Global weak solutions to the two-dimensional Navier-Stokes equations of compressible heat-conducting flows with symmetric data and forces

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February 2014, 34(2): 567-587. doi: 10.3934/dcds.2014.34.567

Global weak solutions to the two-dimensional Navier-Stokes equations of compressible heat-conducting flows with symmetric data and forces

Fei Jiang ^1, , Song Jiang ^2, and Junpin Yin ^3,

1.	College of Mathematics and Computer Science, Fuzhou University, Fuzhou, 361000, China
2.	Institute of Applied Physics and Computational Mathematics, P.O.Box 8009-28, Beijing 100088
3.	Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, China

Received September 2012 Revised May 2013 Published August 2013

Abstract
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We prove the global existence of weak solutions to the Navier-Stokes equations of compressible heat-conducting fluids in two spatial dimensions with initial data and external forces which are large and spherically symmetric. The solutions will be obtained as the limit of the approximate solutions in an annular domain. We first derive a number of regularity results on the approximate physical quantities in the ``fluid region'', as well as the new uniform integrability of the velocity and temperature in the entire space-time domain by exploiting the theory of the Orlicz spaces. By virtue of these a priori estimates we then argue in a manner similar to that in [Arch. Rational Mech. Anal. 173 (2004), 297-343] to pass to the limit and show that the limiting functions are indeed a weak solution which satisfies the mass and momentum equations in the entire space-time domain in the sense of distributions, and the energy equation in any compact subset of the ``fluid region''.

Keywords: spherically symmetric solutions, Orlicz spaces., heat-conducting flows, 2D Navier-Stokes equations, Global weak solutions.

Mathematics Subject Classification: Primary: 76N10, 35D05; Secondary: 35E1.

Citation: Fei Jiang, Song Jiang, Junpin Yin. Global weak solutions to the two-dimensional Navier-Stokes equations of compressible heat-conducting flows with symmetric data and forces. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 567-587. doi: 10.3934/dcds.2014.34.567

References:

[1]	R. A. Adams and J. John, "Sobolev Space," $2^{nd}$ edition, Academic Press, New York, 2005.
[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. (9), 87 (2007), 57-90. doi: 10.1016/j.matpur.2006.11.001.
[3]	R. Erban, On the existence of solutions to the Navier-Stokes equations of a two-dimensional compressible flow, Math. Meth. Appl. Sci., 26 (2003), 489-517. doi: 10.1002/mma.362.
[4]	E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford Lecture Series in Mathematics and its Applications, 26, Oxford Univ. Press, Oxford, 2004.
[5]	E. Feireisl, On compactness of solutions to the compressible isentropic Navier-Stokes equations when the density is not integrable, Comment. Math. Univ. Carolinae, 42 (2001), 83-98.
[6]	E. Feireisl and A. Novotnỳ, "Singular Limits in Thermodynamics of Viscous Fluids," Advances in Mathematical Fluid Mechanics, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8843-0.
[7]	E. Feireisl, A. Novotnỳ and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392. doi: 10.1007/PL00000976.
[8]	H. Frid and V. Shelukhin, Vanishing shear viscosity in the equations of compressible fluids for the flows with the cylinder symmetry, SIAM J. Math. Anal., 31 (2000), 1144-1156. doi: 10.1137/S003614109834394X.
[9]	D. Hoff, Spherically symmetric solutions of the Navier-Stokes equations for compressible, isothermal flow with large, discontinuous initial data, Indiana Univ. Math. J., 41 (1992), 1225-1302. doi: 10.1512/iumj.1992.41.41060.
[10]	D. Hoff and H. K. Jenssen, Symmetric nonbarotropic flows with large data and forces, Arch. Rational Mech. Anal., 173 (2004), 297-343. doi: 10.1007/s00205-004-0318-5.
[11]	F. Jiang and Z. Tan, On the domain dependence of solutions to the Navier-Stokes equations of a two-dimensional compressible flow, Math. Meth. Appl. Sci., 32 (2009), 2350-2367. doi: 10.1002/mma.1138.
[12]	S. Jiang and P. Zhang, Axisymmetric solutions of the 3D Navier-Stokes equations for compressible isentropic fluids, J. Math. Pures Appl. (9), 82 (2003), 949-973. doi: 10.1016/S0021-7824(03)00015-1.
[13]	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.
[14]	S. Jiang and P. Zhang, Remarks on weak solutions to the Navier-Stokes equations for 2-D compressible isothermal fluids with spherically symmetric initial data, Indiana Univ. Math. J., 51 (2002), 345-355. doi: 10.1512/iumj.2002.51.2264.
[15]	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.
[16]	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.
[17]	A. Kufner, O. John and S. Fučik, "Function Spaces," Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis, Noordhoff International Publ., Leyden; Academia, Prague, 1977.
[18]	P.-L. Lions, "Mathematical Topics in Fluid Mechanics. Vol. 2. Compressible Models," Oxford Lecture Series in Mathematics and its Applications, 10, Science Publications, The Clarendon Press, Oxford Univ. Press, New York, 1998.
[19]	J. Zhang, S. Jiang and F. Xie, Global weak solutions of an initial boundary value problem for screw pinches in plasma physics, Math. Models Meth. Appl. Sci., 19 (2009), 833-875. doi: 10.1142/S0218202509003644.

