# American Institute of Mathematical Sciences

March  2016, 9(1): 75-103. doi: 10.3934/krm.2016.9.75

## Global existence of weak solution to the free boundary problem for compressible Navier-Stokes

 1 Center for Nonlinear Studies and School of Mathematics, Northwest University, Xi'an 710069, China 2 School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China

Received  October 2014 Revised  August 2015 Published  October 2015

In this paper, the compressible Navier-Stokes system (CNS) with constant viscosity coefficients is considered in three space dimensions. we prove the global existence of spherically symmetric weak solutions to the free boundary problem for the CNS with vacuum and free boundary separating fluids and vacuum. In addition, the free boundary is shown to expand outward at an algebraic rate in time.
Citation: 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
##### References:
 [1] Y. Cho, H. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids,, J. Math. Pures Appl., 83 (2004), 243.  doi: 10.1016/j.matpur.2003.11.004.  Google Scholar [2] H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids,, J. Differ. Eqs., 190 (2003), 504.  doi: 10.1016/S0022-0396(03)00015-9.  Google Scholar [3] H. J. Choe and H. Kim, Global existence of the radially symmetric solutions of the Navier-Stokes equations for the isentropic compressible fluids,, Math. Methods Appl., 28 (2005), 1.  doi: 10.1002/mma.545.  Google Scholar [4] E. Feireisl, A. Novotny and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations,, J. Math. Fluid Mech., 3 (2001), 358.  doi: 10.1007/PL00000976.  Google Scholar [5] E. Feireisl, On the motion of a viscous, compressible, and heat conducting fluid,, Indiana Univ. Math. J., 53 (2004), 1705.  doi: 10.1512/iumj.2004.53.2510.  Google Scholar [6] E. Feireisl, Dynamics of Viscous Compressible Fluids,, Oxford University Press, (2004).   Google Scholar [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,, Commun. Math. Phys., 309 (2012), 371.  doi: 10.1007/s00220-011-1334-6.  Google Scholar [8] D. Hoff, Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data,, Trans. Amer. Math. Soc., 303 (1987), 169.  doi: 10.2307/2000785.  Google Scholar [9] 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.  doi: 10.1512/iumj.1992.41.41060.  Google Scholar [10] D. Hoff, Global existence of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data,, J. Diff. Eqs., 120 (1995), 215.  doi: 10.1006/jdeq.1995.1111.  Google Scholar [11] D. Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data,, Arch. Rat. Mech. Anal., 132 (1995), 1.  doi: 10.1007/BF00390346.  Google Scholar [12] D. Hoff and H. K. Jenssen, Symmetric nonbarotropic flows with large data and forces,, Arch. Ration. Mech. Anal., 173 (2004), 297.  doi: 10.1007/s00205-004-0318-5.  Google Scholar [13] D. Hoff and D. Serre, The failure of continuous dependence on initial data for the Navier-Stokes equations of compressible flow,, SIAM J. Appl. Math., 51 (1991), 887.  doi: 10.1137/0151043.  Google Scholar [14] 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.  doi: 10.1002/cpa.21382.  Google Scholar [15] S. Jiang and P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations,, Comm. Math. Phys., 215 (2001), 559.  doi: 10.1007/PL00005543.  Google Scholar [16] H. Kong, Global Existence of Spherically Symmetric Weak Solutions to the Free Boundary Value Problem of 3D Isentropic Compressible Navier-Stokes Equations,, Master thesis, (2014).   Google Scholar [17] A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas,, J. Appl. Math. Mech., 41 (1977), 273.   Google Scholar [18] P. L. Lions, Mathematical Topics in Fluid Mechanics,, Vol. 2. Compressible models, (1998).   Google Scholar [19] T. Luo, Z. P. Xin and T. Yang, Interface behavior of compressible Navier-Stokes equations with vacuum,, SIAM J. Math. Anal., 31 (2000), 1175.  doi: 10.1137/S0036141097331044.  Google Scholar [20] A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heatconductive gases,, J. Math. Kyoto Univ., 20 (1980), 67.   Google Scholar [21] J. Nash, Le problème de Cauchy pour les équations différentielles d'un fluide général,, Bull. Soc. Math. France, 90 (1962), 487.   Google Scholar [22] M. Okada, Free boundary problem for the equation of one dimensional motion of viscous gas,, Japan J. Appl. Math., 6 (1989), 161.  doi: 10.1007/BF03167921.  Google Scholar [23] M. Okada and T. Makino, Free boundary problem for the equation of spherically Symmetrical motion of viscous gas,, Japan J. Appl. Math., 10 (1993), 219.  doi: 10.1007/BF03167573.  Google Scholar [24] R. Salvi and I. Straškraba, Global existence for viscous compressible fluids and their behavior as $t\rightarrow\infty$,, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 40 (1993), 17.   Google Scholar [25] J. Serrin, On the uniqueness of compressible fluid motion,, Arch. Rational. Mech. Anal., 3 (1959), 271.   Google Scholar [26] Z. P. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equations with compact density,, Comm. Pure Appl. Math., 51 (1998), 229.  doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C.  Google Scholar

