October  2017, 37(10): 5065-5083. doi: 10.3934/dcds.2017219

The global stability of 2-D viscous axisymmetric circulatory flows

1. 

School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China

2. 

Department of Mathematics and IMS, Nanjing University, Nanjing 210093, China

* Corresponding author: Lin Zhang

Received  February 2015 Revised  May 2017 Published  June 2017

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

In this paper, we study the global existence and stability problem of a perturbed viscous circulatory flow around a disc. This flow is described by two-dimensional Navier-Stokes equations. By introducing some suitable weighted energy space and establishing a priori estimates, we show that the 2-D circulatory flow is globally stable in time when the corresponding initial-boundary values are perturbed sufficiently small.

Citation: Huicheng Yin, Lin Zhang. The global stability of 2-D viscous axisymmetric circulatory flows. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5065-5083. doi: 10.3934/dcds.2017219
References:
[1]

S. Alinhac, Temps de vie des solutions réguliéres des équations d'Euler compressibles axisymétriques en dimension deux, Invent. Math., 111 (1993), 627-670. doi: 10.1007/BF01231301. Google Scholar

[2]

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

[3]

D. Cui and J. Li, On the existence and stability of 2-D perturbed steady subsonic circulatory flows, Sci. China Math., 54 (2011), 1421-1436. doi: 10.1007/s11425-011-4226-5. Google Scholar

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D. Hoff, Spherically symmetric solutions of the Navier-Stokes equations for compressible, isothermal flow with large, discontinuous initial data, Indiana Univ. Math. J., 41 (1992), 1225-1302. doi: 10.1512/iumj.1992.41.41060. Google Scholar

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J. Li and Z. Liang, On local classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier-Stokes equations with vacuum, J. Math. Pures Appl., 102 (2014), 640-671. doi: 10.1016/j.matpur.2014.02.001. Google Scholar

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J. Li and H. Yin, On the blowup problem of unsteady 2-D circulatory flow, Preprint, 2014.Google Scholar

[7]

C. H. Jun and K. Hyunseok, Global existence of the radially symmetric solutions of the Navier-Stokes equations for the isentropic compressible fluids, Math. Methods Appl. Sci., 28 (2005), 1-28. doi: 10.1002/mma.545. Google Scholar

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Y. Kagei and T. Kobayashi, Asymptotic behavior of solutions of the compressible Navier-Stokes equations on the half space, Arch. Ration. Mech. Anal., 177 (2005), 231-330. doi: 10.1007/s00205-005-0365-6. Google Scholar

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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

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Y. Kagei, Asymptotic behavior of solutions of the compressible Navier-Stokes equation around the plane Couette flow, J. Math. Fluid Mech., 13 (2011), 1-31. doi: 10.1007/s00021-009-0019-9. Google Scholar

[11]

Y. Kagei, Asymptotic behavior of solutions to the compressible Navier-Stokes equation around a parallel flow, Arch. Ration. Mech. Anal., 205 (2012), 585-650. doi: 10.1007/s00205-012-0516-5. Google Scholar

[12]

P. L. Lions, Mathematical Topics in Fluid Dynamics Vol. 2, Compressible Models, Oxford Science Publication, Oxford, 1998. Google Scholar

[13]

A. Matsumura and T. Nishida, 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

[14]

Q. JiuY. 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

[15]

S. DingH. WenL. Yao and C. Zhu, Global spherically symmetric classical solution to compressible Navier-Stokes equations with large initial data and vacuum, SIAM J. Math. Anal., 44 (2012), 1257-1278. doi: 10.1137/110836663. Google Scholar

[16]

T. C. Sideris, Delayed singularity formation in 2 D compressible flow, Amer. J. Math., 119 (1997), 371-422. doi: 10.1353/ajm.1997.0014. Google Scholar

[17]

S. Jiang, Global spherically symmetric solutions to the equations of a viscous polytropic ideal gas in an exterior domain, Comm. Math. Phys., 178 (1996), 339-374. doi: 10.1007/BF02099452. Google Scholar

[18]

S. Jiang and P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations, Comm. Math. Phys., 215 (2001), 559-581. doi: 10.1007/PL00005543. Google Scholar

