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Global strong solution and exponential decay for nonhomogeneous Navier-Stokes and magnetohydrodynamic equations
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China |
The present paper concerns an initial boundary value problem of two-dimensional (2D) nonhomogeneous magnetohydrodynamic (MHD) equations with non-negative density. We establish the global existence and exponential decay of strong solutions. In particular, the initial data can be arbitrarily large. The key idea is to use a lemma due to Desjardins (Arch. Rational Mech. Anal. 137:135–158, 1997).
References:
[1] |
H. Abidi and M. Paicu,
Global existence for the magnetohydrodynamic system in critical spaces, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 447-476.
doi: 10.1017/S0308210506001181. |
[2] |
C. Amrouche and V. Girault,
Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czechoslovak Math. J., 44 (1994), 109-140.
|
[3] |
Q. Bie, Q. Wang and Z. Yao,
Global well-posedness of the 3D incompressible MHD equations with variable density, Nonlinear Anal. Real World Appl., 47 (2019), 85-105.
doi: 10.1016/j.nonrwa.2018.10.008. |
[4] |
F. Chen, B. Guo and X. Zhai,
Global solution to the 3-D inhomogeneous incompressible MHD system with discontinuous density, Kinet. Relat. Models, 12 (2019), 37-58.
doi: 10.3934/krm.2019002. |
[5] |
F. Chen, Y. Li and H. Xu,
Global solution to the 3D nonhomogeneous incompressible MHD equations with some large initial data, Discrete Contin. Dyn. Syst., 36 (2016), 2945-2967.
doi: 10.3934/dcds.2016.36.2945. |
[6] |
Q. Chen, Z. Tan and Y. Wang,
Strong solutions to the incompressible magnetohydrodynamic equations, Math. Methods Appl. Sci., 34 (2011), 94-107.
doi: 10.1002/mma.1338. |
[7] |
H. J. Choe and H. Kim,
Strong solutions of the Navier-Stokes equations for nonhomogeneous incompressible fluids, Comm. Partial Differential Equations, 28 (2003), 1183-1201.
doi: 10.1081/PDE-120021191. |
[8] |
R. Danchin and P. B. Mucha,
The incompressible Navier-Stokes equations in vacuum, Comm. Pure Appl. Math., 72 (2019), 1351-1385.
doi: 10.1002/cpa.21806. |
[9] |
B. Desjardins,
Regularity results for two-dimensional flows of multiphase viscous fluids, Arch. Rational Mech. Anal., 137 (1997), 135-158.
doi: 10.1007/s002050050025. |
[10] |
L. C. Evans, Partial Differential Equations, 2$^nd$ edition, American Mathematical Society, Providence, RI, 2010.
doi: 10.1090/gsm/019. |
[11] |
A. Friedman, Partial Differential Equations, Dover Books on Mathematics, New York, 2008. |
[12] |
X. Huang and Y. Wang,
Global strong solution to the 2D nonhomogeneous incompressible MHD system, J. Differential Equations, 254 (2013), 511-527.
doi: 10.1016/j.jde.2012.08.029. |
[13] |
H. Li,
Global strong solution to the three dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density-dependent viscosity and resistivity, Math. Methods Appl. Sci., 41 (2018), 3062-3092.
doi: 10.1002/mma.4801. |
[14] |
J. Li,
Local existence and uniqueness of strong solutions to the Navier-Stokes equations with nonnegative density, J. Differential Equations, 263 (2017), 6512-6536.
doi: 10.1016/j.jde.2017.07.021. |
[15] |
Z. Liang,
Local strong solution and blow-up criterion for the 2D nonhomogeneous incompressible fluids, J. Differential Equations, 258 (2015), 2633-2654.
doi: 10.1016/j.jde.2014.12.015. |
[16] |
P.-L. Lions, Mathematical Topics in Fluid Mechanics, vol. Ⅰ: Incompressible Models, Oxford University Press, Oxford, 1996.
![]() ![]() |
[17] |
Y. Liu, Global existence and exponential decay of strong solutions for the 3D incompressible MHD equations with density-dependent viscosity coefficient, Z. Angew. Math. Phys., 70 (2019), Paper No. 107.
doi: 10.1007/s00033-019-1157-4. |
[18] |
B. Lü, X. Shi and X. Zhong,
Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent Navier-Stokes equations with vacuum, Nonlinearity, 31 (2018), 2617-2632.
doi: 10.1088/1361-6544/aab31f. |
[19] |
B. Lü, Z. Xu and X. Zhong,
Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent magnetohydrodynamic equations with vacuum, J. Math. Pures Appl., 108 (2017), 41-62.
doi: 10.1016/j.matpur.2016.10.009. |
[20] |
M. Paicu, P. Zhang and Z. Zhang,
Global unique solvability of inhomogeneous Navier-Stokes equations with bounded density, Comm. Partial Differential Equations, 38 (2013), 1208-1234.
doi: 10.1080/03605302.2013.780079. |
[21] |
X. Si and X. Ye, Global well-posedness for the incompressible MHD equations with density-dependent viscosity and resistivity coefficients, Z. Angew. Math. Phys., 67 (2016), Paper No. 126.
doi: 10.1007/s00033-016-0722-3. |
[22] |
S. Song, On local strong solutions to the three-dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density-dependent viscosity and vacuum, Z. Angew. Math. Phys., 69 (2018), Paper No. 23.
