Article Contents
Article Contents

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

• Zilai Li, E-mail address: lizilai0917@163.com
• 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.

Mathematics Subject Classification: Primary: 35J05, 35J10; Secondary: 35Q30.

 Citation:

•  [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.