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Global existence and low Mach number limit to a 3D compressible micropolar fluids model in a bounded domain
Asymptotic stability of stationary solutions for magnetohydrodynamic equations
School of Mathematical Science and Fujian Provincial Key Laboratory, on Mathematical Modeling & High Performance Scientific Computing, Xiamen University, Xiamen 361005, China |
In this paper, we are concerned with the compressible magnetohydrodynamic equations with Coulomb force in three-dimensional space. We show the asymptotic stability of solutions to the Cauchy problem near the non-constant equilibrium state provided that the initial perturbation is sufficiently small. Moreover, the convergence rates are obtained by combining the linear Lp-Lq decay estimates and the higher-order energy estimates.
References:
[1] |
M. Alessandro, T. Yuri and T. Paola,
Well-posedness of the linearized problem for MHD contact discontinuities, J. Differential Equations, 258 (2015), 2531-2571.
doi: 10.1016/j.jde.2014.12.018. |
[2] |
Q. Chen and Z. Tan,
Global existence and convergence rates of smooth solutions for the compressible magnetohydroynamics equations, Nonlinear Anal., 72 (2010), 4438-4451.
doi: 10.1016/j.na.2010.02.019. |
[3] |
G. Q. Chen and D. H. Wang,
Global solution of nonlinear magnetohydrodynamics with large initial data, J. Differential Equations, 182 (2002), 344-376.
doi: 10.1006/jdeq.2001.4111. |
[4] |
G. Q. Chen and D. H. Wang,
Existence and continuous dependence of large solutions for the magnetohydrodynamic equations, Z. Angew. Math. Phys., 54 (2003), 608-632.
doi: 10.1007/s00033-003-1017-z. |
[5] |
L. L. Du and D. Q. Zhou,
Global well-posedness of two-dimensional magnetohydrodynamic flows with partial dissipation and magnetic diffusion, SIAM J. Math. Anal., 47 (2015), 1562-1589.
doi: 10.1137/140959821. |
[6] |
J.S. Fan, A. Ahmed, H. Tasawar, N. Gen and Y. Zhou,
On strong solutions to the compressible Hall-magnetohydrodynamic system, Nonlinear Anal. Real World Appl., 22 (2015), 423-434.
doi: 10.1016/j.nonrwa.2014.10.003. |
[7] |
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. |
[8] |
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. |
[9] |
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. |
[10] |
E. Feireisl, A. Novotný and H. Petleltová,
On the existence of globally defined weak solutions to the Navier-Stokes equations of isentropic compressible fluids, J. Math. Fluid Mech., 3 (2001), 358-392.
doi: 10.1007/PL00000976. |
[11] |
Y. Guo and Y. J. Wang,
Decay of dissipative equations and negative Sobolev spaces, Comm. Partial Differential Equations, 37 (2012), 2165-2208.
doi: 10.1080/03605302.2012.696296. |
[12] |
C. C. Hao and H. L. Li,
Global existence for compressible Navier-Stokes-Poisson equations in three and higher dimensions, J. Differential Equations, 246 (2009), 4791-4812.
doi: 10.1016/j.jde.2008.11.019. |
[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 solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys., 283 (2008), 255-284.
doi: 10.1007/s00220-008-0497-2. |
[15] |
X. P. Hu and D. H. Wang,
Low Mach number limit of viscous compressible magnetohydrodynamic flows, SIAM J. Math. Anal., 41 (2009), 1272-1294.
doi: 10.1137/080723983. |
[16] |
X. P. Hu and D. H. Wang,
Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 197 (2010), 203-238.
doi: 10.1007/s00205-010-0295-9. |
[17] |
S. Jiang, Q. C. Ju and F. C. Li,
Incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary condtions, Comm. Math. Phys., 297 (2010), 371-400.
doi: 10.1007/s00220-010-0992-0. |
[18] |
S. Jiang, Q. C. Ju, F. C. Li and Z. P. Xin,
Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data, Adv. Math., 259 (2014), 384-420.
doi: 10.1016/j.aim.2014.03.022. |
[19] |
Q. C. Ju, F. C. Li and Y. Li,
Asymptotic limits of the full comressible magnetohydrodynamic equations, SIAM J. Math. Anal., 45 (2013), 2597-2624.
