Free boundary problem for compressible flows with density--dependent viscosity coefficients

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March 2011, 10(2): 459-478. doi: 10.3934/cpaa.2011.10.459

Free boundary problem for compressible flows with density--dependent viscosity coefficients

Ping Chen ^1, , Daoyuan Fang ^1, and Ting Zhang ^1,

1.	Department of Mathematics, Zhejiang University, Hangzhou 310027, China

Received May 2010 Revised September 2010 Published December 2010

Abstract
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In this paper, we consider the free boundary problem of the spherically symmetric compressible isentropic Navier--Stokes equations in $R^n (n \geq 1)$, with density--dependent viscosity coefficients. Precisely, the viscosity coefficients $\mu$ and $\lambda$ are assumed to be proportional to $\rho^\theta$, $0 < \theta < 1$, where $\rho$ is the density. We obtain the global existence, uniqueness and continuous dependence on initial data of a weak solution, with a Lebesgue initial velocity $u_0\in L^{4 m}$, $4m>n$ and $\theta<\frac{4m-2}{4m+n}$. We weaken the regularity requirement of the initial velocity, and improve some known results of the one-dimensional system.

Keywords: density-dependent viscosity coefficients., Compressible Navier-Stokes equations.

Mathematics Subject Classification: Primary: 76D05, 35R35; Secondary: 35Q35, 76N1.

Citation: 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

References:

[1]	D. Bresch, B. Desjardins and C. K. Lin, On some compressible fluid models: Korteweg, lubrication, and shallow water systems, Comm. Partial Differential Equations, 28 (2003), 843-868. doi: doi:10.1081/PDE-120020499.
[2]	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.
[3]	G. Q. Chen and M. Kratka, Global solutions to the Navier-Stokes equations for compressible heat-conducting flow with symmetry and free boundary, Comm. Partial Differential Equations, 27 (2002), 907-943. doi: doi:10.1081/PDE-120020499.
[4]	G. Q. Chen, Vacuum states and global stability of rarefaction waves for compressible flow, Methods Appl. Anal., 7 (2000), 337-361.
[5]	P. Chen and T. Zhang, A vacuum problem for multidimensional compressible Navier-Stokes equations with degenerate viscosity coefficients, Commun. Pure Appl. Anal., 7 (2008), 987-1016. doi: doi:10.3934/cpaa.2008.7.987.
[6]	D. Y. Fang and T. Zhang, Compressible Navier-Stokes equations with vacuum state in one dimension, Commun. Pure Appl. Anal., 3 (2004), 675-694. doi: doi:10.3934/cpaa.2004.3.675.
[7]	D. Y. Fang and T. Zhang, A note on compressible Navier-Stokes equations with vacuum state in one dimension, Nonlinear Anal., 58 (2004), 719-731. doi: doi:10.1016/j.na.2004.05.016.
[8]	D. Y. Fang and T. Zhang, Global solutions of the Navier-Stokes equations for compressible flow with density-dependent viscosity and discontinuous initial data, J. Differential Equations, 222 (2006), 63-94. doi: doi:10.1016/j.jde.2005.07.011.
[9]	Z. H. Guo, Q. S. Jiu and Z. P. Xin, Spherically symmetric isentropic compressible flows with density-dependent viscosity coefficients, SIAM J. Math. Anal., 39 (2008), 1402-1427. doi: doi:10.1137/070680333.
