Asymptotic behavior of solutions to 1D compressible Navier-Stokes equations with gravity and vacuum

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November 2011, 30(4): 1263-1283. doi: 10.3934/dcds.2011.30.1263

Asymptotic behavior of solutions to 1D compressible Navier-Stokes equations with gravity and vacuum

Changjiang Zhu ^1, and Ruizhao Zi ^1,

1.	Laboratory of Nonlinear Analysis, School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China, China

Received June 2010 Revised August 2010 Published May 2011

Abstract
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In this paper, we study the asymptotic behavior of solutions to one-dimensional compressible Navier-Stokes equations with gravity and vacuum for isentropic flows with density-dependent viscosity $\mu(\rho)=c\rho^{\theta}$. Under some suitable assumptions on the initial date and $\gamma>1$, if $\theta\in(0,\frac{\gamma}{2}]$, we prove the weak solution $(\rho(x,t),u(x,t))$ behavior asymptotically to the stationary one by adapting and modifying the technique of weighted estimates. This result improves the one in [5] where Duan showed that the weak solution converges to the stationary one in the sense of integral for shallow water model. In addition, if $\theta\in(0,\frac{\gamma}{2}]\cap(0,\gamma-1]$, following the same idea in [9], we estimate the stabilization rate of the solution as time tends to infinity in the sense of $L^\infty$ norm, weighted $L^2$ norm and weighted $H^1$ norm.

Keywords: density-dependent viscosity, a priori estimates, Navier-Stokes equations, asymptotic behavior., vacuum.

Mathematics Subject Classification: Primary: 76D05, 35B45, 35B4.

Citation: Changjiang Zhu, Ruizhao Zi. Asymptotic behavior of solutions to 1D compressible Navier-Stokes equations with gravity and vacuum. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1263-1283. doi: 10.3934/dcds.2011.30.1263

References:

[1]	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.
[2]	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.
[3]	S. Chapman and T.-G. Cowling, "The Mathematical Theory of Non-Uniform Gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases," 3^rd edition, prepared in co-operation with D. Burnett, Cambridge University Press, London, 1970.
[4]	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.
[5]	Q. Duan, On the dynamics of Navier-Stokes equations for a shallow water model, J. Differential Equations, 250 (2011), 2687-2714.
[6]	D.-Y. Fang and T. Zhang, Copressible Navier-Stokes equations with vacuum state in the case of general pressure law, Math. Methods Appl. Sci., 29 (2006), 1081-1106. doi: 10.1002/mma.708.
[7]	D.-Y. Fang and T. Zhang, Compressible Navier-Stokes equations with vacuum state in one dimension, Comm. Pure Appl. Anal., 3 (2004), 675-694. doi: 10.3934/cpaa.2004.3.675.
[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.
[9]	D.-Y. Fang and T. Zhang, Global behavior of compressible Navier-Stokes equations with a degenerate viscosity coefficient, Arch. Rational Mech. Anal., 182 (2006), 223-253. doi: 10.1007/s00205-006-0425-6.
[10]	D.-Y. Fang and T. Zhang, A note on spherically symmetric isentropic compressible flows with density-dependent viscosity coefficients, Nonlinear Anal., Real World Appl., 10 (2009), 2272-2285.
[11]	D.-Y. Fang and T. Zhang, Global behavior of spherically symmetric Navier-Stokes-Poisson system with degenerate viscosity coefficients, Arch. Rational Mech. Anal., 191 (2009), 195-243. doi: 10.1007/s00205-008-0183-8.
[12]	E. Feireisl, A. Novotny and H. Petzeltov, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392.
[13]	Z.-H. Guo and C.-J. Zhu, Remarks on one-dimensional compressible Navier-Stokes equations with density-dependent viscosity and vacuum, Acta Math. Sinica, Ser. B, 26 (2010), 2015-2030.
[14]	Z.-H. Guo and C.-J. Zhu, Global weak solutions and asymptotic behavior to 1D compressible Navier-Stokes equations with density-dependent viscosity and vacuum, J. Differential Equations, 248 (2010), 2768-2799.
[15]	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: 10.1137/070680333.
[16]	D. Hoff, Construction of solutions for compressible, isentropic Navier-Stokes equations in one space dimension with nonsmooth initial data, Proc. Roy. Soc. Edinburgh, Sect. A, 103 (1986), 301-315.
[17]	D. Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data, Arch. Rational Mech. Anal., 132 (1995), 1-14. doi: 10.1007/BF00390346.
[18]	D. Hoff and T.-P. Liu, The inviscid limit for the Navier-Stokes equations of compressible isentropic flow with shock data, Indiana Univ. Math. J., 38 (1989), 861-915. doi: 10.1512/iumj.1989.38.38041.
[19]	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: 10.1137/0151043.
[20]	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.
[21]	P.-L. Lions, "Mathematical Topics in Fluid Mechanics," Oxford: Clarendon Press, 2, 1998.
[22]	T.-P. Liu, Z.-P. Xin and T. Yang, Vacuum states of compressible flow, Discrete Contin. Dynam. Systems, 4 (1998), 1-32.
[23]	T. Luo, Z.-P. Xin and T. Yang, Interface behavior of compressible Navier-Stokes equations with vacuum, SIAM J. Math. Anal., 31 (2000), 1175-1191. doi: 10.1137/S0036141097331044.
[24]	A. Matsumura and T. Nishida, The initial-value problem for the equations of motion of compressible viscous and heat conducting fluids, Proc. Japan Acad., Ser. A, 55 (1979), 337-342.
[25]	A. Matsumura and T. Nishida, The initial-value problem for the equations of motion of viscous and heat conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.
[26]	A. Mellet and A. Vasseur, On the isentropic compressible Navier-Stokes equation, Comm. Partial Differential Equations, 32 (2007), 431-452.
[27]	M. Okada, Free boundary value problems for the equation of one-dimensional motion of viscous gas, Japan J. Appl. Math., 6 (1989), 161-177. doi: 10.1007/BF03167921.
[28]	M. Okada and T. Makino, Free boundary problem for the equations of spherically symmetrical motion of viscous gas, Japan J. Indust. Appl. Math., 10 (1993), 219-235.
[29]	M. Okada, Free boundary problem for one-dimensional motion of compressible gas and vaccum, Japan J. Indust. Appl. Math., (2004), 109-128.
[30]	X. Qin, Z.-A. Yao and H. Zhao, One dimensional compressible Navier-Stokes euqaitons with density-dependent viscosity and free boudnaries, Comm. Pure Appl. Anal., 7 (2008), 373-381.
[31]	I. Straskraba and A. Zlotnik, Global behavior of 1D-viscous compressible barotropic fluid with a free boundary and large data, J. Math. Fluid Mech., 5 (2003), 119-143.
[32]	Z.-G. Wu and C.-J. Zhu, Vacuum problem for 1D compressible Navier-Stokes equations with gravity and general pressure law, Z. Angew. Math. Phys., 60 (2009), 246-270. doi: 10.1007/s00033-008-6109-3.
[33]	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.
[34]	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.
[35]	T. Yang and H.-J. Zhao, A vacuum problem for the one-dimensional compressible Navier- Stokes equations with density-dependent viscosity, J. Differential Equations, 184 (2002), 163-184.
[36]	T. Yang and C.-J. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum, Comm. Math. Phys., 230 (2002), 329-363. doi: 10.1007/s00220-002-0703-6.
[37]	C.-J. Zhu, Asymptotic behavior of compressible Navier-Stokes equations with density-dependent viscosity and vacumm, Comm. Math. Phys., 293 (2010), 279-299. doi: 10.1007/s00220-009-0914-1.

show all references

References:

[1]	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.
[2]	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.
[3]	S. Chapman and T.-G. Cowling, "The Mathematical Theory of Non-Uniform Gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases," 3^rd edition, prepared in co-operation with D. Burnett, Cambridge University Press, London, 1970.
[4]	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.
[5]	Q. Duan, On the dynamics of Navier-Stokes equations for a shallow water model, J. Differential Equations, 250 (2011), 2687-2714.
[6]	D.-Y. Fang and T. Zhang, Copressible Navier-Stokes equations with vacuum state in the case of general pressure law, Math. Methods Appl. Sci., 29 (2006), 1081-1106. doi: 10.1002/mma.708.
[7]	D.-Y. Fang and T. Zhang, Compressible Navier-Stokes equations with vacuum state in one dimension, Comm. Pure Appl. Anal., 3 (2004), 675-694. doi: 10.3934/cpaa.2004.3.675.
[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.
[9]	D.-Y. Fang and T. Zhang, Global behavior of compressible Navier-Stokes equations with a degenerate viscosity coefficient, Arch. Rational Mech. Anal., 182 (2006), 223-253. doi: 10.1007/s00205-006-0425-6.
[10]	D.-Y. Fang and T. Zhang, A note on spherically symmetric isentropic compressible flows with density-dependent viscosity coefficients, Nonlinear Anal., Real World Appl., 10 (2009), 2272-2285.
[11]	D.-Y. Fang and T. Zhang, Global behavior of spherically symmetric Navier-Stokes-Poisson system with degenerate viscosity coefficients, Arch. Rational Mech. Anal., 191 (2009), 195-243. doi: 10.1007/s00205-008-0183-8.
[12]	E. Feireisl, A. Novotny and H. Petzeltov, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392.
[13]	Z.-H. Guo and C.-J. Zhu, Remarks on one-dimensional compressible Navier-Stokes equations with density-dependent viscosity and vacuum, Acta Math. Sinica, Ser. B, 26 (2010), 2015-2030.
[14]	Z.-H. Guo and C.-J. Zhu, Global weak solutions and asymptotic behavior to 1D compressible Navier-Stokes equations with density-dependent viscosity and vacuum, J. Differential Equations, 248 (2010), 2768-2799.
[15]	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: 10.1137/070680333.
[16]	D. Hoff, Construction of solutions for compressible, isentropic Navier-Stokes equations in one space dimension with nonsmooth initial data, Proc. Roy. Soc. Edinburgh, Sect. A, 103 (1986), 301-315.
[17]	D. Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data, Arch. Rational Mech. Anal., 132 (1995), 1-14. doi: 10.1007/BF00390346.
[18]	D. Hoff and T.-P. Liu, The inviscid limit for the Navier-Stokes equations of compressible isentropic flow with shock data, Indiana Univ. Math. J., 38 (1989), 861-915. doi: 10.1512/iumj.1989.38.38041.
[19]	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: 10.1137/0151043.
[20]	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.
[21]	P.-L. Lions, "Mathematical Topics in Fluid Mechanics," Oxford: Clarendon Press, 2, 1998.
[22]	T.-P. Liu, Z.-P. Xin and T. Yang, Vacuum states of compressible flow, Discrete Contin. Dynam. Systems, 4 (1998), 1-32.
[23]	T. Luo, Z.-P. Xin and T. Yang, Interface behavior of compressible Navier-Stokes equations with vacuum, SIAM J. Math. Anal., 31 (2000), 1175-1191. doi: 10.1137/S0036141097331044.
[24]	A. Matsumura and T. Nishida, The initial-value problem for the equations of motion of compressible viscous and heat conducting fluids, Proc. Japan Acad., Ser. A, 55 (1979), 337-342.
[25]	A. Matsumura and T. Nishida, The initial-value problem for the equations of motion of viscous and heat conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.
[26]	A. Mellet and A. Vasseur, On the isentropic compressible Navier-Stokes equation, Comm. Partial Differential Equations, 32 (2007), 431-452.
[27]	M. Okada, Free boundary value problems for the equation of one-dimensional motion of viscous gas, Japan J. Appl. Math., 6 (1989), 161-177. doi: 10.1007/BF03167921.
[28]	M. Okada and T. Makino, Free boundary problem for the equations of spherically symmetrical motion of viscous gas, Japan J. Indust. Appl. Math., 10 (1993), 219-235.
[29]	M. Okada, Free boundary problem for one-dimensional motion of compressible gas and vaccum, Japan J. Indust. Appl. Math., (2004), 109-128.
[30]	X. Qin, Z.-A. Yao and H. Zhao, One dimensional compressible Navier-Stokes euqaitons with density-dependent viscosity and free boudnaries, Comm. Pure Appl. Anal., 7 (2008), 373-381.
[31]	I. Straskraba and A. Zlotnik, Global behavior of 1D-viscous compressible barotropic fluid with a free boundary and large data, J. Math. Fluid Mech., 5 (2003), 119-143.
[32]	Z.-G. Wu and C.-J. Zhu, Vacuum problem for 1D compressible Navier-Stokes equations with gravity and general pressure law, Z. Angew. Math. Phys., 60 (2009), 246-270. doi: 10.1007/s00033-008-6109-3.
[33]	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.
[34]	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.
[35]	T. Yang and H.-J. Zhao, A vacuum problem for the one-dimensional compressible Navier- Stokes equations with density-dependent viscosity, J. Differential Equations, 184 (2002), 163-184.
[36]	T. Yang and C.-J. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum, Comm. Math. Phys., 230 (2002), 329-363. doi: 10.1007/s00220-002-0703-6.
[37]	C.-J. Zhu, Asymptotic behavior of compressible Navier-Stokes equations with density-dependent viscosity and vacumm, Comm. Math. Phys., 293 (2010), 279-299. doi: 10.1007/s00220-009-0914-1.

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