# American Institute of Mathematical Sciences

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

 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

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.
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," 3rd 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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##### 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," 3rd 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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