[10]	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), 887-898. doi: doi:10.1137/0151043.
[11]	D. Hoff, Discontinuous solutions of the Navier-Stokes equations for compressible flow, Arch. Rational Mech. Anal., 114 (1991), 15-46. doi: doi:10.1007/BF00375683.
[12]	D. Hoff, Global well-posedness of the Cauchy problem for the Navier-Stokes equations of nonisentropic flow with discontinuous initial data, J. Differential Equations, 95 (1992), 33-74. doi: doi:10.1016/0022-0396(92)90042-L.
[13]	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), 1225-1302. doi: doi:10.1512/iumj.1992.41.41060.
[14]	S. Jiang, Z. P. Xin and P. Zhang, Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity, Methods Appl. Anal., 12 (2005), 239-251.
[15]	S. Jiang and A. A. Zlotnik, Global well-posedness of the Cauchy problem for the equations of a one-dimensional viscous heat-conducting gas with Lebesgue initial data, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 939-960. doi: doi:10.1017/S0308210500003565.
[16]	P.-L. Lions, "Mathematical Topics in Fluid Mechanics," Vol. 1-2. Oxford University Press: New York, 1996, 1998.
[17]	T. P. Liu, Z. P. Xin and T. Yang, Vacuum states for compressible flow, Discrete Contin. Dynam. Systems, 4 (1998), 1-32.
[18]	A. Mellet and A. Vasseur, On the barotropic compressible Navier-Stokes equation, Comm. Partial Differential Equations, 32 (2007), 431-452. doi: doi:10.1080/03605300600857079.
[19]	X. L. Qin, Z. A. Yao and H. X. Zhao, One dimensional compressible Navier-Stokes equations with density-dependent viscosity and free boundaries, Comm. Pure Appl. Anal., 7 (2008), 373-381.
[20]	S. W. Vong, T. Yang and C. J. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum(II), J. Differential Equations, 192 (2003), 475-501. doi: doi:10.1016/S0022-0396(03)00060-3.
[21]	V. A. Vaigant and A. V. Kazhikhov, On existence of global solutions to the two-dimensional Navier-Stokes equations for a compressible viscosity fluid, Siberian Math. J., 2 (1995), 1108-1141. doi: doi:10.1007/BF02106835.
[22]	Z. P. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240. doi: doi:10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C.
[23]	T. Yang, Z. A. Yao and C. J. Zhu, Compressible Navier-Stokes equations with density-dependent viscosity and vacuum, Comm. Partial Differential Equations, 26 (2001), 965-981. doi: doi:10.1081/PDE-100002385.
[24]	T. Yang and C. J. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum, Comm. Math. Phys., 230 (2002), 329-363. doi: doi:10.1007/s00220-002-0703-6.
[25]	T. Zhang and D. Y. Fang, Global behavior of spherically symmetric Navier-Stokes equations with density-dependent viscosity, J. Differential Equations, 236 (2007), 293-341. doi: doi:10.1016/j.jde.2007.01.025.
[26]	T. Zhang and D. Y. Fang, Global behavior of spherically symmetric Navier-Stokes-Poisson system with degenerate viscosity coefficients, Arch. Ration. Mech. Anal., 191 (2009), 195-243. doi: doi:10.1007/s00205-008-0183-8.
[27]	T. Zhang and D. Y. Fang, A note on spherically symmetric isentropic compressible flows with density-dependent viscosity coefficients, Nonlinear Analysis: Real World Applications, 10 (2009), 2272-2285. doi: doi:10.1016/j.nonrwa.2008.04.014.

