# American Institute of Mathematical Sciences

January  2017, 16(1): 1-24. doi: 10.3934/cpaa.2017001

## Global weak solution to 3D compressible flows with density-dependent viscosity and free boundary

 1 School of Sciences, Xi'an University of Technology, Xian 710048, Shaanxi, P. R. China 2 School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, 454000, Henan Province, P. R. China 3 Center for Nonlinear Studies and School of Mathematics, Northwest University, Xian 710069, P. R. China

Received  January 2015 Revised  October 2015 Published  November 2016

In this paper, we obtain the global weak solution to the 3D spherically symmetric compressible isentropic Navier-Stokes equations with arbitrarily large, vacuum data and free boundary when the shear viscosity $\mu$ is a positive constant and the bulk viscosity $\lambda(\rho)=\rho^\beta$ with $\beta>0$. The analysis of the upper and lower bound of the density is based on some well-chosen functionals. In addition, the free boundary can be shown to expand outward at an algebraic rate in time.

Citation: Mei Wang, Zilai Li, Zhenhua Guo. Global weak solution to 3D compressible flows with density-dependent viscosity and free boundary. Communications on Pure and Applied Analysis, 2017, 16 (1) : 1-24. doi: 10.3934/cpaa.2017001
##### References:
 [1] D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 (2003), 211-223. doi: 10.1007/s00220-003-0859-8. [2] D. Bresch and B. Desjardins, On the construction of approximate solutions for 2D viscous shallow water model and for compressible Navier-Stokes models, J. Math. Pures Appl., 86 (2006), 362-368. doi: 10.1016/j.matpur.2006.06.005. [3] Q. L. Chen, C. X. Miao and Z. F. Zhang, Global well-posedness for compressible NavierStokes equations with highly oscillating initial velocity, Comm. Pure Appl. Math., 63 (2010), 1173-1224. doi: 10.1002/cpa.20325. [4] R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614. doi: 10.1007/s002220000078. [5] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004. [6] Z. H. Guo, Q. S. Jiu and Z. P. Xin, Sphericall symmetric isentropic compressible flows with density-dependent viscosity coefficients, SIAM J. Math. Anal., 39 (2008), 1402-1427. doi: 10.1137/070680333. [7] Z. H. Guo, H. L. Li and Z. P. Xin, Lagrange structure and dynamics for solutions to the spherically symmetric compressible Navier-Stokes equations, Comm. Math. Phys., 309 (2012), 371-412. doi: 10.1007/s00220-011-1334-6. [8] Z. H. Guo and Z. L. Li, Global existence of weak solution to the free boundary problem for compressible Navier-Stokes system, to appear. doi: 10.3934/krm.2016.9.75. [9] Z. H. Guo, M. Wang and Y. Wang, Global solution to the 3D spherically symmetric compressible Navier-Stokes equations with large data, Preprint. [10] D. Hoff, Spherically symmetric solutions of the Navier-Stokes equations for compressible, isothermal flow with large, discontinuous initial data, Indaina Univ. Math. J., 41 (1992), 1225-1302. doi: 10.1512/iumj.1992.41.41060. [11] D. Hoff, Discontinoous solution of the Navier-Stokes equations for multi-dimensional heatconducting fluids, Arch. Rat. Mech. Anal., 193 (1997), 303-354. doi: 10.1007/s002050050055. [12] D. Hoff and J. Smoller, Non-formation of vacuum states for compressible Navier-Stokes equations, Comm. Math. Phys., 216 (2001), 255-276. doi: 10.1007/s002200000322. [13] D. Hoff and H. K. Jenssen, Symmetric nonbarotropic flows with large data and forces, Arch. Rational Mech. Anal., 173 (2004), 297-343. doi: 10.1007/s00205-004-0318-5. [14] X. D. Huang and J. Li, Existence and blowup behavior of global strong solutions to the two-dimensional baratropic compressible Navier-Stokes system with vacuum and large initial data, to appear. doi: 10.1016/j.matpur.2016.02.003. [15] X. D. Huang and J. Li, Global well-posedness of classical solutions to the Cauchy problem of two-dimesional baratropic compressible Navier-Stokes system with vacuum and large initial data, preprint, arXiv: 1207.3746. [16] X. D. Huang, J. Li and Z. P. 