# American Institute of Mathematical Sciences

March 2018, 38(3): 1063-1102. doi: 10.3934/dcds.2018045

## The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅱ: 3D Navier-Stokes equations

 1 School of Mathematical Sciences, Jiangsu Provincial Key Laboratory, for Numerical Simulationof Large Scale Complex Systems, Nanjing Normal University, Nanjing 210023, China 2 Department of Mathematics and IMS, Nanjing University, Nanjing 210093, China

* Corresponding author: Huicheng Yin

Received  December 2016 Revised  September 2017 Published  December 2017

Fund Project: The authors were supported by the NSFC (No.11571177, No.11731007) and A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

We concern with the global existence and large time behavior of compressible fluids (including the inviscid gases, viscid gases, and Boltzmann gases) in an infinitely expanding ball. Such a problem is one of the interesting models in studying the theory of global smooth solutions to multidimensional compressible gases with time dependent boundaries and vacuum states at infinite time. Due to the conservation of mass, the fluid in the expanding ball becomes rarefied and eventually tends to a vacuum state meanwhile there are no appearances of vacuum domains in any part of the expansive ball, which is easily observed in finite time. In this paper, as the second part of our three papers, we will confirm this physical phenomenon for the compressible viscid fluids by obtaining the exact lower and upper bound on the density function.

Citation: Huicheng Yin, Lin Zhang. The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅱ: 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1063-1102. doi: 10.3934/dcds.2018045
##### References:
 [1] Y. Cho, H. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pures Appl., 83 (2004), 243-275. doi: 10.1016/j.matpur.2003.11.004. [2] Y. Cho and H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities, Manuscript Math., 120 (2006), 91-129. doi: 10.1007/s00229-006-0637-y. [3] H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids, J. Differential Equations, 190 (2003), 504-523. doi: 10.1016/S0022-0396(03)00015-9. [4] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers Inc., New York, 1948. [5] R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614. doi: 10.1007/s002220000078. [6] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, New York, 2004. [7] E. Feireisl, O. Kreml, S. Necasova, J. Neustupa and J. Stebel, Weak solutions to the barotropic Navier-Stokes system with slip boundary conditions in time dependent domains, J. Differential Equations, 254 (2013), 125-140. doi: 10.1016/j.jde.2012.08.019. [8] E. Feireisl, A. Novotny and H. Petzeltova, On the existence of globally defined weak solutions to the Navier-Stokes equations of compressible isentropic fluids, J. Math. Fluid Mech., 3 (2001), 358-392. doi: 10.1007/PL00000976. [9] D. Hoff, Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data, Trans. Amer. Math. Soc., 303 (1987), 169-181. [10] D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data, J. Differential Equations, 120 (1995), 215-254. doi: 10.1006/jdeq.1995.1111. [11] D. Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data, Arch. Rational Mech. Anal., 132 (1995), 1-14. doi: 10.1007/BF00390346. [12] D. Hoff, Discontinuous solution of the Navier-Stokes equations for multi-dimensional heat-conducting fluids, Arch. Ration. Mech. Anal., 193 (1997), 303-354. doi: 10.1007/s002050050055. [13] 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. [14] X. Huang, J. Li and Z. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Comm. Pure Appl. Math., 65 (2012), 549-585. doi: 10.1002/cpa.21382. [15] S. Jiang and P. Zhang, Global spherically symmetric solutions of the compressible isentropic Navier-Stokes equations, Comm. Math. Phys., 215 (2001), 559-581. doi: 10.1007/PL00005543. [16] Q. Jiu, Y. Wang and Z. Xin, Global well-posedness of the Cauchy problem of two-dimensional compressible Navier-Stokes equations in weighted spaces, J. Differential Equations, 255 (2013), 351-404. doi: 10.1016/j.jde.2013.04.014. [17] Y. Kagei and S. Kawashima, Local solvability of initial boundary value problem for a quasilinear hyperbolic-parabolic system, J. Hyperbolic Differential Equations, 3 (2006), 195-232. doi: 10.1142/S0219891606000768. [18] Y. Kagei and S. Kawashima, Stability of planar stationary solutions to the compressible Navier-Stokes equation on the half space, Comm. Math. Phys., 266 (2006), 401-430. doi: 10.1007/s00220-006-0017-1. [19] H. Li, J. Li and Zhouping Xin, Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations, Comm. Math. Phys., 281 (2008), 401-444. doi: 10.1007/s00220-008-0495-4. [20] P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2. Compressible Models, Oxford University Press, New York, 1998. [21] T. Liu, Z. Xin and T. Yang, Vacuum states of compressible flow, Discrete Contin. Dyn. Syst., 4 (1998), 1-32. [22] A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104. doi: 10.1215/kjm/1250522322. [23] A. Nishida and T. Matsumura, Initial boundary value problems for the equations of motion of compressible viscous and heat conductive fluids, Comm. Math. Phys., 89 (1983), 445-464. [24] A. Mellet and A. Vasseur, On the barotropic compressible Navier-Stokes equation, Comm. Partial Differential Equations, 32 (2007), 431-452. doi: 10.1080/03605300600857079. [25] J. Nash, Le probleme de Cauchy pour les equations differentielles d'un fluide general, Bull. Soc. Math. France., 90 (1962), 487-497. [26] D. Serre, Solutions faibles globales des équations de Navier-Stokes pour un fluide compressible, C. R. Acad. Sci. Paris Sér. I Math., 303 (1986), 639-642. [27] V. A. Vaigant and A. V. Kazhikhov, On the existence of global solutions of two-dimensional Navier-Stokes equations of a compressible viscous fluid, (Russian) Sibirsk. Mat. Zh., 36 (1995), 1283-1316, ⅱ; translation in Siberian Math. J. , 36 (1995), 1108-1141. [28] A. Valli, An existence theorem for compressible visous fluids, Ann. Mat. Pura Appl. (Ⅳ), 130 (1982), 197-213. doi: 10.1007/BF01761495. [29] Z. Xin, Blow-up of smooth solution to the compressible Navier-Stokes equations with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240. doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C. [30] Z. Xin and H. Yin, The transonic shock in a nozzle, 2-D and 3-D complete Euler systems, J. Differential Equations, 245 (2008), 1014-1085. doi: 10.1016/j.jde.2008.04.010. [31] G. Xu and H. Yin, The global existence and large time behaviore and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅰ: 3D Euler equations, Preprint, arXiv: 1706. 01183. [32] T. Yang, Z. Yao and C. Zhu, Compressible Navier-Stokes equations with density dependent viscosity and vacuum, Comm. Partial Differential Equations, 26 (2001), 965-981. doi: 10.1081/PDE-100002385. [33] T. Yang and C. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum, Comm. Math. Phys., 230 (2002), 329-363. doi: 10.1007/s00220-002-0703-6. [34] H. Yin and W. Zhao, The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅲ: 3D Boltzmann equation, J. Differential Equations, 264 (2018), 30-81. doi: 10.1016/j.jde.2017.08.064.

