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

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

1. 

Department of Mathematics, Zhejiang University, Hangzhou 310027, China

Received  May 2010 Revised  September 2010 Published  December 2010

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.
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:
[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

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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

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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