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. [17] N. Itaya, On the Cauchy problem for the system of fundamental equations describing the movement of compressible viscous fluids, Kodia Math. Sem. Rep., 23 (1971), 60-120. [18] S. Jiang and P. Zhang, Global sphericall symmetric solutions of the compressible isentropic Navier-Stokes equations, Comm. Math. Phys., 215 (2001), 559-581. doi: 10.1007/PL00005543. [19] S. Jiang, Z. P. Xin and P. Zhang, Global weak solutions to 1D compressible isentropy NavierStokes with density-dependent viscosity, Methods and Applications of Analysis, 12 (2005), 239-252. doi: 10.4310/MAA.2005.v12.n3.a2. [20] Q. S. Jiu, Y. Wang and Z. P. Xin, Stability of rarefaction waves to the 1D compressible Navier-Stokes equations with density-dependent viscosity, Comm. Part. Diff. Equ., 36 (2011), 602-634. doi: 10.1080/03605302.2010.516785. [21] Q. S. Jiu, Y. Wang and Z. P. Xin, Vacuum behaviors around the rarefaction waves to the 1D compressible Navier-Stokes equations with density-dependent viscosity, http://arxiv.org/abs/1109.0871. doi: 10.1080/03605302.2010.516785. [22] Q. S. Jiu, Y. Wang and Z. P. Xin, Global well-posedness of the Cauchy problem of the twodimensional compressible Navier-Stokes equations in weighted spaces, J. Diff. Equa., 255 (2013), 351-404. doi: 10.1016/j.jde.2013.04.014. [23] Q. S. Jiu, Y. Wang and Z. P. Xin, Global well-posedness of 2D compressible Navier-Stokes equations with large data and vacuum, J. Math. Fluid Mech., 16 (2014), 483-521. doi: 10.1007/s00021-014-0171-8. [24] Q. S. Jiu and Z. P. Xin, The Cauchy problem for 1D compressible flows with density-dependent viscosity coefficients, Kinet. Relat. Models, 1 (2008), 313-330. doi: 10.3934/krm.2008.1.313. [25] J. I. Kanel, A model system of equations for the one-dimensional motion of a gas (in Russian), Diff. Uravn., 4 (1968), 721-734. [26] A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initialboundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech., 41 (1977), 273-282. [27] H. L. Li, J. Li and Z. P. 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. [28] P. L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 2, Compressible Models, Oxford Science Publication, Oxford, 1998. [29] T. P. Liu and J. Smoller, On the vacuum state for the isentropic gas dynamics equations, Adv. in Appl. Math., 1 (1980), 345-359. doi: 10.1016/0196-8858(80)90016-0. [30] T. P. Liu, Z. P. Xin and T. Yang, Vacuum states of compressible flow, Discrete Contin. Dyn. Syst., 4 (1998), 1-32. [31] A. Matsumura and T. Nishida, The initial value peoblem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ., 20 (1980), 67-104. [32] A. Mellet and A. Vasseur, On the barotropic compressible Navier-Stokes equation, Comm. Part. Diff. Equa., 32 (2007), 431-452. doi: 10.1080/03605300600857079. [33] J. Nash, Le problème de Cauchy pour les équations différentielles d'un fluide général, Bull. Soc. Math. France, 90 (1962), 487-497. [34] M. Okada, Free boundary problem for the equation of spherically Symmetrical motion of viscous gas, Japan J. Appl. Math., 10 (1993), 219-235. doi: 10.1007/BF03167573. [35] M. Perepelitsa, On the global existence of weak solutions for the Navier-stokes equations of compressible fluid flows, SIAM J. Math. Anal., 38 (2006), 1126-153. doi: 10.1137/040619119. [36] O. Rozanova, Blow up of smooth solutions to the compressible Navier-Stokes equations with the data highly decreasing at infinity, J. Differ. Eqs., 245 (2008), 1762-1774. doi: 10.1016/j.jde.2008.07.007. [37] I. Straskraba and A. Zlotnik, Global properties of solutions to 1D viscous Straskraba barotropic fluid equations with density dependent viscosity, Z. Angew. Math. Phys., 54 (2003), 593-607. doi: 10.1007/s00033-003-1009-z. [38] A. Tani, On the first initial-boundary value problems of compressible viscous fluid motion, Publ. Res. Inst. Math. Sci. Kytt Univ., 13 (1971), 193-253. [39] Z. P. 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>3.3.CO;2-K. [40] Z. P. Xin and W. Yan, On blowup of classical solutions to the compressible Navier-Stokes equations, preprint, to appear. doi: 10.1007/s00220-012-1610-0.

