March  2013, 12(2): 647-661. doi: 10.3934/cpaa.2013.12.647

The decay estimates of solutions for 1D compressible flows with density-dependent viscosity coefficients

1. 

School of Mathematical and information Science, Henan Polytechnic University, Jiaozuo Henan, 454003, China

2. 

Beijing No. 19 Middle School, Beijing 100089, China

3. 

School of Mathematical Sciences, Capital Normal University, Beijing 100037

Received  February 2010 Revised  April 2012 Published  September 2012

We consider the Cauchy problem for 1D compressible flows with density-dependent viscosity coefficients, we are mainly concerned with the decay estimates of the density and velocity as $t \rightarrow \infty$. Firstly, we obtain the decay estimates of $\rho-\bar{\rho}$ and u in $L^2(R)$ norm, then we obtain the decay estimate of $\rho-\bar{\rho}$ in $L^{\infty}(R)$ norm as $\bar{\rho}>0$. Secondly, we construct a functional and use the energy method to obtain the decay estimate of $\rho$ in $L^{\infty}(R)$ norm as $\bar{\rho}=0$.
Citation: Wuming Li, Xiaojun Liu, Quansen Jiu. The decay estimates of solutions for 1D compressible flows with density-dependent viscosity coefficients. Communications on Pure & Applied Analysis, 2013, 12 (2) : 647-661. doi: 10.3934/cpaa.2013.12.647
References:
[1]

Robert A. Adams and John J. F. Fournier, "Sobolev Spaces," Second Edition,, Academic Press, (2003). doi: 10.1016/S0079-8169(03)80001-6.

[2]

D. Bresch, B. Desjardins and Chi-Kun Lin, On some compressible fluid models: Korteweg, lubrication, and shallow water systems,, Communications in Partial Differential Equations, 28 (2003), 843. doi: 10.1081/PDE-120020499.

[3]

Eduard Feireisl, On the motion of a viscous, compressible, and heat conducting fluid,, Indiana Univ. Math. J., 53 (2004), 1705. doi: 10.1512/iumj.2004.53.2510.

[4]

E. Feireisl, A. Novotný and H.Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations,, J. Math. Fluid Mech., 3 (2001), 358. doi: 10.1007/PL00000976.

[5]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I,, Archive for Rational Mechanics and Analysis, 16 (1964), 269. doi: 10.1007/BF00276188.

[6]

J. F. Gerbeau and B. Perthame, Derivation of viscous Saint-Venant system for laminar shallow water; Numerical validation,, Discrete and Continuous Dynamical Systems, 1 (2001), 89. doi: 10.3934/dcdsb.2001.1.89.

[7]

Y. Giga and T. Miyakawa, Solutions in $L_r$ of the Navier-Stokes initial value problem,, Archive for Rational Mechanics and Analysis, 89 (1985), 267. doi: 10.1007/BF00276875.

[8]

David Gilbarg and Neil S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer, (1998).

[9]

Zhenhua Guo, Quansen Jiu and Zhouping Xin, Spherically symmetric isentropic compressible flows with density-dependent viscosity coefficients,, SIAM Journal on Mathematical Analysis, 39 (2008), 1402. doi: 10.1137/070680333.

[10]

Zhenhua Guo and Changjiang Zhu, Remarks on one-dimensional Compressible Navier-Stokes equations with density-dependent viscosity and vacuum,, Acta Mathematica Sinica, 26 (2010), 2015. doi: 10.1007/s10114-009-7559-z.

[11]

Cheng He and Zhouping Xin, On the regularity of weak solutions to the Magnetohydrodynamic equations,, Journal of Differential Equations, 213 (2005), 235. doi: 10.1016/j.jde.2004.07.002.

[12]

David Hoff, Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data,, Trans. Amer. Math. Soc., 303 (1987), 169. doi: 10.1090/S0002-9947-1987-0896014-6.

[13]

D. Hoff, Global solutions of the equations of one-dimensional, compressible flow with large data and forces, and with differing end states,, ZAMP, 49 (1998), 774. doi: 10.1007/PL00001488.

[14]

David Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data,, Archive for Rational Mechanics and Analysis, 132 (1995), 1. doi: 10.1007/BF00390346.

[15]

David Hoff and Denis 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: 10.1137/0151043.

[16]

Quansen Jiu and Zhouping Xin, The Cauchy problem for 1D compressible flows with density-dependent viscosity coefficients,, Kinetic and Related Models, 1 (2008), 313. doi: 10.3934/krm.2008.1.313.

