# American Institute of Mathematical Sciences

March  2019, 18(2): 751-794. doi: 10.3934/cpaa.2019037

## Compressible viscous flows in a symmetric domain with complete slip boundary: The nonlinear stability of uniformly rotating states with small angular velocities

 Department of Mathematics, Texas A & M University, College Station, Texas, USA

Received  February 2018 Revised  July 2018 Published  October 2018

Fund Project: This work is part of the doctoral dissertation of the author under the supervision of Professor Zhouping Xin at the Institute of Mathematical Sciences of the Chinese University of Hong Kong, Hong Kong. The author would like to express great gratitude to Prof. Xin for his kindly support and professional guidance.

This work is devoted to studying the global behavior of viscous flows contained in a symmetric domain with complete slip boundary. In such a scenario, the boundary no longer provides friction and therefore the perturbation of the angular velocity lacks decaying structure. In fact, we show the existence of uniformly rotating solutions as steady states for the compressible Navier-Stokes equations. By manipulating the conservation law of angular momentum, we establish a suitable Korn's type inequality to control the perturbation and show the stability of the uniformly rotating solutions with a small angular velocity. In particular, the initial perturbation which preserves the angular momentum will be stable in the sense that the global strong solution to the Navier-Stokes equations exists and the perturbation is uniformly bounded and small in time.

Citation: Xin Liu. Compressible viscous flows in a symmetric domain with complete slip boundary: The nonlinear stability of uniformly rotating states with small angular velocities. Communications on Pure and Applied Analysis, 2019, 18 (2) : 751-794. doi: 10.3934/cpaa.2019037
##### References:
 [1] R. A. Adams and J. J. Fournier, Sobolev Spaces, 2nd edition, Academic Press, 2003. [2] S. N. Antontsev and H. B. de Oliveira, Navier-Stokes equations with absorption under slip boundary conditions: existence, uniqueness and extinction in time, RIMS Kôkyûroku Bessatsu B, 1 (2007), 21-42. [3] N. T. Bishop, The Poincaré inequality for a vector field with zero tangential or normal component on the boundary, Quaest. Math., 11 (1988), 195-199. [4] H. Brezis and P. Mironescu, Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces, J. Evol. Equations, 1 (2001), 387-404.  doi: 10.1007/PL00001378. [5] G.-Q. Chen and Z. Qian, A study of the Navier-Stokes equations with the kinematic and Navier boundary conditions, Indiana Univ. Math. J., 59 (2010), 721-760.  doi: 10.1512/iumj.2010.59.3898. [6] Y. Cho and B. J. Jin, Blow-up of viscous heat-conducting compressible flows, J. Math. Anal. Appl., 320 (2006), 819-826.  doi: 10.1016/j.jmaa.2005.08.005. [7] L. Desvillettes and C. Villani, On a variant of Korn's inequality arising in statistical mechanics, ESAIM Control. Optim. Calc. Var., 8 (2002), 603-619.  doi: 10.1051/cocv:2002036. [8] S. Ding, Q. Li and Z. Xin, Stability analysis for the incompressible Navier-Stokes equations with Navier boundary conditions, Journal of Mathematical Fluid Mechanics, 20 (2018), 603-629 doi: 10.1007/s00021-017-0337-2. [9] Z. Ding, A proof of the trace theorem of Sobolev spaces on lipschitz domains, Proc. Am. Math. Soc., 124 (1996), 591-601.  doi: 10.1090/S0002-9939-96-03132-2. [10] R. Farwig, Stationary solutions of compressible Navier-Stokes equations with slip boundary condition, Commun. Partial Differ. Equations, 14 (1989), 1579-1606.  doi: 10.1080/03605308908820667. [11] K. O. Friedrichs, On the boundary-value problems of the theory of elasticity and Korn's Inequality, Ann. Math., 48 (1947), 441. doi: 10.2307/1969180. [12] Y. Guo and I. Tice, Compressible, inviscid Rayleigh-Taylor instability, Indiana Univ. Math. J., 60 (2011), 677-711.  