show all references

##### References:
 [1] D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 (2003), 211-223. doi: 10.1007/s00220-003-0859-8. [2] D. Bresch and B. Desjardins, On the construction of approximate solutions for 2D viscous shallow water model and for compressible Navier-Stokes models, J. Math. Pures Appl., 86 (2006), 362-368. doi: 10.1016/j.matpur.2006.06.005. [3] Q. L. Chen, C. X. Miao and Z. F. Zhang, Global well-posedness for compressible NavierStokes equations with highly oscillating initial velocity, Comm. Pure Appl. Math., 63 (2010), 1173-1224. doi: 10.1002/cpa.20325. [4] R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614. doi: 10.1007/s002220000078. [5] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004. [6] Z. H. Guo, Q. S. Jiu and Z. P. Xin, Sphericall symmetric isentropic compressible flows with density-dependent viscosity coefficients, SIAM J. Math. Anal., 39 (2008), 1402-1427. doi: 10.1137/070680333. [7] Z. H. Guo, H. L. Li and Z. P. Xin, Lagrange structure and dynamics for solutions to the spherically symmetric compressible Navier-Stokes equations, Comm. Math. Phys., 309 (2012), 371-412. doi: 10.1007/s00220-011-1334-6. [8] Z. H. Guo and Z. L. Li, Global existence of weak solution to the free boundary problem for compressible Navier-Stokes system, to appear. doi: 10.3934/krm.2016.9.75. [9] Z. H. Guo, M. Wang and Y. Wang, Global solution to the 3D spherically symmetric compressible Navier-Stokes equations with large data, Preprint. [10] D. Hoff, Spherically symmetric solutions of the Navier-Stokes equations for compressible, isothermal flow with large, discontinuous initial data, Indaina Univ. Math. J., 41 (1992), 1225-1302. doi: 10.1512/iumj.1992.41.41060. [11] D. Hoff, Discontinoous solution of the Navier-Stokes equations for multi-dimensional heatconducting fluids, Arch. Rat. Mech. Anal., 193 (1997), 303-354. doi: 10.1007/s002050050055. [12] D. Hoff and J. Smoller, Non-formation of vacuum states for compressible Navier-Stokes equations, Comm. Math. Phys., 216 (2001), 255-276. doi: 10.1007/s002200000322. [13] D. Hoff and H. K. Jenssen, Symmetric nonbarotropic flows with large data and forces, Arch. Rational Mech. Anal., 173 (2004), 297-343. doi: 10.1007/s00205-004-0318-5. [14] X. D. Huang and J. Li, Existence and blowup behavior of global strong solutions to the two-dimensional baratropic compressible Navier-Stokes system with vacuum and large initial data, to appear. doi: 10.1016/j.matpur.2016.02.003. [15] X. D. Huang and J. Li, Global well-posedness of classical solutions to the Cauchy problem of two-dimesional baratropic compressible Navier-Stokes system with vacuum and large initial data, preprint, arXiv: 1207.3746. [16] X. D. Huang, J. Li and Z. P. 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. [17] N. Itaya, On the Cauchy problem for the system of fundamental equations describing the movement of compressible viscous fluids, Kodia Math. Sem. Rep., 23 (1971), 60-120. [18] S. Jiang and P. Zhang, Global sphericall symmetric solutions of the compressible isentropic Navier-Stokes equations, Comm. Math. Phys., 215 (2001), 559-581. doi: 10.1007/PL00005543. [19] S. Jiang, Z. P. Xin and P. Zhang, Global weak solutions to 1D compressible isentropy NavierStokes with density-dependent viscosity, Methods and Applications of Analysis, 12 (2005), 239-252. doi: 10.4310/MAA.2005.v12.n3.a2. [20] Q. S. Jiu, Y. Wang and Z. P. Xin, Stability of rarefaction waves to the 1D compressible Navier-Stokes equations with density-dependent viscosity, Comm. Part. Diff. Equ., 36 (2011), 602-634. doi: 