[17]

A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas,, J. Appl. Math. Mech., 41 (1977), 273. doi: 10.1016/0021-8928(77)90011-9.

[18]

H. Kozono and T. Ogawa, Some $L^p$ estimate for the exterior Stokes flow and an application to non-stationary Navier-Stokes equations,, Indiana Univ. Math. J., 41 (1992), 789. doi: 10.1512/iumj.1992.41.41041.

[19]

H. Kozono and H. Sohr, Density properties for solenoidal vector fields, with applications to the Navier-Stokes equations in exterior domains,, Journal of the Mathematical Society of Japan, 44 (1992), 307. doi: 10.2969/jmsj/04420307.

[20]

Hai-liang Li, Jing Li and Zhouping Xin, Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations,, Communications in Mathematical Physics, 281 (2008), 401. doi: 10.1007/s00220-008-0495-4.

[21]

P. L. Lions, "Mathematical Topics in Fluid Mechanics," Volume 2, Compressible Models,, Oxford Science Publications, (1998).

[22]

T. P. Liu, Z. P. Xin and T. Yang, Vacuum states for compressible flow,, Discrete and Continuous Dynamical Systems, 4 (1998), 1. doi: 10.3934/dcds.1998.4.1.

[23]

K. Masuda, Weak solutions of Navier-Stokes equations,, Tohoku Mathematical Journal, 36 (1984), 623. doi: 10.2748/tmj/1178228767.

[24]

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.

[25]

A. Mellet and A. Vasseur, On the barotropic compressible Navier-Stokes equation,, Comm. Partial Differential Equatiions, 32 (2007), 431. doi: 10.1080/03605300600857079.

[26]

J. Serrin, The initial value problem for the Navier-Stokes equations,, Nonlinear Problems, (1963), 69.

[27]

J. Simon, Compact sets in the space $L^p(0,T ;B)$,, Ann. Mat. Pura Appl., 146 (1986), 65. doi: 10.1007/BF01762360.

[28]

H. Sohr, "The Navier-Stokes Equations: An Elementary Functional Analytic Approach,", Birkh$\ddota$user Advanced Texts, (2001).

[29]

Roger Temam, "Navier-Stokes Equations: Theory And Numerical Analysis,", North-Holland Publishing Company-Amsterdam, (1977).

[30]

Zhouping Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density,, Communications on Pure and Applied Mathematics, 51 (1998), 229. doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C.

[31]

Ting zhang and Daoyuan Fang, Global behavior of compressible Navier-Stokes equations with a degenerate viscosity coefficient,, Archive for Rational Mechanics and Analysis, 182 (2006), 223. doi: 10.1007/s00205-006-0425-6.

[32]

Changjiang Zhu, Asymptotic behavior of compressible Navier-Stokes equations with density-dependent viscosity and vacuum,, Communications in Mathematical Physics, 293 (2010), 279. doi: 10.1007/s00220-009-0914-1.

show all references

References:
[1]

Robert A. Adams and John J. F. Fournier, "Sobolev Spaces," Second Edition,, Academic Press, (2003). doi: 10.1016/S0079-8169(03)80001-6.

[2]

D. Bresch, B. Desjardins and Chi-Kun Lin, On some compressible fluid models: Korteweg, lubrication, and shallow water systems,, Communications in Partial Differential Equations, 28 (2003), 843. doi: 10.1081/PDE-120020499.

[3]

Eduard Feireisl, On the motion of a viscous, compressible, and heat conducting fluid,, Indiana Univ. Math. J., 53 (2004), 1705. doi: 10.1512/iumj.2004.53.2510.

[4]

E. Feireisl, A. Novotný and H.Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations,, J. Math. Fluid Mech., 3 (2001), 358. doi: 10.1007/PL00000976.

[5]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I,, Archive for Rational Mechanics and Analysis, 16 (1964), 269. doi: 10.1007/BF00276188.

[6]

J. F. Gerbeau and B. Perthame, Derivation of viscous Saint-Venant system for laminar shallow water; Numerical validation,, Discrete and Continuous Dynamical Systems, 1 (2001), 89. doi: 10.3934/dcdsb.2001.1.89.

[7]

Y. Giga and T. Miyakawa, Solutions in $L_r$ of the Navier-Stokes initial value problem,, Archive for Rational Mechanics and Analysis, 89 (1985), 267. doi: 10.1007/BF00276875.

[8]

David Gilbarg and Neil S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer, (1998).