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##### References:
 [1] Y. Cho, H. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids,, J. Math. Pures Appl., 83 (2004), 243.  doi: 10.1016/j.matpur.2003.11.004.  Google Scholar [2] H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids,, J. Differ. Eqs., 190 (2003), 504.  doi: 10.1016/S0022-0396(03)00015-9.  Google Scholar [3] H. J. Choe and H. Kim, Global existence of the radially symmetric solutions of the Navier-Stokes equations for the isentropic compressible fluids,, Math. Methods Appl., 28 (2005), 1.  doi: 10.1002/mma.545.  Google Scholar [4] E. Feireisl, A. Novotny and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations,, J. Math. Fluid Mech., 3 (2001), 358.  doi: 10.1007/PL00000976.  Google Scholar [5] E. Feireisl, On the motion of a viscous, compressible, and heat conducting fluid,, Indiana Univ. Math. J., 53 (2004), 1705.  doi: 10.1512/iumj.2004.53.2510.  Google Scholar [6] E. Feireisl, Dynamics of Viscous Compressible Fluids,, Oxford University Press, (2004).   Google Scholar [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,, Commun. Math. Phys., 309 (2012), 371.  doi: 10.1007/s00220-011-1334-6.  Google Scholar [8] D. Hoff, Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data,, Trans. Amer. Math. Soc., 303 (1987), 169.  doi: 10.2307/2000785.  Google Scholar [9] 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.  doi: 10.1512/iumj.1992.41.41060.  Google Scholar [10] D. Hoff, Global existence of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data,, J. Diff. Eqs., 120 (1995), 215.  doi: 10.1006/jdeq.1995.1111.  Google Scholar [11] D. Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data,, Arch. Rat. Mech. Anal., 132 (1995), 1.  doi: 10.1007/BF00390346.  Google Scholar [12] D. Hoff and H. K. Jenssen, Symmetric nonbarotropic flows with large data and forces,, Arch. Ration. Mech. Anal., 173 (2004), 297.  doi: 10.1007/s00205-004-0318-5.  Google Scholar [13] D. Hoff and D. Serre, The failure of continuous dependence on initial data for the Navier-Stokes equations of compressible flow,, SIAM J. Appl. Math., 51 (1991), 887.  doi: 10.1137/0151043.  Google Scholar [14] 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.  doi: 10.1002/cpa.21382.  Google Scholar [15] S. Jiang and P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations,, Comm. Math. Phys., 215 (2001), 559.  doi: 10.1007/PL00005543.  Google Scholar [16] H. Kong, Global Existence of Spherically Symmetric Weak Solutions to the Free Boundary Value Problem of 3D Isentropic Compressible Navier-Stokes Equations,, Master thesis, (2014).   Google Scholar [17] A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas,, J. Appl. Math. Mech., 41 (1977), 273.   Google Scholar [18] P. L. Lions, Mathematical Topics in Fluid Mechanics,, Vol. 2. Compressible models, (1998).   Google Scholar [19] T. Luo, Z. P. Xin and T. Yang, Interface behavior of compressible Navier-Stokes equations with vacuum,, SIAM J. Math. Anal., 31 (2000), 1175.  doi: 10.1137/S0036141097331044.  Google Scholar [20] A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heatconductive gases,, J. Math. Kyoto Univ., 20 (1980), 67.   Google Scholar [21] J. Nash, Le problème de Cauchy pour les équations différentielles d'un fluide général,, Bull. Soc. Math. France, 90 (1962), 487.   Google Scholar [22] M. Okada, Free boundary problem for the equation of one dimensional motion of viscous gas,, Japan J. Appl. Math., 6 (1989), 161.  doi: 10.1007/BF03167921.  Google Scholar [23] M. Okada and T. Makino, Free boundary problem for the equation of spherically Symmetrical motion of viscous gas,, Japan J. Appl. Math., 10 (1993), 219.  doi: 10.1007/BF03167573.  Google Scholar [24] R. Salvi and I. Straškraba, Global existence for viscous compressible fluids and their behavior as $t\rightarrow\infty$,, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 40 (1993), 17.   Google Scholar [25] J. Serrin, On the uniqueness of compressible fluid motion,, Arch. Rational. Mech. Anal., 3 (1959), 271.   Google Scholar [26] Z. P. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equations with compact density,, Comm. Pure Appl. Math., 51 (1998), 229.  doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C.  Google Scholar
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