doi: 10.1512/iumj.2011.60.4193. [13] Y. Guo and I. Tice, Almost exponential decay of periodic viscous surface waves without surface tension, Arch. Ration. Mech. Anal., 207 (2013), 459-531.  doi: 10.1007/s00205-012-0570-z. [14] D. Hoff, Local solutions of a compressible flow problem with Navier boundary conditions in general three-dimensional domains, SIAM J. Math. Anal., 44 (2012), 633-650.  doi: 10.1137/110827065. [15] C. Horgan and L. Payne, On inequalities of Korn, Friedrichs and Babu|s|ka-Aziz, Arch. Ration. Mech. Anal., 82 (1983), 165-179.  doi: 10.1007/BF00250935. [16] F. Huang, J. Li and Z. Xin, Convergence to equilibria and blowup behavior of global strong solutions to the Stokes approximation equations for two-dimensional compressible flows with large data, J. Math. Pures Appl., 86 (2006), 471-491.  doi: 10.1016/j.matpur.2006.10.001. [17] 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, Commun. Pure Appl. Math., 65 (2012), 549-585.  doi: 10.1002/cpa.21382. [18] N. Itaya, On the Cauchy problems for the system of fundamental equations describing the movement of compressible viscous fluid, Kodai Math. Sem. Rep., 23 (1971), 60-120. [19] J. Jang, I. Tice and Y. Wang, The compressible viscous surface-internal wave problem: Stability and vanishing surface tension limit, Commun. Math. Phys., 343 (2016), 1039-1113.  doi: 10.1007/s00220-016-2603-1. [20] O. A. Ladyzhenskaya and R. Silverman, The Mathematical Theory of Viscous Incompressible Flow, 2nd edition, Gordon & Breach Science Publishers Ltd, New York, NY, 1969. [21] Z. Lei and Z. Xin, On scaling invariance and type-Ⅰ singularities for the compressible Navier-Stokes equations, preprint, arXiv: 1710.02253. [22] H. Li and X. Zhang, Stability of plane Couette flow for the compressible Navier-Stokes equations with Navier-slip boundary, Journal of Differential Equations, 263 (2017), 1160-1187.  doi: 10.1016/j.jde.2017.03.009. [23] T.-P. Liu, Compressible flow with damping and vacuum, Japan J. Indust. Appl. Math., 13 (1996), 25-32.  doi: 10.1007/BF03167296. [24] 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. [25] A. Matsumura and T. Nishida, Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Commun. Math. Phys., 89 (1983), 445-464. [26] C.-L. Navier, Sur les lois de léquilibre et du mouvement des corps élastiques, Mem. Acad. R. Sci. Inst. Fr., 6 (1827), 369. [27] J. Serrin, On the uniqueness of compressible fluid motions, Arch. Ration. Mech. Anal., 3-3 (1959), 271-288.  doi: 10.1007/BF00284180. [28] V. Solonnikov and V. Ščadilov, On a boundary value problem for a stationary system of Navier-Stokes equations, Proc. Steklov Inst. Math., 125 (1973), 186-199. [29] A. Tani, On the first initial-boundary value problem of compressible viscous fluid motion, Publ. Res. Inst. Math. Sci., 13 (1977), 193-253. [30] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, American Mathematical Soc., 1984. [31] Y. Wang, I. Tice and C. Kim, The viscous surface-internal wave problem: global well-posedness and decay, Arch. Ration. Mech. Anal., 212 (2014), 1-92.  doi: 10.1007/s00205-013-0700-2. [32] J. Watanabe, On incompressible viscous fluid flows with slip boundary conditions, J. Comput. Appl. Math., 159 (2003), 161-172.  doi: 10.1016/S0377-0427(03)00568-5. [33] Z. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Commun. Pure Appl. Math., LI (1998), 229-240.  doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.3.CO;2-K. [34] Z. Xin and W. Yan, On blowup of classical solutions to the compressible Navier-Stokes equations, Commun. Math. Phys., 321 (2013), 529-541.  doi: 10.1007/s00220-012-1610-0. [35] W. Zajączkowski, On nonstationary motion of a compressible barotropic viscous fluid bounded by a free surface, Institute of Mathematics, Polish Academy of Sciences, 1993. [36] W. Zajączkowski, On nonstationary motion of a compressible barotropic viscous fluid with boundary slip condition, J. Appl. Anal., 4 (1998), 167-204.  doi: 10.1515/JAA.1998.167.

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##### References:
 [1] R. A. Adams and J. J. Fournier, Sobolev Spaces, 2nd edition, Academic Press, 2003. [2] S. N. Antontsev and H. B. de Oliveira, Navier-Stokes equations with absorption under slip boundary conditions: existence, uniqueness and extinction in time, RIMS Kôkyûroku Bessatsu B, 1 (2007), 21-42. [3] N. T. Bishop, The Poincaré inequality for a vector field with zero tangential or normal component on the boundary, Quaest. Math., 11 (1988), 195-199. [4] H. Brezis and P. Mironescu, Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces, J. Evol. Equations, 1 (2001), 387-404.  doi: 10.1007/PL00001378. [5] G.-Q. Chen and Z. Qian, A study of the Navier-Stokes equations with the kinematic and Navier boundary conditions, Indiana Univ. Math. J., 59 (2010), 721-760.  doi: 10.1512/iumj.2010.59.3898. [6] Y. Cho and B. J. Jin, Blow-up of viscous heat-conducting compressible flows, J. Math. Anal. Appl., 320 (2006), 819-826.  doi: 10.1016/j.jmaa.2005.08.005. [7] L. Desvillettes and C. Villani, On a variant of Korn's inequality arising in statistical mechanics, ESAIM Control. Optim. Calc. Var., 8 (2002), 603-619.  doi: 10.1051/cocv:2002036. [8] S. Ding, Q. Li and Z. Xin, Stability analysis for the incompressible Navier-Stokes equations with Navier boundary conditions, Journal of Mathematical Fluid Mechanics, 20 (2018), 603-629 doi: 10.1007/s00021-017-0337-2. [9] Z. Ding, A proof of the trace theorem of Sobolev spaces on lipschitz domains, Proc. Am. Math. Soc., 124 (1996), 591-601.  doi: 10.1090/S0002-9939-96-03132-2. [10] R. Farwig, Stationary solutions of compressible Navier-Stokes equations with slip boundary condition, Commun. Partial Differ. Equations, 14 (1989), 1579-1606.  doi: 10.1080/03605308908820667. [11] K. O. Friedrichs, On the boundary-value problems of the theory of elasticity and Korn's Inequality, Ann. Math., 48 (1947), 441. doi: 10.2307/1969180. [12] Y. Guo and I. Tice, Compressible, inviscid Rayleigh-Taylor instability, Indiana Univ. Math. J., 60 (2011), 677-711.  doi: 10.1512/iumj.2011.60.4193. [13] Y. Guo and I. Tice, Almost exponential decay of periodic viscous surface waves without surface tension, Arch. Ration. Mech. Anal., 207 (2013), 459-531.  doi: 10.1007/s00205-012-0570-z. [14] D. Hoff, Local solutions of a compressible flow problem with Navier boundary conditions in general three-dimensional domains, SIAM J. Math. Anal., 44 (2012), 633-650.  doi: 10.1137/110827065. [15] C. Horgan and L. Payne, On inequalities of Korn, Friedrichs and Babu|s|ka-Aziz, Arch. Ration. Mech. Anal., 82 (1983), 165-179.  doi: 10.1007/BF00250935. [16] F. Huang, J. Li and Z. Xin, Convergence to equilibria and blowup behavior of global strong solutions to the Stokes approximation equations for two-dimensional compressible flows with large data, J. Math. Pures Appl., 86 (2006), 471-491.  doi: 10.1016/j.matpur.2006.10.001. [17] 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, Commun. Pure Appl. Math., 65 (2012), 549-585.  doi: 10.1002/cpa.21382. [18] N. Itaya, On the Cauchy problems for the system of fundamental equations describing the movement of compressible viscous fluid, Kodai Math. Sem. Rep., 23 (1971), 60-120. [19] J. Jang, I. Tice and Y. Wang, The compressible viscous surface-internal wave problem: Stability and vanishing surface tension limit, Commun. Math. Phys., 343 (2016), 1039-1113.  doi: 10.1007/s00220-016-2603-1. [20] O. A. Ladyzhenskaya and R. Silverman, The Mathematical Theory of Viscous Incompressible Flow, 2nd edition, Gordon & Breach Science Publishers Ltd, New York, NY, 1969. [21] Z. Lei and Z. Xin, On scaling invariance and type-Ⅰ singularities for the compressible Navier-Stokes equations, preprint, arXiv: 1710.02253. [22] H. Li and X. Zhang, Stability of plane Couette flow for the compressible Navier-Stokes equations with Navier-slip boundary, Journal of Differential Equations, 263 (2017), 1160-1187.  doi: 10.1016/j.jde.2017.03.009. [23] T.-P. Liu, Compressible flow with damping and vacuum, Japan J. Indust. Appl. Math., 13 (1996), 25-32.  doi: 10.1007/BF03167296. [24] 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. [25] A. Matsumura and T. Nishida, Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Commun. Math. Phys., 89 (1983), 445-464. [26] C.-L. Navier, Sur les lois de léquilibre et du mouvement des corps élastiques, Mem. Acad. R. Sci. Inst. Fr., 6 (1827), 369. [27] J. Serrin, On the uniqueness of compressible fluid motions, Arch. Ration. Mech. Anal., 3-3 (1959), 271-288.  doi: 10.1007/BF00284180. [28] V. Solonnikov and V. Ščadilov, On a boundary value problem for a stationary system of Navier-Stokes equations, Proc. Steklov Inst. Math., 125 (1973), 186-199. [29] A. Tani, On the first initial-boundary value problem of compressible viscous fluid motion, Publ. Res. Inst. Math. Sci., 13 (1977), 193-253. [30] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, American Mathematical Soc., 1984. [31] Y. Wang, I. Tice and C. Kim, The viscous surface-internal wave problem: global well-posedness and decay, Arch. Ration. Mech. Anal., 212 (2014), 1-92.  doi: 10.1007/s00205-013-0700-2. [32] J. Watanabe, On incompressible viscous fluid flows with slip boundary conditions, J. Comput. Appl. Math., 159 (2003), 161-172.  doi: 10.1016/S0377-0427(03)00568-5. [33] Z. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Commun. Pure Appl. Math., LI (1998), 229-240.  doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.3.CO;2-K. [34] Z. Xin and W. Yan, On blowup of classical solutions to the compressible Navier-Stokes equations, Commun. Math. Phys., 321 (2013), 529-541.  doi: 10.1007/s00220-012-1610-0. [35] W. Zajączkowski, On nonstationary motion of a compressible barotropic viscous fluid bounded by a free surface, Institute of Mathematics, Polish Academy of Sciences, 1993. [36] W. Zajączkowski, On nonstationary motion of a compressible barotropic viscous fluid with boundary slip condition, J. Appl. Anal., 4 (1998), 167-204.  doi: 10.1515/JAA.1998.167.
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