doi: 10.1137/130913390. |
[20] |
N. Ju,
Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the Sobolev space, Comm. Math. Phys., 251 (2004), 365-376.
doi: 10.1007/s00220-004-1062-2. |
[21] |
S. Kawashima,
Smooth global solutions for two-dimensional equations of electromagneto fluid dynamics, Japan J. Appl. Math., 1 (1984), 207-222.
doi: 10.1007/BF03167869. |
[22] |
S. Kawashima and M. Okada,
Smooth global solutions for the one-dimensional equations in magnetohydrodynamics, Proc. Japan Acad., 58 (1982), 384-387.
doi: 10.3792/pjaa.58.384. |
[23] |
T. Kobayashi and T. Suzuki,
Weak solutions to the Navier-Stokes-Poisson equation, Adv. Math. Sci. Appl., 18 (2008), 141-168.
|
[24] |
Z. Lei,
On axially symmetric incompressible magnetohydrodynamics in three dimensions, J. Differential Equations, 259 (2015), 3202-3215.
doi: 10.1016/j.jde.2015.04.017. |
[25] |
H. L. Li, A. Matsumura and G. J. Zhang,
Optimal decay rate of the compressible Navier-Stokes-Poisson system in ℝ3, Arch. Ration. Mech. Anal., 196 (2010), 681-713.
doi: 10.1007/s00205-009-0255-4. |
[26] |
H. L. Li, X. Y. Xu and J. W. Zhang,
Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillationa and vacuum, SIAM J. Math. Anal., 45 (2013), 1356-1387.
doi: 10.1137/120893355. |
[27] |
F. C. Li and H. J. Yu,
Optimal decay rate of classical solutions to the compressible magnetohydrodynamic equations, Proc. Roy. Soc. Edinburgh Sect., 141 (2011), 109-126.
doi: 10.1017/S0308210509001632. |
[28] |
P. L. Lions, Mathematical Topics in Fluids Mechanics. Oxford Lecture Ser. Math. Appl, Clarendon Press, Oxford University Press, New York, 1998.
![]() ![]() |
[29] |
A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2002.
![]() ![]() |
[30] |
L. Nirenberg,
On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.
|
[31] |
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. |
[32] |
Z. Tan, L. L. Tong and Y. Wang,
Large time behavior of the compressible magnetohydrodynamic equations with Coulomb force, J. Math. Anal. Appl., 427 (2015), 600-617.
doi: 10.1016/j.jmaa.2015.02.077. |
[33] |
Z. Tan and Y. J. Wang,
Global existence and large-time behavior of weak solutions to the compressible magnetohydrodynamic equations with Coulomb force, Nonlinear Anal., 71 (2009), 5866-5884.
doi: 10.1016/j.na.2009.05.012. |
[34] |
Z. Tan and G. C. Wu,
Global existence for the non-isentropic compressible Navier-Stokes-Poisson system in three and higher dimensions, Nonlinear Anal. Real World Appl., 13 (2012), 650-664.
doi: 10.1016/j.nonrwa.2011.08.005. |
[35] |
Z. Tan and H. Q. Wang,
Optimal decay rates of the compressible magnetohydrodynamic equations, Nonlinear Anal. Real World Appl., 14 (2013), 188-201.
doi: 10.1016/j.nonrwa.2012.05.012. |
[36] |
Z. Tan, Y. J. Wang and Y. Wang,
Stability of steady states of the Navier-Stokes-Poisson equations with non-flat doping profile, SIAM J. Math. Anal., 47 (2015), 179-209.
doi: 10.1137/130950069. |
[37] |
Z. Tan, Y. Wang and X. Zhang,
Large time behavior of solutions to the non-isentropic compressible Navier-Stokes-Poisson system in ℝ3, Kinet. Relat. Models, 5 (2012), 615-638.
doi: 10.3934/krm.2012.5.615. |
[38] |
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. |
[39] |
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. |
[40] |
Y. J. Wang,
Decay of the Navier-Stokes-Poisson equations, J. Differential Equations, 253 (2012), 273-297.
doi: 10.1016/j.jde.2012.03.006. |
[41] |
L. Zhang and S. Li,
A regularity criterion for 2D MHD flows with horizontal dissipation and horizontal magnetic diffusion, Nonlinear Anal. Real World Appl., 21 (2015), 197-206.
