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Cucker-Smale model with normalized communication weights and time delay
Global strong solutions to the planar compressible magnetohydrodynamic equations with large initial data and vacuum
1. | Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China |
2. | School of Mathematics, Shandong University, Jinan 250100, China |
3. | Department of Mathematics, Nanjing University, Nanjing 210093, China |
$κ(θ)$ |
$C_{1}(1+\theta^q)\leq \kappa(\theta)\leq C_2(1+\theta^q)$ |
$q>0$ |
$C_1,C_2>0$ |
References:
[1] |
A. A. Amosov and A. A. Zlotnik,
Global generalized solutions of the equations of the one-dimensional motion of a viscous heat-conducting gas, Sov. Mat. Dokl., 38 (1989), 1-5.
|
[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] |
H. Cabannes,
Theoretical Magnetofluiddynamics Academic Press, New York, London, 1970. |
[4] |
G.-Q. Chen and D.-H. Wang,
Globla solutions of nonlinear magnetohydrodynamics with large initial data, J. Differential Equations, 182 (2002), 344-376.
doi: 10.1006/jdeq.2001.4111. |
[5] |
G.-Q. Chen and D.-H. Wang,
Existence and continuous dependence of large solutions for the magnetohydrodynamics equations, Z. Angew. Math. Phys., 54 (2003), 608-632.
doi: 10.1007/s00033-003-1017-z. |
[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] |
B. Ducomet and E. Feireisl,
The equations of Magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars, Commun. Math. Phys., 226 (2006), 595-629.
doi: 10.1007/s00220-006-0052-y. |
[8] |
J.-S. Fan, S. Jiang and G. Nakamura,
Vanishing shear viscosity limit in the magnetohydrodynamic equations, Comm. Math. Phys., 270 (2007), 691-708.
doi: 10.1007/s00220-006-0167-1. |
[9] |
J.-S. Fan, S. Jiang and G. Nakamura,
Stability of weak solutions to equations of magnetohydrodynamics with Lebesgue initial data, J. Differential Equations, 251 (2011), 2025-2036.
doi: 10.1016/j.jde.2011.06.019. |
[10] |
J.-S. Fan and W.-H. Yu,
Global variational solutions to the compressible magnetohydrodynamic equations, Nonlinear Anal., 69 (2008), 3637-3660.
doi: 10.1016/j.na.2007.10.005. |
[11] |
J.-S Fan and W.-H. Yu,
Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. Real World Appl., 10 (2009), 392-409.
doi: 10.1016/j.nonrwa.2007.10.001. |
[12] |
E. Feireisl,
Dynamics Of Viscous Compressible Fluids, Oxford Lecture Series in Mathematics and its Applications, 26 Oxford University Press, Oxford, 2004. |
[13] |
D. Hoff and E. Tsyganov,
Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics, Z. Angew. Math. Phys., 56 (2005), 791-804.
doi: 10.1007/s00033-005-4057-8. |
[14] |
X.-P. Hu and D.-H. Wang,
Global existence and large-time behavior of solutions to the threedimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 197 (2010), 203-238.
doi: 10.1007/s00205-010-0295-9. |
[15] |
X.-P. Hu and D.-H. Wang,
Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys., 283 (2008), 255-284.
doi: 10.1007/s00220-008-0497-2. |
[16] |
X.-P. Hu and D.-H. Wang,
Compactness of weak solutions to the three-dimensional compressible magnetohydrodynamic equations, J. Differential Equations, 245 (2008), 2176-2198.
doi: 10.1016/j.jde.2008.07.019. |
[17] |
Y.-X. Hu and Q.-C. Ju,
Global large solutions of magnetohydrodynamics with temperature-dependent heat conductivity, Z. Angew. Math. Phys., 66 (2015), 865-889.
doi: 10.1007/s00033-014-0446-1. |
[18] |
X. -D. Huang and J. Li,
Global Classical and Weak Solutions to the Three-Dimensional Full Compressible Navier-Stokes System with Vacuum and Large Oscillations arXiv: 1107.4655v2, [math. ph]. |
[19] |
A. Jeffrey and T. Taniuti,
Non-Linear Wave Propagation. With Applications To Physics And Magnetohydrodynamics, Academic Press, New York, 1964. |
[20] |
S. Kawashima,
Smooth global solutions for two-dimensional equations of electromagnetofluid dynamics, Japan J. Appl. Math., 1 (1984), 207-222.
doi: 10.1007/BF03167869. |
[21] |
S. Kawashima and M. Okada,
Smooth global solutions for the one-dimensional equations in magnetohydrodynamics, Proc. Japan Acad. Ser. A Math. Sci., 58 (1982), 384-387.
doi: 10.3792/pjaa.58.384. |
[22] |
B. Kawohl,
Global existence of large solutions to initial boundary value problems for a viscous, heat-conducting, one-dimensional real gas, J. Differential Equations, 58 (1985), 76-103.
doi: 10.1016/0022-0396(85)90023-3. |
[23] |
A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initialboundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math.
