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Local uniqueness of steady spherical transonic shock-fronts for the three--dimensional full Euler equations
Dynamics of vacuum states for one-dimensional full compressible Navier-Stokes equations
1. | Pohang Mathematics Institute, Pohang University of Science and Technology, Pohang, Kyungbuk 790-784, South Korea |
2. | School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, China |
References:
[1] |
A. Amosov and A. Zlotnik, A semidiscrete method for solving equations of the one dimensional motion of a non homogeneous viscous heat conducting gas with nonsmooth data,, Izv. Vyssh. Uchebn. Zaved. Mat., 41 (1997), 3.
|
[2] |
D. Bresch and B. Desjardins, On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids,, J. Math. Pures Appl., 87 (2007), 57.
doi: 10.1016/j.matpur.2006.11.001. |
[3] |
T. Chang and L. Hsiao, "The Riemann Problem and Interaction of Waves in Gas Dynamics,", Longman Scientific and Technical, (1989).
|
[4] |
G. Chen, D. Hoff and K. Trivisa, Global solutions of the compressible Navier-Stokes equations with large discontinuous initial data,, Comm. Partial Differential Equations, 25 (2000), 2233.
|
[5] |
G. Chen and J. Glimm, Global solutions to the compressible Euler equations with geometrical structure,, Comm. Math. Phys., 180 (1996), 153.
doi: 10.1007/BF02101185. |
[6] |
Y. Cho and H. Kim, Existence results for viscous polytropic fluids with vacuum,, J. Differential Equations, 228 (2006), 377.
doi: 10.1016/j.jde.2006.05.001. |
[7] |
R. Duan and Y. Zhao, A note on the non-formation of vacuum states for compressible Navier-Stokes equations,, J. Math. Anal. Appl., 311 (2005), 744.
doi: 10.1016/j.jmaa.2005.02.065. |
[8] |
D. 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. |
[9] |
E. Feireisl, "Dynamics of Viscous Compressible Fluids,", Oxford University Press, (2004).
|
[10] |
H. Fujita-Yashima, M. Padula and A. Novotny, équation monodimensionnelle d'un gaz vizqueux et calorifére avec des conditions initiales moins restrictives,, Ric. Mat., 42 (1993), 199.
|
[11] |
D. Hoff, Global well-posedness of the Cauchy problem for the Navier-Stokes equations of nonisentropic flow with discontinuous initial data,, J. Differential Equations, 95 (1992), 33.
doi: 10.1016/0022-0396(92)90042-L. |
[12] |
D. Hoff, Discontinuous solutions of the Navier-Stokes equations for compressible flow,, Arch. Ration. Mech. Anal., 114 (1991), 15.
doi: 10.1007/BF00375683. |
[13] |
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. |
[14] |
D. Hoff and J. Smoller, Non-formation of vacuum states for compressible Navier-Stokes equations,, Comm. Math. Phys., 216 (2001), 255.
doi: 10.1007/s002200000322. |
[15] |
X. Huang and J. Li, Global classical and weak solutions to the three-dimensional full compressible Navier-Stokes system with vacuum and large oscillations, preprint,, \arXiv{1107.4655}., (). Google Scholar |
[16] |
X. Huang, J. Li and Z. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations,, Comm. Pure Appl. Math., 65 (2012), 549.
doi: 10.1002/cpa.21382. |
[17] |
S. Jiang and P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations,, Comm. Math. Phys., 215 (2001), 559.
doi: 10.1007/PL00005543. |
[18] |
S. Jiang and P. Zhang, Global weak solutions to the Navier-Stokes equations for a 1D viscous polytropic ideal gas,, Quart. Appl. Math., 61 (2003), 435.
|
[19] |
Y. Kanel, The Cauchy problem for equations of gas dynamics with viscosity,, Siberian Math. J., 20 (1979), 208.
|
[20] |
A. Kazhikhov, On the Cauchy problem for the equations of a viscous gas,, Siberian Math. J., 23 (1982), 44.
doi: 10.1007/BF00971419. |
[21] |
A. Kazhikhov and 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. |
[22] |
H. Li, J. Li and Z. Xin, Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations,, Comm. Math. Phys., 281 (2008), 401.
doi: 10.1007/s00220-008-0495-4. |
[23] |
P. Lions, "Mathematical Topics in Fluid Mechanics. II. Compressible Models,", The Clarendon Press, (1998).
|
[24] |
T. Luo, Z. Xin and T. Yang, Interface behavior of compressible Navier-Stokes equations with vacuum,, SIAM J. Math. Anal., 31 (2000), 1175.
doi: 10.1137/S0036141097331044. |
[25] |
Z. Luo, Local existence of classical solutions to the two-dimensional viscous compressible flows with vacuum,, Comm. Math. Sci., 10 (2012), 527.
|
[26] |
A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases,, Math. Kyoto Univ., 20 (1980), 67.
|
[27] |
J. Nash, Le problème de Cauchy pour les équations différentielles d'un fluide général,, Bull. Soc. Math. France., 90 (1962), 487.
|
[28] |
J. Serrin, On the uniqueness of compressible fluid motions,, Arch. Ration. Mech. Anal., 3 (1959), 271.
doi: 10.1007/BF00284180. |
[29] |
Z. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density,, Comm. Pure Appl. Math., 51 (1998), 229.
