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

References:

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[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. Google Scholar
[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. doi: doi:10.1081/PDE-120020499. Google Scholar
[4]	G. Q. Chen, Vacuum states and global stability of rarefaction waves for compressible flow,, Methods Appl. Anal., 7 (2000), 337. Google Scholar
[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. doi: doi:10.3934/cpaa.2008.7.987. Google Scholar
[6]	D. Y. Fang and T. Zhang, Compressible Navier-Stokes equations with vacuum state in one dimension,, Commun. Pure Appl. Anal., 3 (2004), 675. doi: doi:10.3934/cpaa.2004.3.675. Google Scholar
[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. doi: doi:10.1016/j.na.2004.05.016. Google Scholar
[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. doi: doi:10.1016/j.jde.2005.07.011. Google Scholar
[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. doi: doi:10.1137/070680333. Google Scholar
[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. doi: doi:10.1137/0151043. Google Scholar
[11]	D. Hoff, Discontinuous solutions of the Navier-Stokes equations for compressible flow,, Arch. Rational Mech. Anal., 114 (1991), 15. doi: doi:10.1007/BF00375683. Google Scholar
[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. doi: doi:10.1016/0022-0396(92)90042-L. Google Scholar
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[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. Google Scholar
[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. doi: doi:10.1017/S0308210500003565. Google Scholar
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[17]	T. P. Liu, Z. P. Xin and T. Yang, Vacuum states for compressible flow,, Discrete Contin. Dynam. Systems, 4 (1998), 1. Google Scholar
[18]	A. Mellet and A. Vasseur, On the barotropic compressible Navier-Stokes equation,, Comm. Partial Differential Equations, 32 (2007), 431. doi: doi:10.1080/03605300600857079. Google Scholar
[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. Google Scholar
[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. doi: doi:10.1016/S0022-0396(03)00060-3. Google Scholar
[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. doi: doi:10.1007/BF02106835. Google Scholar
[22]	Z. P. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density,, Comm. Pure Appl. Math., 51 (1998), 229. doi: doi:10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C. Google Scholar
[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. doi: doi:10.1081/PDE-100002385. Google Scholar
[24]	T. Yang and C. J. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum,, Comm. Math. Phys., 230 (2002), 329. doi: doi:10.1007/s00220-002-0703-6. Google Scholar
[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. doi: doi:10.1016/j.jde.2007.01.025. Google Scholar
[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. doi: doi:10.1007/s00205-008-0183-8. Google Scholar
[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. doi: doi:10.1016/j.nonrwa.2008.04.014. Google Scholar

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. doi: doi:10.1081/PDE-120020499. Google Scholar
[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. Google Scholar
[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. doi: doi:10.1081/PDE-120020499. Google Scholar
[4]	G. Q. Chen, Vacuum states and global stability of rarefaction waves for compressible flow,, Methods Appl. Anal., 7 (2000), 337. Google Scholar
[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. doi: doi:10.3934/cpaa.2008.7.987. Google Scholar
[6]	D. Y. Fang and T. Zhang, Compressible Navier-Stokes equations with vacuum state in one dimension,, Commun. Pure Appl. Anal., 3 (2004), 675. doi: doi:10.3934/cpaa.2004.3.675. Google Scholar
[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. doi: doi:10.1016/j.na.2004.05.016. Google Scholar
[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. doi: doi:10.1016/j.jde.2005.07.011. Google Scholar
[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. doi: doi:10.1137/070680333. Google Scholar
[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. doi: doi:10.1137/0151043. Google Scholar
[11]	D. Hoff, Discontinuous solutions of the Navier-Stokes equations for compressible flow,, Arch. Rational Mech. Anal., 114 (1991), 15. doi: doi:10.1007/BF00375683. Google Scholar
[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. doi: doi:10.1016/0022-0396(92)90042-L. Google Scholar
[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. doi: doi:10.1512/iumj.1992.41.41060. Google Scholar
[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. Google Scholar
[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. doi: doi:10.1017/S0308210500003565. Google Scholar
[16]	P.-L. Lions, "Mathematical Topics in Fluid Mechanics," Vol. 1-2., Oxford University Press: New York, (1996). Google Scholar
[17]	T. P. Liu, Z. P. Xin and T. Yang, Vacuum states for compressible flow,, Discrete Contin. Dynam. Systems, 4 (1998), 1. Google Scholar
[18]	A. Mellet and A. Vasseur, On the barotropic compressible Navier-Stokes equation,, Comm. Partial Differential Equations, 32 (2007), 431. doi: doi:10.1080/03605300600857079. Google Scholar
[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. Google Scholar
[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. doi: doi:10.1016/S0022-0396(03)00060-3. Google Scholar
[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. doi: doi:10.1007/BF02106835. Google Scholar
[22]	Z. P. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density,, Comm. Pure Appl. Math., 51 (1998), 229. doi: doi:10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C. Google Scholar
[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. doi: doi:10.1081/PDE-100002385. Google Scholar
[24]	T. Yang and C. J. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum,, Comm. Math. Phys., 230 (2002), 329. doi: doi:10.1007/s00220-002-0703-6. Google Scholar
[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. doi: doi:10.1016/j.jde.2007.01.025. Google Scholar
[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. doi: doi:10.1007/s00205-008-0183-8. Google Scholar
[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. doi: doi:10.1016/j.nonrwa.2008.04.014. Google Scholar

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