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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

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 & 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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford Lecture Series in Mathematics and its Applications, 26, Oxford Univ. Press, Oxford, 2004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

R. A. Adams and J. John, "Sobolev Space," $2^{nd}$ edition, Academic Press, New York, 2005. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford Lecture Series in Mathematics and its Applications, 26, Oxford Univ. Press, Oxford, 2004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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