doi: 10.1007/s00033-018-0915-z. |
[23] |
M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 4$^th$ edition, Springer-Verlag, Berlin, 2008. |
show all references
References:
[1] |
H. Abidi and M. Paicu,
Global existence for the magnetohydrodynamic system in critical spaces, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 447-476.
doi: 10.1017/S0308210506001181. |
[2] |
C. Amrouche and V. Girault,
Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czechoslovak Math. J., 44 (1994), 109-140.
|
[3] |
Q. Bie, Q. Wang and Z. Yao,
Global well-posedness of the 3D incompressible MHD equations with variable density, Nonlinear Anal. Real World Appl., 47 (2019), 85-105.
doi: 10.1016/j.nonrwa.2018.10.008. |
[4] |
F. Chen, B. Guo and X. Zhai,
Global solution to the 3-D inhomogeneous incompressible MHD system with discontinuous density, Kinet. Relat. Models, 12 (2019), 37-58.
doi: 10.3934/krm.2019002. |
[5] |
F. Chen, Y. Li and H. Xu,
Global solution to the 3D nonhomogeneous incompressible MHD equations with some large initial data, Discrete Contin. Dyn. Syst., 36 (2016), 2945-2967.
doi: 10.3934/dcds.2016.36.2945. |
[6] |
Q. Chen, Z. Tan and Y. Wang,
Strong solutions to the incompressible magnetohydrodynamic equations, Math. Methods Appl. Sci., 34 (2011), 94-107.
doi: 10.1002/mma.1338. |
[7] |
H. J. Choe and H. Kim,
Strong solutions of the Navier-Stokes equations for nonhomogeneous incompressible fluids, Comm. Partial Differential Equations, 28 (2003), 1183-1201.
doi: 10.1081/PDE-120021191. |
[8] |
R. Danchin and P. B. Mucha,
The incompressible Navier-Stokes equations in vacuum, Comm. Pure Appl. Math., 72 (2019), 1351-1385.
doi: 10.1002/cpa.21806. |
[9] |
B. Desjardins,
Regularity results for two-dimensional flows of multiphase viscous fluids, Arch. Rational Mech. Anal., 137 (1997), 135-158.
doi: 10.1007/s002050050025. |
[10] |
L. C. Evans, Partial Differential Equations, 2$^nd$ edition, American Mathematical Society, Providence, RI, 2010.
doi: 10.1090/gsm/019. |
[11] |
A. Friedman, Partial Differential Equations, Dover Books on Mathematics, New York, 2008. |
[12] |
X. Huang and Y. Wang,
Global strong solution to the 2D nonhomogeneous incompressible MHD system, J. Differential Equations, 254 (2013), 511-527.
doi: 10.1016/j.jde.2012.08.029. |
[13] |
H. Li,
Global strong solution to the three dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density-dependent viscosity and resistivity, Math. Methods Appl. Sci., 41 (2018), 3062-3092.
doi: 10.1002/mma.4801. |
[14] |
J. Li,
Local existence and uniqueness of strong solutions to the Navier-Stokes equations with nonnegative density, J. Differential Equations, 263 (2017), 6512-6536.
doi: 10.1016/j.jde.2017.07.021. |
[15] |
Z. Liang,
Local strong solution and blow-up criterion for the 2D nonhomogeneous incompressible fluids, J. Differential Equations, 258 (2015), 2633-2654.
doi: 10.1016/j.jde.2014.12.015. |
[16] |
P.-L. Lions, Mathematical Topics in Fluid Mechanics, vol. Ⅰ: Incompressible Models, Oxford University Press, Oxford, 1996.
![]() ![]() |
[17] |
Y. Liu, Global existence and exponential decay of strong solutions for the 3D incompressible MHD equations with density-dependent viscosity coefficient, Z. Angew. Math. Phys., 70 (2019), Paper No. 107.
doi: 10.1007/s00033-019-1157-4. |
[18] |
B. Lü, X. Shi and X. Zhong,
Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent Navier-Stokes equations with vacuum, Nonlinearity, 31 (2018), 2617-2632.
doi: 10.1088/1361-6544/aab31f. |
[19] |
B. Lü, Z. Xu and X. Zhong,
Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent magnetohydrodynamic equations with vacuum, J. Math. Pures Appl., 108 (2017), 41-62.
doi: 10.1016/j.matpur.2016.10.009. |
[20] |
M. Paicu, P. Zhang and Z. Zhang,
Global unique solvability of inhomogeneous Navier-Stokes equations with bounded density, Comm. Partial Differential Equations, 38 (2013), 1208-1234.
doi: 10.1080/03605302.2013.780079. |
[21] |
X. Si and X. Ye, Global well-posedness for the incompressible MHD equations with density-dependent viscosity and resistivity coefficients, Z. Angew. Math. Phys., 67 (2016), Paper No. 126.
doi: 10.1007/s00033-016-0722-3. |
[22] |
S. Song, On local strong solutions to the three-dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density-dependent viscosity and vacuum, Z. Angew. Math. Phys., 69 (2018), Paper No. 23.
doi: 10.1007/s00033-018-0915-z. |
[23] |
M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 4$^th$ edition, Springer-Verlag, Berlin, 2008. |
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