10.1080/03605302.2010.516785. [21] Q. S. Jiu, Y. Wang and Z. P. Xin, Vacuum behaviors around the rarefaction waves to the 1D compressible Navier-Stokes equations with density-dependent viscosity, http://arxiv.org/abs/1109.0871. doi: 10.1080/03605302.2010.516785. [22] Q. S. Jiu, Y. Wang and Z. P. Xin, Global well-posedness of the Cauchy problem of the twodimensional compressible Navier-Stokes equations in weighted spaces, J. Diff. Equa., 255 (2013), 351-404. doi: 10.1016/j.jde.2013.04.014. [23] Q. S. Jiu, Y. Wang and Z. P. Xin, Global well-posedness of 2D compressible Navier-Stokes equations with large data and vacuum, J. Math. Fluid Mech., 16 (2014), 483-521. doi: 10.1007/s00021-014-0171-8. [24] Q. S. Jiu and Z. P. Xin, The Cauchy problem for 1D compressible flows with density-dependent viscosity coefficients, Kinet. Relat. Models, 1 (2008), 313-330. doi: 10.3934/krm.2008.1.313. [25] J. I. Kanel, A model system of equations for the one-dimensional motion of a gas (in Russian), Diff. Uravn., 4 (1968), 721-734. [26] A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initialboundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech., 41 (1977), 273-282. [27] H. L. Li, J. Li and Z. P. 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. [28] P. L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 2, Compressible Models, Oxford Science Publication, Oxford, 1998. [29] T. P. Liu and J. Smoller, On the vacuum state for the isentropic gas dynamics equations, Adv. in Appl. Math., 1 (1980), 345-359. doi: 10.1016/0196-8858(80)90016-0. [30] T. P. Liu, Z. P. Xin and T. Yang, Vacuum states of compressible flow, Discrete Contin. Dyn. Syst., 4 (1998), 1-32. [31] A. Matsumura and T. Nishida, The initial value peoblem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ., 20 (1980), 67-104. [32] A. Mellet and A. Vasseur, On the barotropic compressible Navier-Stokes equation, Comm. Part. Diff. Equa., 32 (2007), 431-452. doi: 10.1080/03605300600857079. [33] J. Nash, Le problème de Cauchy pour les équations différentielles d'un fluide général, Bull. Soc. Math. France, 90 (1962), 487-497. [34] M. Okada, Free boundary problem for the equation of spherically Symmetrical motion of viscous gas, Japan J. Appl. Math., 10 (1993), 219-235. doi: 10.1007/BF03167573. [35] M. Perepelitsa, On the global existence of weak solutions for the Navier-stokes equations of compressible fluid flows, SIAM J. Math. Anal., 38 (2006), 1126-153. doi: 10.1137/040619119. [36] O. Rozanova, Blow up of smooth solutions to the compressible Navier-Stokes equations with the data highly decreasing at infinity, J. Differ. Eqs., 245 (2008), 1762-1774. doi: 10.1016/j.jde.2008.07.007. [37] I. Straskraba and A. Zlotnik, Global properties of solutions to 1D viscous Straskraba barotropic fluid equations with density dependent viscosity, Z. Angew. Math. Phys., 54 (2003), 593-607. doi: 10.1007/s00033-003-1009-z. [38] A. Tani, On the first initial-boundary value problems of compressible viscous fluid motion, Publ. Res. Inst. Math. Sci. Kytt Univ., 13 (1971), 193-253. [39] Z. P. 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>3.3.CO;2-K. [40] Z. P. Xin and W. Yan, On blowup of classical solutions to the compressible Navier-Stokes equations, preprint, to appear. doi: 10.1007/s00220-012-1610-0.
