. [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. [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. [5] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford Lecture Series in Mathematics and its Applications, 26. Oxford University Press, Oxford, 2004. [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. [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. [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. [9] H.-L. Li, A. 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. [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. [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. [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. [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. [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. [15] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970. [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. [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. [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. [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. [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. [21] Y. Z. Sun, C. 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. [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. [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.
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