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

Previous Article
Asymptotic preserving scheme for a kinetic model describing incompressible fluids
KRM Home
This Issue
Next Article
Kinetic derivation of fractional Stokes and Stokes-Fourier systems

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
Related Papers

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. 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

show all references

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

[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 & Applied Analysis, 2010, 9 (4) : 1041-1052. doi: 10.3934/cpaa.2010.9.1041
[2]	Hantaek Bae. Solvability of the free boundary value problem of the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 769-801. doi: 10.3934/dcds.2011.29.769
[3]	Daoyuan Fang, Bin Han, Matthias Hieber. Local and global existence results for the Navier-Stokes equations in the rotational framework. Communications on Pure & Applied Analysis, 2015, 14 (2) : 609-622. doi: 10.3934/cpaa.2015.14.609
[4]	Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085
[5]	Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations II: Global existence of small solutions. Evolution Equations & Control Theory, 2012, 1 (1) : 217-234. doi: 10.3934/eect.2012.1.217
[6]	Feimin Huang, Xiaoding Shi, Yi Wang. Stability of viscous shock wave for compressible Navier-Stokes equations with free boundary. Kinetic & Related Models, 2010, 3 (3) : 409-425. doi: 10.3934/krm.2010.3.409
[7]	Zilai Li, Zhenhua Guo. On free boundary problem for compressible navier-stokes equations with temperature-dependent heat conductivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3903-3919. doi: 10.3934/dcdsb.2017201
[8]	Yoshihiro Shibata. On the local wellposedness of free boundary problem for the Navier-Stokes equations in an exterior domain. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1681-1721. doi: 10.3934/cpaa.2018081
[9]	Joel Avrin. Global existence and regularity for the Lagrangian averaged Navier-Stokes equations with initial data in $H^{1//2}$. Communications on Pure & Applied Analysis, 2004, 3 (3) : 353-366. doi: 10.3934/cpaa.2004.3.353
[10]	Yuming Qin, Lan Huang, Zhiyong Ma. Global existence and exponential stability in $H^4$ for the nonlinear compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1991-2012. doi: 10.3934/cpaa.2009.8.1991
[11]	Michele Campiti, Giovanni P. Galdi, Matthias Hieber. Global existence of strong solutions for $2$-dimensional Navier-Stokes equations on exterior domains with growing data at infinity. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1613-1627. doi: 10.3934/cpaa.2014.13.1613
[12]	Evrad M. D. Ngom, Abdou Sène, Daniel Y. Le Roux. Global stabilization of the Navier-Stokes equations around an unstable equilibrium state with a boundary feedback controller. Evolution Equations & Control Theory, 2015, 4 (1) : 89-106. doi: 10.3934/eect.2015.4.89
[13]	Yoshikazu Giga. A remark on a Liouville problem with boundary for the Stokes and the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1277-1289. doi: 10.3934/dcdss.2013.6.1277
[14]	Joanna Rencławowicz, Wojciech M. Zajączkowski. Global regular solutions to the Navier-Stokes equations with large flux. Conference Publications, 2011, 2011 (Special) : 1234-1243. doi: 10.3934/proc.2011.2011.1234
[15]	Keyan Wang. On global regularity of incompressible Navier-Stokes equations in $\mathbf R^3$. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1067-1072. doi: 10.3934/cpaa.2009.8.1067
[16]	Jing Wang, Lining Tong. Stability of boundary layers for the inflow compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2595-2613. doi: 10.3934/dcdsb.2012.17.2595
[17]	Chérif Amrouche, Nour El Houda Seloula. $L^p$-theory for the Navier-Stokes equations with pressure boundary conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1113-1137. doi: 10.3934/dcdss.2013.6.1113
[18]	Sylvie Monniaux. Various boundary conditions for Navier-Stokes equations in bounded Lipschitz domains. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1355-1369. doi: 10.3934/dcdss.2013.6.1355
[19]	Huicheng Yin, Lin Zhang. The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅱ: 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1063-1102. doi: 10.3934/dcds.2018045
[20]	Luigi C. Berselli. An elementary approach to the 3D Navier-Stokes equations with Navier boundary conditions: Existence and uniqueness of various classes of solutions in the flat boundary case.. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 199-219. doi: 10.3934/dcdss.2010.3.199

2018 Impact Factor: 1.38

American Institute of Mathematical Sciences