 [1] 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 and Applied Analysis, 2010, 9 (4) : 1041-1052. doi: 10.3934/cpaa.2010.9.1041 [2] Guangwu Wang, Boling Guo. Global weak solution to the quantum Navier-Stokes-Landau-Lifshitz equations with density-dependent viscosity. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6141-6166. doi: 10.3934/dcdsb.2019133 [3] Xulong Qin, Zheng-An Yao, Hongxing Zhao. One dimensional compressible Navier-Stokes equations with density-dependent viscosity and free boundaries. Communications on Pure and Applied Analysis, 2008, 7 (2) : 373-381. doi: 10.3934/cpaa.2008.7.373 [4] Wenjun Wang, Lei Yao. Spherically symmetric Navier-Stokes equations with degenerate viscosity coefficients and vacuum. Communications on Pure and Applied Analysis, 2010, 9 (2) : 459-481. doi: 10.3934/cpaa.2010.9.459 [5] Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163 [6] Zhenhua Guo, Zilai Li. Global existence of weak solution to the free boundary problem for compressible Navier-Stokes. Kinetic and Related Models, 2016, 9 (1) : 75-103. doi: 10.3934/krm.2016.9.75 [7] Jianwei Yang, Peng Cheng, Yudong Wang. Asymptotic limit of a Navier-Stokes-Korteweg system with density-dependent viscosity. Electronic Research Announcements, 2015, 22: 20-31. doi: 10.3934/era.2015.22.20 [8] Xinhua Zhao, Zilai Li. Asymptotic behavior of spherically or cylindrically symmetric solutions to the compressible Navier-Stokes equations with large initial data. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1421-1448. doi: 10.3934/cpaa.2020052 [9] 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 and Related Models, 2016, 9 (3) : 469-514. doi: 10.3934/krm.2016004 [10] J. Huang, Marius Paicu. Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647 [11] Zilai Li, Zhenhua Guo. On free boundary problem for compressible navier-stokes equations with temperature-dependent heat conductivity. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3903-3919. doi: 10.3934/dcdsb.2017201 [12] Chérif Amrouche, María Ángeles Rodríguez-Bellido. On the very weak solution for the Oseen and Navier-Stokes equations. Discrete and Continuous Dynamical Systems - S, 2010, 3 (2) : 159-183. doi: 10.3934/dcdss.2010.3.159 [13] Jishan Fan, Fucai Li, Gen Nakamura. Global strong solution to the two-dimensional density-dependent magnetohydrodynamic equations with vaccum. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1481-1490. doi: 10.3934/cpaa.2014.13.1481 [14] Fei Jiang, Song Jiang, Junpin Yin. Global weak solutions to the two-dimensional Navier-Stokes equations of compressible heat-conducting flows with symmetric data and forces. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 567-587. doi: 10.3934/dcds.2014.34.567 [15] Hantaek Bae. Solvability of the free boundary value problem of the Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 769-801. doi: 10.3934/dcds.2011.29.769 [16] Ping Chen, Daoyuan Fang, Ting Zhang. Free boundary problem for compressible flows with density--dependent viscosity coefficients. Communications on Pure and Applied Analysis, 2011, 10 (2) : 459-478. doi: 10.3934/cpaa.2011.10.459 [17] Feimin Huang, Xiaoding Shi, Yi Wang. Stability of viscous shock wave for compressible Navier-Stokes equations with free boundary. Kinetic and Related Models, 2010, 3 (3) : 409-425. doi: 10.3934/krm.2010.3.409 [18] Yoshihiro Shibata. On the local wellposedness of free boundary problem for the Navier-Stokes equations in an exterior domain. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1681-1721. doi: 10.3934/cpaa.2018081 [19] Ciprian Foias, Ricardo Rosa, Roger Temam. Topological properties of the weak global attractor of the three-dimensional Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1611-1631. doi: 10.3934/dcds.2010.27.1611 [20] Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure and Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35

2020 Impact Factor: 1.916