Mech. , 41 (1977), 273–282, translated from Prikl. Mat. Meh. , 41 (1977), 282–291 (Russian).
doi: 10.1016/0021-8928(77)90011-9. |
[24] |
A. G. Kulikovskiy and G. A. Lyubimov,
Magnetohydrodynamics Addison-Wesley, Reading, Massachusetts, 1965. |
[25] |
L. D. Landau and E. M. Lifshitz,
Electrodynamics of Continuous Media Oxford-London-New York-Paris; Addison-Wesley Publishing Co. , Inc. , Reading, Mass. 1960. |
[26] |
F.-C. Li and H.-Y. Yu,
Optimal decay rate of classical solutions to the compressible magnetohydrodynamic equations, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 109-126.
doi: 10.1017/S0308210509001632. |
[27] |
X.-L. Li, N. Su and D.-H. Wang,
Local strong solution to the compressible magnetohydrodynamic flow with large data, J. Hyperbolic Differ. Equ., 8 (2011), 415-436.
doi: 10.1142/S0219891611002457. |
[28] |
T.-P Liu and Y. Zeng,
Large-time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws, Mem. Amer. Math. Soc., 125 (1997), viii+120 pp.
doi: 10.1090/memo/0599. |
[29] |
R.-H. Pan and W.-Z. Zhang,
Compressible Navier-Stokes equations with temperature dependent heat conductivity, Commun. Math. Sci., 13 (2015), 401-425.
doi: 10.4310/CMS.2015.v13.n2.a7. |
[30] |
R. V. Polovin and V. P. Demutskii,
Fundamentals Of Magnetohydrodynamics, Consultants Bureau, New York, 1990. |
[31] |
G. Ströhmer,
About compressible viscous fluid flow in a bounded region, Pacific J. Math., 143 (1990), 359-375.
doi: 10.2140/pjm.1990.143.359. |
[32] |
A. Suen and D. Hoff,
About compressible viscous fluid flow in a bounded region, Pacific J. Math., 143 (1990), 359-375.
doi: 10.1007/s00205-012-0498-3. |
[33] |
E. Tsyganov and D. Hoff,
Systems of partial differential equations of mixed hyperbolic-parabolic type, J. Differential Equations, 204 (2004), 163-201.
doi: 10.1016/j.jde.2004.03.010. |
[34] |
T. Umeda, S. Kawashima and Y. Shizuta,
On the decay of solutions to the linearized equations of electromagnetofluid dynamics, Japan J. Appl. Math., 1 (1984), 435-457.
doi: 10.1007/BF03167068. |
[35] |
A. I. Vol'pert and S. I. Hudjaev,
On the Cauchy problem for composite systems of nonlinear
differential equations, (Russian), Mat. Sb. (N.S.), 87(129) (1972), 504-528.
|
[36] |
D.-H. Wang,
Large solutions to the initial-boundary value problem for planar magnetohydrodynamics, SIAM J. Appl. Math., 63 (2003), 1424-1441.
doi: 10.1137/S0036139902409284. |
[37] |
H. Wen and C. Zhu,
Global classical large solutions to Navier-Stokes equations for viscous compressible and heat conducting fluids with vacuum, SIAM J. Math. Anal., 45 (2013), 431-468.
doi: 10.1137/120877829. |
[38] |
H.-Y. Wen and C.-J. Zhu,
Global symmetric classical and strong solutions of the full compressible Navier-Stokes equations with vacuum and large initial data, J. Math. Pures Appl.(9), 102 (2014), 498-545.
doi: 10.1016/j.matpur.2013.12.003. |
[39] |
J.-W. Zhang, S. Jiang and F. Xie,
Global weak solutions of an initial boundary value problem for screw pinches in plasma physics, Math. Models Methods Appl. Sci., 19 (2009), 833-875.
doi: 10.1142/S0218202509003644. |
[40] |
A. A. Zlotnik and A. A. Amosov,
On stability of generalized solutions to the equations of one-dimensional motion of a viscous heat-conducting gas, Sib. Math. J., 38 (1997), 663-684.