doi: 10.1016/j.nonrwa.2014.07.005. |
[42] |
G. J. Zhang, H. L. Li and C. J. Zhu,
Optimal decay rate of the non-isentropic compressible Navier-Stokes-Poisson system in ℝ3, J. Differential Equations, 250 (2011), 866-891.
doi: 10.1016/j.jde.2010.07.035. |
show all references
References:
[1] |
M. Alessandro, T. Yuri and T. Paola,
Well-posedness of the linearized problem for MHD contact discontinuities, J. Differential Equations, 258 (2015), 2531-2571.
doi: 10.1016/j.jde.2014.12.018. |
[2] |
Q. Chen and Z. Tan,
Global existence and convergence rates of smooth solutions for the compressible magnetohydroynamics equations, Nonlinear Anal., 72 (2010), 4438-4451.
doi: 10.1016/j.na.2010.02.019. |
[3] |
G. Q. Chen and D. H. Wang,
Global solution of nonlinear magnetohydrodynamics with large initial data, J. Differential Equations, 182 (2002), 344-376.
doi: 10.1006/jdeq.2001.4111. |
[4] |
G. Q. Chen and D. H. Wang,
Existence and continuous dependence of large solutions for the magnetohydrodynamic equations, Z. Angew. Math. Phys., 54 (2003), 608-632.
doi: 10.1007/s00033-003-1017-z. |
[5] |
L. L. Du and D. Q. Zhou,
Global well-posedness of two-dimensional magnetohydrodynamic flows with partial dissipation and magnetic diffusion, SIAM J. Math. Anal., 47 (2015), 1562-1589.
doi: 10.1137/140959821. |
[6] |
J.S. Fan, A. Ahmed, H. Tasawar, N. Gen and Y. Zhou,
On strong solutions to the compressible Hall-magnetohydrodynamic system, Nonlinear Anal. Real World Appl., 22 (2015), 423-434.
doi: 10.1016/j.nonrwa.2014.10.003. |
[7] |
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. |
[8] |
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. |
[9] |
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. |
[10] |
E. Feireisl, A. Novotný and H. Petleltová,
On the existence of globally defined weak solutions to the Navier-Stokes equations of isentropic compressible fluids, J. Math. Fluid Mech., 3 (2001), 358-392.
doi: 10.1007/PL00000976. |
[11] |
Y. Guo and Y. J. Wang,
Decay of dissipative equations and negative Sobolev spaces, Comm. Partial Differential Equations, 37 (2012), 2165-2208.
doi: 10.1080/03605302.2012.696296. |
[12] |
C. C. Hao and H. L. Li,
Global existence for compressible Navier-Stokes-Poisson equations in three and higher dimensions, J. Differential Equations, 246 (2009), 4791-4812.
doi: 10.1016/j.jde.2008.11.019. |
[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 solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys., 283 (2008), 255-284.
doi: 10.1007/s00220-008-0497-2. |
[15] |
X. P. Hu and D. H. Wang,
Low Mach number limit of viscous compressible magnetohydrodynamic flows, SIAM J. Math. Anal., 41 (2009), 1272-1294.
doi: 10.1137/080723983. |
[16] |
X. P. Hu and D. H. Wang,
Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 197 (2010), 203-238.
doi: 10.1007/s00205-010-0295-9. |
[17] |
S. Jiang, Q. C. Ju and F. C. Li,
Incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary condtions, Comm. Math. Phys., 297 (2010), 371-400.
doi: 10.1007/s00220-010-0992-0. |
[18] |
S. Jiang, Q. C. Ju, F. C. Li and Z. P. Xin,
Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data, Adv. Math., 259 (2014), 384-420.
doi: 10.1016/j.aim.2014.03.022. |
[19] |
Q. C. Ju, F. C. Li and Y. Li,
Asymptotic limits of the full comressible magnetohydrodynamic equations, SIAM J. Math. Anal., 45 (2013), 2597-2624.
doi: 10.1137/130913390. |
[20] |
N. Ju,
Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the Sobolev space, Comm. Math. Phys., 251 (2004), 365-376.
doi: 10.1007/s00220-004-1062-2. |
[21] |
S. Kawashima,
Smooth global solutions for two-dimensional equations of electromagneto fluid dynamics, Japan J. Appl. Math., 1 (1984), 207-222.
