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Multiple solutions for superlinear elliptic systems of Hamiltonian type
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 |
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.
|
[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.
|
[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, (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.
|
[5] |
Q. Duan, On the dynamics of Navier-Stokes equations for a shallow water model,, J. Differential Equations, 250 (2011), 2687. Google Scholar |
[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.
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.
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.
|
[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.
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., 10 (2009), 2272.
|
[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.
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.
|
[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, 26 (2010), 2015. Google Scholar |
[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. Google Scholar |
[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.
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, 103 (1986), 301.
|
[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.
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.
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.
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.
|
[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. Google Scholar |
[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.
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., 55 (1979), 337.
|
[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.
|
[26] |
A. Mellet and A. Vasseur, On the isentropic compressible Navier-Stokes equation,, Comm. Partial Differential Equations, 32 (2007), 431.
|
[27] |
M. Okada, Free boundary value problems for the equation of one-dimensional motion of viscous gas,, Japan J. Appl. Math., 6 (1989), 161.
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.
|
[29] |
M. Okada, Free boundary problem for one-dimensional motion of compressible gas and vaccum,, Japan J. Indust. Appl. Math., (2004), 109.
|
[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.
|
[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.
|
[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.
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.
|
[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.
|
[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.
|
[36] |
T. Yang and C.-J. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum,, Comm. Math. Phys., 230 (2002), 329.
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.
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.
|
[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.
|
[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, (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.
|
[5] |
Q. Duan, On the dynamics of Navier-Stokes equations for a shallow water model,, J. Differential Equations, 250 (2011), 2687. Google Scholar |
[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.
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.
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.
|
[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.
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., 10 (2009), 2272.
|
[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.
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.
|
[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, 26 (2010), 2015. Google Scholar |
[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. Google Scholar |
[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.
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, 103 (1986), 301.
|
[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.
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.
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.
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.
|
[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. Google Scholar |
[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.
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., 55 (1979), 337.
|
[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.
|
[26] |
A. Mellet and A. Vasseur, On the isentropic compressible Navier-Stokes equation,, Comm. Partial Differential Equations, 32 (2007), 431.
|
[27] |
M. Okada, Free boundary value problems for the equation of one-dimensional motion of viscous gas,, Japan J. Appl. Math., 6 (1989), 161.
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.
|
[29] |
M. Okada, Free boundary problem for one-dimensional motion of compressible gas and vaccum,, Japan J. Indust. Appl. Math., (2004), 109.
|
[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.
|
[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.
|
[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.
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.
|
[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.
|
[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.
|
[36] |
T. Yang and C.-J. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum,, Comm. Math. Phys., 230 (2002), 329.
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.
doi: 10.1007/s00220-009-0914-1. |
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