show all references

References:

[1]	D. Bresch, B. Desjardins and C. K. Lin, On some compressible fluid models: Korteweg, lubrication, and shallow water systems, Comm. Partial Differential Equations, 28 (2003), 843-868. doi: doi:10.1081/PDE-120020499.
[2]	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.
[3]	G. Q. Chen and M. Kratka, Global solutions to the Navier-Stokes equations for compressible heat-conducting flow with symmetry and free boundary, Comm. Partial Differential Equations, 27 (2002), 907-943. doi: doi:10.1081/PDE-120020499.
[4]	G. Q. Chen, Vacuum states and global stability of rarefaction waves for compressible flow, Methods Appl. Anal., 7 (2000), 337-361.
[5]	P. Chen and T. Zhang, A vacuum problem for multidimensional compressible Navier-Stokes equations with degenerate viscosity coefficients, Commun. Pure Appl. Anal., 7 (2008), 987-1016. doi: doi:10.3934/cpaa.2008.7.987.
[6]	D. Y. Fang and T. Zhang, Compressible Navier-Stokes equations with vacuum state in one dimension, Commun. Pure Appl. Anal., 3 (2004), 675-694. doi: doi:10.3934/cpaa.2004.3.675.
[7]	D. Y. Fang and T. Zhang, A note on compressible Navier-Stokes equations with vacuum state in one dimension, Nonlinear Anal., 58 (2004), 719-731. doi: doi:10.1016/j.na.2004.05.016.
[8]	D. Y. Fang and T. Zhang, Global solutions of the Navier-Stokes equations for compressible flow with density-dependent viscosity and discontinuous initial data, J. Differential Equations, 222 (2006), 63-94. doi: doi:10.1016/j.jde.2005.07.011.
[9]	Z. H. Guo, Q. S. Jiu and Z. P. Xin, Spherically symmetric isentropic compressible flows with density-dependent viscosity coefficients, SIAM J. Math. Anal., 39 (2008), 1402-1427. doi: doi:10.1137/070680333.
[10]	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), 887-898. doi: doi:10.1137/0151043.
[11]	D. Hoff, Discontinuous solutions of the Navier-Stokes equations for compressible flow, Arch. Rational Mech. Anal., 114 (1991), 15-46. doi: doi:10.1007/BF00375683.
[12]	D. Hoff, Global well-posedness of the Cauchy problem for the Navier-Stokes equations of nonisentropic flow with discontinuous initial data, J. Differential Equations, 95 (1992), 33-74. doi: doi:10.1016/0022-0396(92)90042-L.
[13]	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), 1225-1302. doi: doi:10.1512/iumj.1992.41.41060.
[14]	S. Jiang, Z. P. Xin and P. Zhang, Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity, Methods Appl. Anal., 12 (2005), 239-251.
[15]	S. Jiang and A. A. Zlotnik, Global well-posedness of the Cauchy problem for the equations of a one-dimensional viscous heat-conducting gas with Lebesgue initial data, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 939-960. doi: doi:10.1017/S0308210500003565.
[16]	P.-L. Lions, "Mathematical Topics in Fluid Mechanics," Vol. 1-2. Oxford University Press: New York, 1996, 1998.
[17]	T. P. Liu, Z. P. Xin and T. Yang, Vacuum states for compressible flow, Discrete Contin. Dynam. Systems, 4 (1998), 1-32.
[18]	A. Mellet and A. Vasseur, On the barotropic compressible Navier-Stokes equation, Comm. Partial Differential Equations, 32 (2007), 431-452. doi: doi:10.1080/03605300600857079.
[19]	X. L. Qin, Z. A. Yao and H. X. Zhao, One dimensional compressible Navier-Stokes equations with density-dependent viscosity and free boundaries, Comm. Pure Appl. Anal., 7 (2008), 373-381.
[20]	S. W. Vong, T. Yang and C. J. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum(II), J. Differential Equations, 192 (2003), 475-501. doi: doi:10.1016/S0022-0396(03)00060-3.
[21]	V. A. Vaigant and A. V. Kazhikhov, On existence of global solutions to the two-dimensional Navier-Stokes equations for a compressible viscosity fluid, Siberian Math. J., 2 (1995), 1108-1141. doi: doi:10.1007/BF02106835.
[22]	Z. P. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240. doi: doi:10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C.
[23]	T. Yang, Z. A. Yao and C. J. Zhu, Compressible Navier-Stokes equations with density-dependent viscosity and vacuum, Comm. Partial Differential Equations, 26 (2001), 965-981. doi: doi:10.1081/PDE-100002385.
[24]	T. Yang and C. J. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum, Comm. Math. Phys., 230 (2002), 329-363. doi: doi:10.1007/s00220-002-0703-6.
[25]	T. Zhang and D. Y. Fang, Global behavior of spherically symmetric Navier-Stokes equations with density-dependent viscosity, J. Differential Equations, 236 (2007), 293-341. doi: doi:10.1016/j.jde.2007.01.025.
[26]	T. Zhang and D. Y. Fang, Global behavior of spherically symmetric Navier-Stokes-Poisson system with degenerate viscosity coefficients, Arch. Ration. Mech. Anal., 191 (2009), 195-243. doi: doi:10.1007/s00205-008-0183-8.
[27]	T. Zhang and D. Y. Fang, A note on spherically symmetric isentropic compressible flows with density-dependent viscosity coefficients, Nonlinear Analysis: Real World Applications, 10 (2009), 2272-2285. doi: doi:10.1016/j.nonrwa.2008.04.014.

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