show all references

##### References:
 [1] Y. Cho, H. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pures Appl., 83 (2004), 243-275. doi: 10.1016/j.matpur.2003.11.004. [2] Y. Cho and H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities, Manuscript Math., 120 (2006), 91-129. doi: 10.1007/s00229-006-0637-y. [3] H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids, J. Differential Equations, 190 (2003), 504-523. doi: 10.1016/S0022-0396(03)00015-9. [4] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers Inc., New York, 1948. [5] R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614. doi: 10.1007/s002220000078. [6] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, New York, 2004. [7] E. Feireisl, O. Kreml, S. Necasova, J. Neustupa and J. Stebel, Weak solutions to the barotropic Navier-Stokes system with slip boundary conditions in time dependent domains, J. Differential Equations, 254 (2013), 125-140. doi: 10.1016/j.jde.2012.08.019. [8] E. Feireisl, A. Novotny and H. Petzeltova, On the existence of globally defined weak solutions to the Navier-Stokes equations of compressible isentropic fluids, J. Math. Fluid Mech., 3 (2001), 358-392. doi: 10.1007/PL00000976. [9] D. Hoff, Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data, Trans. Amer. Math. Soc., 303 (1987), 169-181. [10] D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data, J. Differential Equations, 120 (1995), 215-254. doi: 10.1006/jdeq.1995.1111. [11] D. Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data, Arch. Rational Mech. Anal., 132 (1995), 1-14. doi: 10.1007/BF00390346. [12] D. Hoff, Discontinuous solution of the Navier-Stokes equations for multi-dimensional heat-conducting fluids, Arch. Ration. Mech. Anal., 193 (1997), 303-354. doi: 10.1007/s002050050055. [13] 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. [14] X. Huang, J. Li and Z. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Comm. Pure Appl. Math., 65 (2012), 549-585. doi: 10.1002/cpa.21382. [15] S. Jiang and P. Zhang, Global spherically symmetric solutions of the compressible isentropic Navier-Stokes equations, Comm. Math. Phys., 215 (2001), 559-581. doi: 10.1007/PL00005543. [16] Q. Jiu, Y. Wang and Z. Xin, Global well-posedness of the Cauchy problem of two-dimensional compressible Navier-Stokes equations in weighted spaces, J. Differential Equations, 255 (2013), 351-404. doi: 10.1016/j.jde.2013.04.014. [17] Y. Kagei and S. Kawashima, Local solvability of initial boundary value problem for a quasilinear hyperbolic-parabolic system, J. Hyperbolic Differential Equations, 3 (2006), 195-232. doi: 10.1142/S0219891606000768. [18] Y. Kagei and S. Kawashima, Stability of planar stationary solutions to the compressible Navier-Stokes equation on the half space, Comm. Math. Phys., 266 (2006), 401-430. doi: 10.1007/s00220-006-0017-1. [19] H. Li, J. Li and Zhouping Xin, Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations, Comm. Math. Phys., 281 (2008), 401-444. doi: 10.1007/s00220-008-0495-4. [20] P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2. Compressible Models, Oxford University Press, New York, 1998. [21] T. Liu, Z. Xin and T. Yang, Vacuum states of compressible flow, Discrete Contin. Dyn. Syst., 4 (1998), 1-32. [22] A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104. doi: 10.1215/kjm/1250522322. [23] A. Nishida and T. Matsumura, Initial boundary value problems for the equations of motion of compressible viscous and heat conductive fluids, Comm. Math. Phys., 89 (1983), 445-464. [24] A. Mellet and A. Vasseur, On the barotropic compressible Navier-Stokes equation, Comm. Partial Differential Equations, 32 (2007), 431-452. doi: 10.1080/03605300600857079. [25] J. Nash, Le probleme de Cauchy pour les equations differentielles d'un fluide general, Bull. Soc. Math. France., 90 (1962), 487-497. [26] D. Serre, Solutions faibles globales des équations de Navier-Stokes pour un fluide compressible, C. R. Acad. Sci. Paris Sér. I Math., 303 (1986), 639-642. [27] V. A. Vaigant and A. V. Kazhikhov, On the existence of global solutions of two-dimensional Navier-Stokes equations of a compressible viscous fluid, (Russian) Sibirsk. Mat. Zh., 36 (1995), 1283-1316, ⅱ; translation in Siberian Math. J. , 36 (1995), 1108-1141. [28] A. Valli, An existence theorem for compressible visous fluids, Ann. Mat. Pura Appl. (Ⅳ), 130 (1982), 197-213. doi: 10.1007/BF01761495. [29] Z. Xin, Blow-up of smooth solution to the compressible Navier-Stokes equations with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240. doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C. [30] Z. Xin and H. Yin, The transonic shock in a nozzle, 2-D and 3-D complete Euler systems, J. Differential Equations, 245 (2008), 1014-1085. doi: 10.1016/j.jde.2008.04.010. [31] G. Xu and H. Yin, The global existence and large time behaviore and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅰ: 3D Euler equations, Preprint, arXiv: 1706. 01183. [32] T. Yang, Z. Yao and C. Zhu, Compressible Navier-Stokes equations with density dependent viscosity and vacuum, Comm. Partial Differential Equations, 26 (2001), 965-981. doi: 10.1081/PDE-100002385. [33] T. Yang and C. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum, Comm. Math. Phys., 230 (2002), 329-363. doi: 10.1007/s00220-002-0703-6. [34] H. Yin and W. Zhao, The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅲ: 3D Boltzmann equation, J. Differential Equations, 264 (2018), 30-81. doi: 10.1016/j.jde.2017.08.064.
A viscous fluid in a 3-D expanding ball
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