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

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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

Abstract
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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.

Keywords: global existence, free boundary., Navier-Stokes equations.

Mathematics Subject Classification: Primary: 35Q35; Secondary: 76N1.

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.
[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.
[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.
[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.
[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.
[6]	E. Feireisl, Dynamics of Viscous Compressible Fluids,, Oxford University Press, (2004).
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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).
[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.
[18]	P. L. Lions, Mathematical Topics in Fluid Mechanics,, Vol. 2. Compressible models, (1998).
[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.
[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.
[21]	J. Nash, Le problème de Cauchy pour les équations différentielles d'un fluide général,, Bull. Soc. Math. France, 90 (1962), 487.
[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.
[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.
[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.
[25]	J. Serrin, On the uniqueness of compressible fluid motion,, Arch. Rational. Mech. Anal., 3 (1959), 271.
[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.

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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.
[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.
[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.
[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.
[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.
[6]	E. Feireisl, Dynamics of Viscous Compressible Fluids,, Oxford University Press, (2004).
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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).
[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.
[18]	P. L. Lions, Mathematical Topics in Fluid Mechanics,, Vol. 2. Compressible models, (1998).
[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.
[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.
[21]	J. Nash, Le problème de Cauchy pour les équations différentielles d'un fluide général,, Bull. Soc. Math. France, 90 (1962), 487.
[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.
[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.
[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.
[25]	J. Serrin, On the uniqueness of compressible fluid motion,, Arch. Rational. Mech. Anal., 3 (1959), 271.
[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.

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