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November  2013, 12(6): 2543-2564. doi: 10.3934/cpaa.2013.12.2543

## 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

Received  June 2012 Revised  March 2013 Published  May 2013

In this paper, we consider the properties of the vacuum states for weak solutions to one-dimensional full compressible Navier-Stokes system with viscosity and heat conductivities for general equation of states. Under weak conditions on initial data, we prove that if there is no vacuum initially then the vacuum states do not occur in a finite time. In particular, the temperature variation has no immediate effects on the formation of the vacuum. There are no assumptions on density in large sets. Furthermore, we prove that two initially non interacting vacuum regions will never touch in the future.
Citation: Ben Duan, Zhen Luo. Dynamics of vacuum states for one-dimensional full compressible Navier-Stokes equations. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2543-2564. doi: 10.3934/cpaa.2013.12.2543
##### 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), 3-19. [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), 57-90. 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, Harlow; copublished in the United States with John Wiley and Sons, Inc., New York, 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), 2233-2257. [5] G. Chen and J. Glimm, Global solutions to the compressible Euler equations with geometrical structure, Comm. Math. Phys., 180 (1996), 153-193. doi: 10.1007/BF02101185. [6] Y. Cho and H. Kim, Existence results for viscous polytropic fluids with vacuum, J. Differential Equations, 228 (2006), 377-411. 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), 744-754. 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), 675-694. doi: 10.3934/cpaa.2004.3.675. [9] E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford University Press, Oxford, 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), 199-248. [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), 33-74. 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), 15-46. 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), 887-898. 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), 255-276. 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}., (). [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), 549-585. 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), 559-581. 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), 435-449. [19] Y. Kanel, The Cauchy problem for equations of gas dynamics with viscosity, Siberian Math. J., 20 (1979), 208-218. [20] A. Kazhikhov, On the Cauchy problem for the equations of a viscous gas, Siberian Math. J., 23 (1982), 44-49. 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), 273-282. 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), 401-444. doi: 10.1007/s00220-008-0495-4. [23] P. Lions, "Mathematical Topics in Fluid Mechanics. II. Compressible Models," The Clarendon Press, Oxford University Press, New York, 1998. [24] T. Luo, Z. Xin and T. Yang, Interface behavior of compressible Navier-Stokes equations with vacuum, SIAM J. Math. Anal., 31 (2000), 1175-1191. 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), 527-554. [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), 67-104. [27] J. Nash, Le problème de Cauchy pour les équations différentielles d'un fluide général, Bull. Soc. Math. France., 90 (1962), 487-497. [28] J. Serrin, On the uniqueness of compressible fluid motions, Arch. Ration. Mech. Anal., 3 (1959), 271-288. 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), 229-240. [30] Z. Xin and H. Yuan, Vacuum state for spherically symmetric solutions of the compressible Navier-Stokes equations, J. Hyperbolic Differential Equations, 3 (2006), 403-442. doi: 10.1142/S0219891606000847.

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##### 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), 3-19. [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), 57-90. 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, Harlow; copublished in the United States with John Wiley and Sons, Inc., New York, 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), 2233-2257. [5] G. Chen and J. Glimm, Global solutions to the compressible Euler equations with geometrical structure, Comm. Math. Phys., 180 (1996), 153-193. doi: 10.1007/BF02101185. [6] Y. Cho and H. Kim, Existence results for viscous polytropic fluids with vacuum, J. Differential Equations, 228 (2006), 377-411. 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), 744-754. 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), 675-694. doi: 10.3934/cpaa.2004.3.675. [9] E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford University Press, Oxford, 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), 199-248. [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), 33-74. 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), 15-46. 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), 887-898. 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), 255-276. 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}., (). [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), 549-585. 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), 559-581. 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), 435-449. [19] Y. Kanel, The Cauchy problem for equations of gas dynamics with viscosity, Siberian Math. J., 20 (1979), 208-218. [20] A. Kazhikhov, On the Cauchy problem for the equations of a viscous gas, Siberian Math. J., 23 (1982), 44-49. 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), 273-282. 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), 401-444. doi: 10.1007/s00220-008-0495-4. [23] P. Lions, "Mathematical Topics in Fluid Mechanics. II. Compressible Models," The Clarendon Press, Oxford University Press, New York, 1998. [24] T. Luo, Z. Xin and T. Yang, Interface behavior of compressible Navier-Stokes equations with vacuum, SIAM J. Math. Anal., 31 (2000), 1175-1191. 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), 527-554. [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), 67-104. [27] J. Nash, Le problème de Cauchy pour les équations différentielles d'un fluide général, Bull. Soc. Math. France., 90 (1962), 487-497. [28] J. Serrin, On the uniqueness of compressible fluid motions, Arch. Ration. Mech. Anal., 3 (1959), 271-288. 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), 229-240. [30] Z. Xin and H. Yuan, Vacuum state for spherically symmetric solutions of the compressible Navier-Stokes equations, J. Hyperbolic Differential Equations, 3 (2006), 403-442. doi: 10.1142/S0219891606000847.
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