Figure 1.
Subsonic case of a viscous flow around a disc
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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 |
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.
Keywords:
Compressible Navier-Stokes equations,
circulatory flow,
weighted energy space,
global existence.
Mathematics Subject Classification:
Primary: 35L70, 35L65; Secondary: 35L67, 76N15.
Citation:
Huicheng Yin, Lin Zhang. The global stability of 2-D viscous axisymmetric circulatory flows. Discrete and Continuous Dynamical Systems,
2017, 37
(10)
: 5065-5083.
doi: 10.3934/dcds.2017219
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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. |
| [2] |
R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves Interscience Publishers Inc., New York, 1948. |
| [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. |
| [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. |
| [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. |
| [6] |
J. Li and H. Yin, On the blowup problem of unsteady 2-D circulatory flow, Preprint, 2014. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [12] |
P. L. Lions, Mathematical Topics in Fluid Dynamics Vol. 2, Compressible Models, Oxford Science Publication, Oxford, 1998. |
| [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.
|
| [14] |
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. |
| [15] |
S. Ding, H. Wen, L. 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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [20] |
C. Yonggeun, C. 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. |
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. |
| [2] |
R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves Interscience Publishers Inc., New York, 1948. |
| [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. |
| [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. |
| [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. |
| [6] |
J. Li and H. Yin, On the blowup problem of unsteady 2-D circulatory flow, Preprint, 2014. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [12] |
P. L. Lions, Mathematical Topics in Fluid Dynamics Vol. 2, Compressible Models, Oxford Science Publication, Oxford, 1998. |
| [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.
|
| [14] |
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. |
| [15] |
S. Ding, H. Wen, L. 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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [20] |
C. Yonggeun, C. 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. |
Figure 2.
Supersonic-sonic-subsonic case of a viscous flow around a disc
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