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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), 211-223. |
[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. |
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," 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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