|
[30] |
Z. Xin and H. Yuan, Vacuum state for spherically symmetric solutions of the compressible Navier-Stokes equations,, J. Hyperbolic Differential Equations, 3 (2006), 403.
doi: 10.1142/S0219891606000847. |
show all references
References:
[1] |
A. Amosov and A. Zlotnik, A semidiscrete method for solving equations of the one dimensional motion of a non homogeneous viscous heat conducting gas with nonsmooth data,, Izv. Vyssh. Uchebn. Zaved. Mat., 41 (1997), 3.
|
[2] |
D. Bresch and B. Desjardins, On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids,, J. Math. Pures Appl., 87 (2007), 57.
doi: 10.1016/j.matpur.2006.11.001. |
[3] |
T. Chang and L. Hsiao, "The Riemann Problem and Interaction of Waves in Gas Dynamics,", Longman Scientific and Technical, (1989).
|
[4] |
G. Chen, D. Hoff and K. Trivisa, Global solutions of the compressible Navier-Stokes equations with large discontinuous initial data,, Comm. Partial Differential Equations, 25 (2000), 2233.
|
[5] |
G. Chen and J. Glimm, Global solutions to the compressible Euler equations with geometrical structure,, Comm. Math. Phys., 180 (1996), 153.
doi: 10.1007/BF02101185. |
[6] |
Y. Cho and H. Kim, Existence results for viscous polytropic fluids with vacuum,, J. Differential Equations, 228 (2006), 377.
doi: 10.1016/j.jde.2006.05.001. |
[7] |
R. Duan and Y. Zhao, A note on the non-formation of vacuum states for compressible Navier-Stokes equations,, J. Math. Anal. Appl., 311 (2005), 744.
doi: 10.1016/j.jmaa.2005.02.065. |
[8] |
D. 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. |
[9] |
E. Feireisl, "Dynamics of Viscous Compressible Fluids,", Oxford University Press, (2004).
|
[10] |
H. Fujita-Yashima, M. Padula and A. Novotny, équation monodimensionnelle d'un gaz vizqueux et calorifére avec des conditions initiales moins restrictives,, Ric. Mat., 42 (1993), 199.
|
[11] |
D. Hoff, Global well-posedness of the Cauchy problem for the Navier-Stokes equations of nonisentropic flow with discontinuous initial data,, J. Differential Equations, 95 (1992), 33.
doi: 10.1016/0022-0396(92)90042-L. |
[12] |
D. Hoff, Discontinuous solutions of the Navier-Stokes equations for compressible flow,, Arch. Ration. Mech. Anal., 114 (1991), 15.
doi: 10.1007/BF00375683. |
[13] |
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. |
[14] |
D. Hoff and J. Smoller, Non-formation of vacuum states for compressible Navier-Stokes equations,, Comm. Math. Phys., 216 (2001), 255.
doi: 10.1007/s002200000322. |
[15] |
X. Huang and J. Li, Global classical and weak solutions to the three-dimensional full compressible Navier-Stokes system with vacuum and large oscillations, preprint,, \arXiv{1107.4655}., (). Google Scholar |
[16] |
X. Huang, J. Li and Z. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations,, Comm. Pure Appl. Math., 65 (2012), 549.
doi: 10.1002/cpa.21382. |
[17] |
S. Jiang and P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations,, Comm. Math. Phys., 215 (2001), 559.
doi: 10.1007/PL00005543. |
[18] |
S. Jiang and P. Zhang, Global weak solutions to the Navier-Stokes equations for a 1D viscous polytropic ideal gas,, Quart. Appl. Math., 61 (2003), 435.
|
[19] |
Y. Kanel, The Cauchy problem for equations of gas dynamics with viscosity,, Siberian Math. J., 20 (1979), 208.
|
[20] |
A. Kazhikhov, On the Cauchy problem for the equations of a viscous gas,, Siberian Math. J., 23 (1982), 44.
doi: 10.1007/BF00971419. |
[21] |
A. Kazhikhov and 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. |
[22] |
H. Li, J. Li and Z. Xin, Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations,, Comm. Math. Phys., 281 (2008), 401.
doi: 10.1007/s00220-008-0495-4. |
[23] |
P. Lions, "Mathematical Topics in Fluid Mechanics. II. Compressible Models,", The Clarendon Press, (1998).
|
[24] |
T. Luo, Z. Xin and T. Yang, Interface behavior of compressible Navier-Stokes equations with vacuum,, SIAM J. Math. Anal., 31 (2000), 1175.
doi: 10.1137/S0036141097331044. |
[25] |
Z. Luo, Local existence of classical solutions to the two-dimensional viscous compressible flows with vacuum,, Comm. Math. Sci., 10 (2012), 527.
|
[26] |
A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases,, Math. Kyoto Univ., 20 (1980), 67.
|
[27] |
J. Nash, Le problème de Cauchy pour les équations différentielles d'un fluide général,, Bull. Soc. Math. France., 90 (1962), 487.
|
[28] |
J. Serrin, On the uniqueness of compressible fluid motions,, Arch. Ration. Mech. Anal., 3 (1959), 271.
doi: 10.1007/BF00284180. |
[29] |
Z. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density,, Comm. Pure Appl. Math., 51 (1998), 229.
|
[30] |
Z. Xin and H. Yuan, Vacuum state for spherically symmetric solutions of the compressible Navier-Stokes equations,, J. Hyperbolic Differential Equations, 3 (2006), 403.
doi: 10.1142/S0219891606000847. |
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