American Institute of Mathematical Sciences

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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

Huicheng Yin 1,, and  Lin Zhang 2,

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.

Keywords: Compressible Navier-Stokes equations, expanding ball, weighted energy estimates, global existence, vacuum state.
Mathematics Subject Classification: 35L70, 35L65, 35L67, 76N15.
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, 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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T. Liu, Z. Xin and T. Yang, Vacuum states of compressible flow, Discrete Contin. Dyn. Syst., 4 (1998), 1-32.   Google Scholar

[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.  Google Scholar

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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.   Google Scholar

[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.  Google Scholar

[25]

J. Nash, Le probleme de Cauchy pour les equations differentielles d'un fluide general, Bull. Soc. Math. France., 90 (1962), 487-497.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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A. Valli, An existence theorem for compressible visous fluids, Ann. Mat. Pura Appl. (Ⅳ), 130 (1982), 197-213.  doi: 10.1007/BF01761495.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers Inc., New York, 1948.   Google Scholar
[5]

R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614.  doi: 10.1007/s002220000078.  Google Scholar

[6] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, New York, 2004.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[9]

D. Hoff, Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data, Trans. Amer. Math. Soc., 303 (1987), 169-181.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20] P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2. Compressible Models, Oxford University Press, New York, 1998.   Google Scholar
[21]

T. Liu, Z. Xin and T. Yang, Vacuum states of compressible flow, Discrete Contin. Dyn. Syst., 4 (1998), 1-32.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[25]

J. Nash, Le probleme de Cauchy pour les equations differentielles d'un fluide general, Bull. Soc. Math. France., 90 (1962), 487-497.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[28]

A. Valli, An existence theorem for compressible visous fluids, Ann. Mat. Pura Appl. (Ⅳ), 130 (1982), 197-213.  doi: 10.1007/BF01761495.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  A viscous fluid in a 3-D expanding ball
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