[9]

Zhenhua Guo, Quansen Jiu and Zhouping Xin, Spherically symmetric isentropic compressible flows with density-dependent viscosity coefficients,, SIAM Journal on Mathematical Analysis, 39 (2008), 1402. doi: 10.1137/070680333.

[10]

Zhenhua Guo and Changjiang Zhu, Remarks on one-dimensional Compressible Navier-Stokes equations with density-dependent viscosity and vacuum,, Acta Mathematica Sinica, 26 (2010), 2015. doi: 10.1007/s10114-009-7559-z.

[11]

Cheng He and Zhouping Xin, On the regularity of weak solutions to the Magnetohydrodynamic equations,, Journal of Differential Equations, 213 (2005), 235. doi: 10.1016/j.jde.2004.07.002.

[12]

David Hoff, Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data,, Trans. Amer. Math. Soc., 303 (1987), 169. doi: 10.1090/S0002-9947-1987-0896014-6.

[13]

D. Hoff, Global solutions of the equations of one-dimensional, compressible flow with large data and forces, and with differing end states,, ZAMP, 49 (1998), 774. doi: 10.1007/PL00001488.

[14]

David Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data,, Archive for Rational Mechanics and Analysis, 132 (1995), 1. doi: 10.1007/BF00390346.

[15]

David Hoff and Denis 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: 10.1137/0151043.

[16]

Quansen Jiu and Zhouping Xin, The Cauchy problem for 1D compressible flows with density-dependent viscosity coefficients,, Kinetic and Related Models, 1 (2008), 313. doi: 10.3934/krm.2008.1.313.

[17]

A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas,, J. Appl. Math. Mech., 41 (1977), 273. doi: 10.1016/0021-8928(77)90011-9.

[18]

H. Kozono and T. Ogawa, Some $L^p$ estimate for the exterior Stokes flow and an application to non-stationary Navier-Stokes equations,, Indiana Univ. Math. J., 41 (1992), 789. doi: 10.1512/iumj.1992.41.41041.

[19]

H. Kozono and H. Sohr, Density properties for solenoidal vector fields, with applications to the Navier-Stokes equations in exterior domains,, Journal of the Mathematical Society of Japan, 44 (1992), 307. doi: 10.2969/jmsj/04420307.

[20]

Hai-liang Li, Jing Li and Zhouping Xin, Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations,, Communications in Mathematical Physics, 281 (2008), 401. doi: 10.1007/s00220-008-0495-4.

[21]

P. L. Lions, "Mathematical Topics in Fluid Mechanics," Volume 2, Compressible Models,, Oxford Science Publications, (1998).

[22]

T. P. Liu, Z. P. Xin and T. Yang, Vacuum states for compressible flow,, Discrete and Continuous Dynamical Systems, 4 (1998), 1. doi: 10.3934/dcds.1998.4.1.

[23]

K. Masuda, Weak solutions of Navier-Stokes equations,, Tohoku Mathematical Journal, 36 (1984), 623. doi: 10.2748/tmj/1178228767.

[24]

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.

[25]

A. Mellet and A. Vasseur, On the barotropic compressible Navier-Stokes equation,, Comm. Partial Differential Equatiions, 32 (2007), 431. doi: 10.1080/03605300600857079.

[26]

J. Serrin, The initial value problem for the Navier-Stokes equations,, Nonlinear Problems, (1963), 69.

[27]

J. Simon, Compact sets in the space $L^p(0,T ;B)$,, Ann. Mat. Pura Appl., 146 (1986), 65. doi: 10.1007/BF01762360.

[28]

H. Sohr, "The Navier-Stokes Equations: An Elementary Functional Analytic Approach,", Birkh$\ddota$user Advanced Texts, (2001).

[29]

Roger Temam, "Navier-Stokes Equations: Theory And Numerical Analysis,", North-Holland Publishing Company-Amsterdam, (1977).

[30]

Zhouping Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density,, Communications on Pure and Applied Mathematics, 51 (1998), 229. doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C.

[31]

Ting zhang and Daoyuan Fang, Global behavior of compressible Navier-Stokes equations with a degenerate viscosity coefficient,, Archive for Rational Mechanics and Analysis, 182 (2006), 223. doi: 10.1007/s00205-006-0425-6.

[32]

Changjiang Zhu, Asymptotic behavior of compressible Navier-Stokes equations with density-dependent viscosity and vacuum,, Communications in Mathematical Physics, 293 (2010), 279. doi: 10.1007/s00220-009-0914-1.

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