doi: 10.1007/BF02674573. |
[41] |
A. A. Zlotnik and A. A. Amosov,
Stability of generalized solutions to equations of one-dimensional motion of viscous heat conducting gases, Math. Notes, 63 (1998), 736-746.
doi: 10.1007/BF02312766. |
show all references
References:
[1] |
A. A. Amosov and A. A. Zlotnik,
Global generalized solutions of the equations of the one-dimensional motion of a viscous heat-conducting gas, Sov. Mat. Dokl., 38 (1989), 1-5.
|
[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] |
H. Cabannes,
Theoretical Magnetofluiddynamics Academic Press, New York, London, 1970. |
[4] |
G.-Q. Chen and D.-H. Wang,
Globla solutions of nonlinear magnetohydrodynamics with large initial data, J. Differential Equations, 182 (2002), 344-376.
doi: 10.1006/jdeq.2001.4111. |
[5] |
G.-Q. Chen and D.-H. Wang,
Existence and continuous dependence of large solutions for the magnetohydrodynamics equations, Z. Angew. Math. Phys., 54 (2003), 608-632.
doi: 10.1007/s00033-003-1017-z. |
[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] |
B. Ducomet and E. Feireisl,
The equations of Magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars, Commun. Math. Phys., 226 (2006), 595-629.
doi: 10.1007/s00220-006-0052-y. |
[8] |
J.-S. Fan, S. Jiang and G. Nakamura,
Vanishing shear viscosity limit in the magnetohydrodynamic equations, Comm. Math. Phys., 270 (2007), 691-708.
doi: 10.1007/s00220-006-0167-1. |
[9] |
J.-S. Fan, S. Jiang and G. Nakamura,
Stability of weak solutions to equations of magnetohydrodynamics with Lebesgue initial data, J. Differential Equations, 251 (2011), 2025-2036.
doi: 10.1016/j.jde.2011.06.019. |
[10] |
J.-S. Fan and W.-H. Yu,
Global variational solutions to the compressible magnetohydrodynamic equations, Nonlinear Anal., 69 (2008), 3637-3660.
doi: 10.1016/j.na.2007.10.005. |
[11] |
J.-S Fan and W.-H. Yu,
Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. Real World Appl., 10 (2009), 392-409.
doi: 10.1016/j.nonrwa.2007.10.001. |
[12] |
E. Feireisl,
Dynamics Of Viscous Compressible Fluids, Oxford Lecture Series in Mathematics and its Applications, 26 Oxford University Press, Oxford, 2004. |
[13] |
D. Hoff and E. Tsyganov,
Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics, Z. Angew. Math. Phys., 56 (2005), 791-804.
doi: 10.1007/s00033-005-4057-8. |
[14] |
X.-P. Hu and D.-H. Wang,
Global existence and large-time behavior of solutions to the threedimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 197 (2010), 203-238.
doi: 10.1007/s00205-010-0295-9. |
[15] |
X.-P. Hu and D.-H. Wang,
Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys., 283 (2008), 255-284.
doi: 10.1007/s00220-008-0497-2. |
[16] |
X.-P. Hu and D.-H. Wang,
Compactness of weak solutions to the three-dimensional compressible magnetohydrodynamic equations, J. Differential Equations, 245 (2008), 2176-2198.
doi: 10.1016/j.jde.2008.07.019. |
[17] |
Y.-X. Hu and Q.-C. Ju,
Global large solutions of magnetohydrodynamics with temperature-dependent heat conductivity, Z. Angew. Math. Phys., 66 (2015), 865-889.
doi: 10.1007/s00033-014-0446-1. |
[18] |
X. -D. Huang and J. Li,
Global Classical and Weak Solutions to the Three-Dimensional Full Compressible Navier-Stokes System with Vacuum and Large Oscillations arXiv: 1107.4655v2, [math. ph]. |
[19] |
A. Jeffrey and T. Taniuti,
Non-Linear Wave Propagation. With Applications To Physics And Magnetohydrodynamics, Academic Press, New York, 1964. |
[20] |
S. Kawashima,
Smooth global solutions for two-dimensional equations of electromagnetofluid dynamics, Japan J. Appl. Math., 1 (1984), 207-222.
doi: 10.1007/BF03167869. |
[21] |
S. Kawashima and M. Okada,
Smooth global solutions for the one-dimensional equations in magnetohydrodynamics, Proc. Japan Acad. Ser. A Math. Sci., 58 (1982), 384-387.
doi: 10.3792/pjaa.58.384. |
[22] |
B. Kawohl,
Global existence of large solutions to initial boundary value problems for a viscous, heat-conducting, one-dimensional real gas, J. Differential Equations, 58 (1985), 76-103.
doi: 10.1016/0022-0396(85)90023-3. |
[23] |
A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initialboundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math.