[19]

A. Tani, On the first initial-boundary value problem of compressible viscous fluid motion, Publ. Res. Inst. Math. Sci. Kyoto Univ., 13 (1977), 193-253. Google Scholar

[20]

C. YonggeunC. H. Jun and K. Hyunseok, 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

show all references

References:
[1]

S. Alinhac, Temps de vie des solutions réguliéres des équations d'Euler compressibles axisymétriques en dimension deux, Invent. Math., 111 (1993), 627-670. doi: 10.1007/BF01231301. Google Scholar

[2]

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

[3]

D. Cui and J. Li, On the existence and stability of 2-D perturbed steady subsonic circulatory flows, Sci. China Math., 54 (2011), 1421-1436. doi: 10.1007/s11425-011-4226-5. Google Scholar

[4]

D. Hoff, Spherically symmetric solutions of the Navier-Stokes equations for compressible, isothermal flow with large, discontinuous initial data, Indiana Univ. Math. J., 41 (1992), 1225-1302. doi: 10.1512/iumj.1992.41.41060. Google Scholar

[5]

J. Li and Z. Liang, On local classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier-Stokes equations with vacuum, J. Math. Pures Appl., 102 (2014), 640-671. doi: 10.1016/j.matpur.2014.02.001. Google Scholar

[6]

J. Li and H. Yin, On the blowup problem of unsteady 2-D circulatory flow, Preprint, 2014.Google Scholar

[7]

C. H. Jun and K. Hyunseok, Global existence of the radially symmetric solutions of the Navier-Stokes equations for the isentropic compressible fluids, Math. Methods Appl. Sci., 28 (2005), 1-28. doi: 10.1002/mma.545. Google Scholar

[8]

Y. Kagei and T. Kobayashi, Asymptotic behavior of solutions of the compressible Navier-Stokes equations on the half space, Arch. Ration. Mech. Anal., 177 (2005), 231-330. doi: 10.1007/s00205-005-0365-6. Google Scholar

[9]

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

[10]

Y. Kagei, Asymptotic behavior of solutions of the compressible Navier-Stokes equation around the plane Couette flow, J. Math. Fluid Mech., 13 (2011), 1-31. doi: 10.1007/s00021-009-0019-9. Google Scholar

[11]

Y. Kagei, Asymptotic behavior of solutions to the compressible Navier-Stokes equation around a parallel flow, Arch. Ration. Mech. Anal., 205 (2012), 585-650. doi: 10.1007/s00205-012-0516-5. Google Scholar

[12]

P. L. Lions, Mathematical Topics in Fluid Dynamics Vol. 2, Compressible Models, Oxford Science Publication, Oxford, 1998. Google Scholar

[13]

A. Matsumura and T. Nishida, 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

[14]

Q. JiuY. 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

[15]

S. DingH. WenL. Yao and C. Zhu, Global spherically symmetric classical solution to compressible Navier-Stokes equations with large initial data and vacuum, SIAM J. Math. Anal., 44 (2012), 1257-1278. doi: 10.1137/110836663. Google Scholar

[16]

T. C. Sideris, Delayed singularity formation in 2 D compressible flow, Amer. J. Math., 119 (1997), 371-422. doi: 10.1353/ajm.1997.0014. Google Scholar

[17]

S. Jiang, Global spherically symmetric solutions to the equations of a viscous polytropic ideal gas in an exterior domain, Comm. Math. Phys., 178 (1996), 339-374. doi: 10.1007/BF02099452. Google Scholar

[18]

S. Jiang and P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations, Comm. Math. Phys., 215 (2001), 559-581. doi: 10.1007/PL00005543. Google Scholar

[19]

A. Tani, On the first initial-boundary value problem of compressible viscous fluid motion, Publ. Res. Inst. Math. Sci. Kyoto Univ., 13 (1977), 193-253. Google Scholar

[20]

C. YonggeunC. H. Jun and K. Hyunseok, 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

Figure 1.  Subsonic case of a viscous flow around a disc
Figure 2.  Supersonic-sonic-subsonic case of a viscous flow around a disc
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