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References:

[1]	R. A. Adams and J. John, "Sobolev Space," $2^{nd}$ edition, Academic Press, New York, 2005.
[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. (9), 87 (2007), 57-90. doi: 10.1016/j.matpur.2006.11.001.
[3]	R. Erban, On the existence of solutions to the Navier-Stokes equations of a two-dimensional compressible flow, Math. Meth. Appl. Sci., 26 (2003), 489-517. doi: 10.1002/mma.362.
[4]	E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford Lecture Series in Mathematics and its Applications, 26, Oxford Univ. Press, Oxford, 2004.
[5]	E. Feireisl, On compactness of solutions to the compressible isentropic Navier-Stokes equations when the density is not integrable, Comment. Math. Univ. Carolinae, 42 (2001), 83-98.
[6]	E. Feireisl and A. Novotnỳ, "Singular Limits in Thermodynamics of Viscous Fluids," Advances in Mathematical Fluid Mechanics, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8843-0.
[7]	E. Feireisl, A. Novotnỳ and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392. doi: 10.1007/PL00000976.
[8]	H. Frid and V. Shelukhin, Vanishing shear viscosity in the equations of compressible fluids for the flows with the cylinder symmetry, SIAM J. Math. Anal., 31 (2000), 1144-1156. doi: 10.1137/S003614109834394X.
[9]	D. Hoff, Spherically symmetric solutions of the Navier-Stokes equations for compressible, isothermal flow with large, discontinuous initial data, Indiana Univ. Math. J., 41 (1992), 1225-1302. doi: 10.1512/iumj.1992.41.41060.
[10]	D. Hoff and H. K. Jenssen, Symmetric nonbarotropic flows with large data and forces, Arch. Rational Mech. Anal., 173 (2004), 297-343. doi: 10.1007/s00205-004-0318-5.
[11]	F. Jiang and Z. Tan, On the domain dependence of solutions to the Navier-Stokes equations of a two-dimensional compressible flow, Math. Meth. Appl. Sci., 32 (2009), 2350-2367. doi: 10.1002/mma.1138.
[12]	S. Jiang and P. Zhang, Axisymmetric solutions of the 3D Navier-Stokes equations for compressible isentropic fluids, J. Math. Pures Appl. (9), 82 (2003), 949-973. doi: 10.1016/S0021-7824(03)00015-1.
[13]	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.
[14]	S. Jiang and P. Zhang, Remarks on weak solutions to the Navier-Stokes equations for 2-D compressible isothermal fluids with spherically symmetric initial data, Indiana Univ. Math. J., 51 (2002), 345-355. doi: 10.1512/iumj.2002.51.2264.
[15]	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.
[16]	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.
[17]	A. Kufner, O. John and S. Fučik, "Function Spaces," Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis, Noordhoff International Publ., Leyden; Academia, Prague, 1977.
[18]	P.-L. Lions, "Mathematical Topics in Fluid Mechanics. Vol. 2. Compressible Models," Oxford Lecture Series in Mathematics and its Applications, 10, Science Publications, The Clarendon Press, Oxford Univ. Press, New York, 1998.
[19]	J. Zhang, S. Jiang and F. Xie, Global weak solutions of an initial boundary value problem for screw pinches in plasma physics, Math. Models Meth. Appl. Sci., 19 (2009), 833-875. doi: 10.1142/S0218202509003644.

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