Mech. , 41 (1977), 273–282, translated from Prikl. Mat. Meh. , 41 (1977), 282–291 (Russian).
doi: 10.1016/0021-8928(77)90011-9. |
[24] |
A. G. Kulikovskiy and G. A. Lyubimov,
Magnetohydrodynamics Addison-Wesley, Reading, Massachusetts, 1965. |
[25] |
L. D. Landau and E. M. Lifshitz,
Electrodynamics of Continuous Media Oxford-London-New York-Paris; Addison-Wesley Publishing Co. , Inc. , Reading, Mass. 1960. |
[26] |
F.-C. Li and H.-Y. Yu,
Optimal decay rate of classical solutions to the compressible magnetohydrodynamic equations, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 109-126.
doi: 10.1017/S0308210509001632. |
[27] |
X.-L. Li, N. Su and D.-H. Wang,
Local strong solution to the compressible magnetohydrodynamic flow with large data, J. Hyperbolic Differ. Equ., 8 (2011), 415-436.
doi: 10.1142/S0219891611002457. |
[28] |
T.-P Liu and Y. Zeng,
Large-time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws, Mem. Amer. Math. Soc., 125 (1997), viii+120 pp.
doi: 10.1090/memo/0599. |
[29] |
R.-H. Pan and W.-Z. Zhang,
Compressible Navier-Stokes equations with temperature dependent heat conductivity, Commun. Math. Sci., 13 (2015), 401-425.
doi: 10.4310/CMS.2015.v13.n2.a7. |
[30] |
R. V. Polovin and V. P. Demutskii,
Fundamentals Of Magnetohydrodynamics, Consultants Bureau, New York, 1990. |
[31] |
G. Ströhmer,
About compressible viscous fluid flow in a bounded region, Pacific J. Math., 143 (1990), 359-375.
doi: 10.2140/pjm.1990.143.359. |
[32] |
A. Suen and D. Hoff,
About compressible viscous fluid flow in a bounded region, Pacific J. Math., 143 (1990), 359-375.
doi: 10.1007/s00205-012-0498-3. |
[33] |
E. Tsyganov and D. Hoff,
Systems of partial differential equations of mixed hyperbolic-parabolic type, J. Differential Equations, 204 (2004), 163-201.
doi: 10.1016/j.jde.2004.03.010. |
[34] |
T. Umeda, S. Kawashima and Y. Shizuta,
On the decay of solutions to the linearized equations of electromagnetofluid dynamics, Japan J. Appl. Math., 1 (1984), 435-457.
doi: 10.1007/BF03167068. |
[35] |
A. I. Vol'pert and S. I. Hudjaev,
On the Cauchy problem for composite systems of nonlinear
differential equations, (Russian), Mat. Sb. (N.S.), 87(129) (1972), 504-528.
|
[36] |
D.-H. Wang,
Large solutions to the initial-boundary value problem for planar magnetohydrodynamics, SIAM J. Appl. Math., 63 (2003), 1424-1441.
doi: 10.1137/S0036139902409284. |
[37] |
H. Wen and C. Zhu,
Global classical large solutions to Navier-Stokes equations for viscous compressible and heat conducting fluids with vacuum, SIAM J. Math. Anal., 45 (2013), 431-468.
doi: 10.1137/120877829. |
[38] |
H.-Y. Wen and C.-J. Zhu,
Global symmetric classical and strong solutions of the full compressible Navier-Stokes equations with vacuum and large initial data, J. Math. Pures Appl.(9), 102 (2014), 498-545.
doi: 10.1016/j.matpur.2013.12.003. |
[39] |
J.-W. Zhang, S. Jiang and F. Xie,
Global weak solutions of an initial boundary value problem for screw pinches in plasma physics, Math. Models Methods Appl. Sci., 19 (2009), 833-875.
doi: 10.1142/S0218202509003644. |
[40] |
A. A. Zlotnik and A. A. Amosov,
On stability of generalized solutions to the equations of one-dimensional motion of a viscous heat-conducting gas, Sib. Math. J., 38 (1997), 663-684.
doi: 10.1007/BF02674573. |
[41] |
A. A. Zlotnik and A. A. Amosov,
Stability of generalized solutions to equations of one-dimensional motion of viscous heat conducting gases, Math. Notes, 63 (1998), 736-746.
doi: 10.1007/BF02312766. |
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