doi: 10.1007/BF03167869. |
[22] |
S. Kawashima and M. Okada,
Smooth global solutions for the one-dimensional equations in magnetohydrodynamics, Proc. Japan Acad., 58 (1982), 384-387.
doi: 10.3792/pjaa.58.384. |
[23] |
T. Kobayashi and T. Suzuki,
Weak solutions to the Navier-Stokes-Poisson equation, Adv. Math. Sci. Appl., 18 (2008), 141-168.
|
[24] |
Z. Lei,
On axially symmetric incompressible magnetohydrodynamics in three dimensions, J. Differential Equations, 259 (2015), 3202-3215.
doi: 10.1016/j.jde.2015.04.017. |
[25] |
H. L. Li, A. Matsumura and G. J. Zhang,
Optimal decay rate of the compressible Navier-Stokes-Poisson system in ℝ3, Arch. Ration. Mech. Anal., 196 (2010), 681-713.
doi: 10.1007/s00205-009-0255-4. |
[26] |
H. L. Li, X. Y. Xu and J. W. Zhang,
Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillationa and vacuum, SIAM J. Math. Anal., 45 (2013), 1356-1387.
doi: 10.1137/120893355. |
[27] |
F. C. Li and H. J. Yu,
Optimal decay rate of classical solutions to the compressible magnetohydrodynamic equations, Proc. Roy. Soc. Edinburgh Sect., 141 (2011), 109-126.
doi: 10.1017/S0308210509001632. |
[28] |
P. L. Lions, Mathematical Topics in Fluids Mechanics. Oxford Lecture Ser. Math. Appl, Clarendon Press, Oxford University Press, New York, 1998.
![]() ![]() |
[29] |
A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2002.
![]() ![]() |
[30] |
L. Nirenberg,
On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.
|
[31] |
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. |
[32] |
Z. Tan, L. L. Tong and Y. Wang,
Large time behavior of the compressible magnetohydrodynamic equations with Coulomb force, J. Math. Anal. Appl., 427 (2015), 600-617.
doi: 10.1016/j.jmaa.2015.02.077. |
[33] |
Z. Tan and Y. J. Wang,
Global existence and large-time behavior of weak solutions to the compressible magnetohydrodynamic equations with Coulomb force, Nonlinear Anal., 71 (2009), 5866-5884.
doi: 10.1016/j.na.2009.05.012. |
[34] |
Z. Tan and G. C. Wu,
Global existence for the non-isentropic compressible Navier-Stokes-Poisson system in three and higher dimensions, Nonlinear Anal. Real World Appl., 13 (2012), 650-664.
doi: 10.1016/j.nonrwa.2011.08.005. |
[35] |
Z. Tan and H. Q. Wang,
Optimal decay rates of the compressible magnetohydrodynamic equations, Nonlinear Anal. Real World Appl., 14 (2013), 188-201.
doi: 10.1016/j.nonrwa.2012.05.012. |
[36] |
Z. Tan, Y. J. Wang and Y. Wang,
Stability of steady states of the Navier-Stokes-Poisson equations with non-flat doping profile, SIAM J. Math. Anal., 47 (2015), 179-209.
doi: 10.1137/130950069. |
[37] |
Z. Tan, Y. Wang and X. Zhang,
Large time behavior of solutions to the non-isentropic compressible Navier-Stokes-Poisson system in ℝ3, Kinet. Relat. Models, 5 (2012), 615-638.
doi: 10.3934/krm.2012.5.615. |
[38] |
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. |
[39] |
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. |
[40] |
Y. J. Wang,
Decay of the Navier-Stokes-Poisson equations, J. Differential Equations, 253 (2012), 273-297.
doi: 10.1016/j.jde.2012.03.006. |
[41] |
L. Zhang and S. Li,
A regularity criterion for 2D MHD flows with horizontal dissipation and horizontal magnetic diffusion, Nonlinear Anal. Real World Appl., 21 (2015), 197-206.
doi: 10.1016/j.nonrwa.2014.07.005. |
[42] |
G. J. Zhang, H. L. Li and C. J. Zhu,
Optimal decay rate of the non-isentropic compressible Navier-Stokes-Poisson system in ℝ3, J. Differential Equations, 250 (2011), 866-891.
doi: 10.1016/j.jde.2010.07.035. |
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