American Institute of Mathematical Sciences

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

The global stability of 2-D viscous axisymmetric circulatory flows

Huicheng Yin 1, and  Lin Zhang 2,,

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.

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 & Continuous Dynamical Systems, 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

[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. 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

[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.  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. 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.  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. 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

[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.  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. 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.  Google Scholar

Figure 1.  Subsonic case of a viscous flow around a disc
Figure Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

[1]

Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085
[2]

Ling-Bing He, Li Xu. On the compressible Navier-Stokes equations in the whole space: From non-isentropic flow to isentropic flow. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3489-3530. doi: 10.3934/dcds.2021005
[3]

Yuming Qin, Lan Huang, Zhiyong Ma. Global existence and exponential stability in $H^4$ for the nonlinear compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1991-2012. doi: 10.3934/cpaa.2009.8.1991
[4]

Sun-Ho Choi. Weighted energy method and long wave short wave decomposition on the linearized compressible Navier-Stokes equation. Networks & Heterogeneous Media, 2013, 8 (2) : 465-479. doi: 10.3934/nhm.2013.8.465
[5]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602
[6]

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
[7]

Daoyuan Fang, Bin Han, Matthias Hieber. Local and global existence results for the Navier-Stokes equations in the rotational framework. Communications on Pure & Applied Analysis, 2015, 14 (2) : 609-622. doi: 10.3934/cpaa.2015.14.609
[8]

Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations II: Global existence of small solutions. Evolution Equations & Control Theory, 2012, 1 (1) : 217-234. doi: 10.3934/eect.2012.1.217
[9]

Zhenhua Guo, Zilai Li. Global existence of weak solution to the free boundary problem for compressible Navier-Stokes. Kinetic & Related Models, 2016, 9 (1) : 75-103. doi: 10.3934/krm.2016.9.75
[10]

Pavel I. Plotnikov, Jan Sokolowski. Optimal shape control of airfoil in compressible gas flow governed by Navier-Stokes equations. Evolution Equations & Control Theory, 2013, 2 (3) : 495-516. doi: 10.3934/eect.2013.2.495
[11]

Shuguang Shao, Shu Wang, Wen-Qing Xu, Bin Han. Global existence for the 2D Navier-Stokes flow in the exterior of a moving or rotating obstacle. Kinetic & Related Models, 2016, 9 (4) : 767-776. doi: 10.3934/krm.2016015
[12]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5383-5405. doi: 10.3934/dcdsb.2020348
[13]

Bingkang Huang, Lusheng Wang, Qinghua Xiao. Global nonlinear stability of rarefaction waves for compressible Navier-Stokes equations with temperature and density dependent transport coefficients. Kinetic & Related Models, 2016, 9 (3) : 469-514. doi: 10.3934/krm.2016004
[14]

Xulong Qin, Zheng-An Yao. Global solutions of the free boundary problem for the compressible Navier-Stokes equations with density-dependent viscosity. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1041-1052. doi: 10.3934/cpaa.2010.9.1041
[15]

Teng Wang, Yi Wang. Large-time behaviors of the solution to 3D compressible Navier-Stokes equations in half space with Navier boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (7&8) : 2811-2838. doi: 10.3934/cpaa.2021080
[16]

Daniel Coutand, Steve Shkoller. Turbulent channel flow in weighted Sobolev spaces using the anisotropic Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations. Communications on Pure & Applied Analysis, 2004, 3 (1) : 1-23. doi: 10.3934/cpaa.2004.3.1
[17]

Yingshan Chen, Shijin Ding, Wenjun Wang. Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5287-5307. doi: 10.3934/dcds.2016032
[18]

Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675
[19]

Jing Wang, Lining Tong. Stability of boundary layers for the inflow compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2595-2613. doi: 10.3934/dcdsb.2012.17.2595
[20]

Misha Perepelitsa. An ill-posed problem for the Navier-Stokes equations for compressible flows. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 609-623. doi: 10.3934/dcds.2010.26.609

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (155)
  • HTML views (73)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy