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Existence and uniqueness of very weak solution of the MHD type system
School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China |
This paper studies the very weak solution to the steady MHD type system in a bounded domain. We prove the existence of very weak solutions to the MHD type system for arbitrary large external forces $ ({\bf f},{\bf{g}}) $ in $ L^r({\Omega})\times [X_{\theta',q'}({\Omega})]' $ and suitable boundary data $ ({\mathcal B}^0,{\mathcal U}^0) $ in $ W^{-1/p,p}({\partial}{\Omega})\times W^{-1/q,q}({\partial}{\Omega}) $, under certain assumptions on $ p,q,r,\theta $. The uniqueness of very weak solution for small data $ ({\bf f},{\bf{g}},{\mathcal B}^0,{\mathcal U}^0) $ is also studied.
References:
[1] |
G. V. Alekseev,
Solvability of control problems for stationary equations of the magnetohydrodynamics of a viscous fluid., Siberian Math. J., 45 (2004), 197-213.
doi: 10.1023/B:SIMJ.0000021277.82617.3b. |
[2] |
K. A. Ames and J. C. Song,
Decay bounds for magnetohydrodynamic geophysical flow, Nonlinear Anal., 65 (2006), 1318-1333.
doi: 10.1016/j.na.2005.10.013. |
[3] |
C. Amrouche and V. Girault,
Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czechoslov. Math. J., 44 (1994), 109-140.
|
[4] |
C. Amrouche and U. Razafison,
Weighted Sobolev spaces for a scalar model of the stationary Oseen equations in $\Bbb R^3$, J. Math. Fluid Mech., 9 (2007), 181-210.
doi: 10.1007/s00021-005-0195-1. |
[5] |
C. Amrouche, $ \rm\check{S} $. Ne$ \rm\check{c} $asová and Y. Raudin,
Very weak, generalized and strong solutions to the Stokes system in the half-space, J. Differ. Equ., 244 (2008), 887-915.
doi: 10.1016/j.jde.2007.10.007. |
[6] |
C. Amrouche and M. Á. Rodríguez-Bellido,
Very weak solutions for the stationary Stokes equations, C. R. Math. Acad. Sci. Paris, 348 (2010), 223-228.
doi: 10.1016/j.crma.2009.12.020. |
[7] |
C. Amrouche and M. Á. Rodríguez-Bellido,
Very weak solutions for the stationary Oseen and Navier-Stokes equations, C. R. Math. Acad. Sci. Paris, 348 (2010), 335-339.
doi: 10.1016/j.crma.2009.12.021. |
[8] |
C. Amrouche and M. Á. Rodríguez-Bellido,
Stationary Stokes, Oseen and Navier-Stokes equations with singular data, Arch. Ration. Mech. Anal., 199 (2011), 597-651.
doi: 10.1007/s00205-010-0340-8. |
[9] |
F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, 183. Springer, New York, 2003.
doi: 10.1007/978-1-4614-5975-0. |
[10] |
M. Bulí$ \rm\check{c} $ek, J. Burczak and S. Schwarzacher,
A unified theory for some non Newtonian fluids under singular forcing, SIAM J. Math. Anal., 48 (2016), 4241-4267.
doi: 10.1137/16M1073881. |
[11] |
C. Cao and J. Wu,
Two regularity criteria for the 3D MHD equations, J. Diff. Equations, 248 (2010), 2263-2274.
doi: 10.1016/j.jde.2009.09.020. |
[12] |
L. Cattabriga,
Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Sem. Mat. Univ. Padova, 31 (1961), 308-340.
|
[13] |
E. V. Chizhonkov,
A system of equations of magnetohydrodynamic type, Dokl. Akad. Nauk SSSR, 278 (1984), 1074-1077.
|
[14] |
R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Volume 3 Spectral Theory and Applications, Springer-Verlag Berlin Heidelberg, 1990. |
[15] |
G. Duvaut and J.-L. Lions,
Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Ration. Mech. Anal., 46 (1972), 241-279.
doi: 10.1007/BF00250512. |
[16] |
G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems, Second Edition, Springer, New York, 2011. |
[17] |
G. P. Galdi, C. G. Simader and H. Sohr,
A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in $W^{-1/q,q}$, Math. Ann., 331 (2005), 41-74.
doi: 10.1007/s00208-004-0573-7. |
[18] |
C. Gerhardt,
Stationary solutions to the Navier-Stokes equations in dimension four, Math. Z., 165 (1979), 193-197.
doi: 10.1007/BF01182469. |
[19] |
D. Glibarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. |
[20] |
F. Guillénâ€"González, M. A. Rodríguez-Bellido and M. A. Rojas-Medar,
Hydrostatic Stokes equations with non-smooth data for mixed boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 807-826.
doi: 10.1016/j.anihpc.2003.11.002. |
[21] |
M. D. Gunzburger, A. J. Meir and J. S. Peterson,
On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics, Math. Comp., 56 (1991), 523-563.
doi: 10.1090/S0025-5718-1991-1066834-0. |
[22] |
C. He and Z. Xin,
On the regularity of weak solutions to the magnetohydrodynamic equations, J. Diff. Equations, 213 (2005), 235-254.
doi: 10.1016/j.jde.2004.07.002. |
[23] |
R. Hide, On planetary atmospheres and interiors, In Mathematical Problems in the Gcophysical Sciences Ⅰ. Amer. Math. Soc., Providence, RI, (1971), 229-353. |
[24] |
H. Kim,
Existence and regularity of very weak solutions of the stationary Navier-Stokes equations, Arch. Ration. Mech. Anal., 193 (2009), 117-152.
doi: 10.1007/s00205-008-0168-7. |
[25] |
I. Kondrashuk, E. A. Notte-Cuello and M. A. Rojas-Medar,
Magnetohydrodynamics's type equations over Clifford algebras, J. Nonlinear Math. Phys., 17 (2010), 337-347.
doi: 10.1142/S1402925110000933. |
[26] |
H. Li and C. Lin,
Spatial decay bounds for time dependent magnetohydrodynamic geophysical flow, Nonlinear Anal. Real World Appl., 11 (2010), 665-682.
doi: 10.1016/j.nonrwa.2009.01.013. |
[27] |
Y. Li and C. Zhao,
Existence, uniqueness and decay properties of strong solutions to an evolutionary system of MHD type in $\Bbb R^3$, J. Dynam. Differential Equations, 18 (2006), 393-426.
doi: 10.1007/s10884-006-9012-7. |
[28] |
E. Maru$ \rm\check{s} $ić-Paloka,
Solvability of the Navier-Stokes system with $L^2$ boundary data, Appl. Math. Optim., 41 (2000), 365-375.
doi: 10.1007/s002459911018. |
[29] |
M. A. Rojas-Medar and J. L. Boldrini,
The weak solutions and reproductive property for a system of evolution equations of magnetohydrodynamic type, Proyecciones, 13 (1994), 85-97.
doi: 10.22199/S07160917.1994.0002.00002. |
[30] |
M. A. Rojas-Medar and J. L. Boldrini,
Global strong solutions of equations of magnetohydrodynamic type, J. Austral. Math. Soc., Ser. B, 38 (1997), 291-306.
doi: 10.1017/S0334270000000680. |
[31] |
M. Sermange and R. Temam,
Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664.
doi: 10.1002/cpa.3160360506. |
[32] |
E. J. Villamizar-Roa, H. Lamos-Díaz and G. Arenas-Díaz,
Very weak solutions for the magnetohydrodynamic type equations, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 957-972.
doi: 10.3934/dcdsb.2008.10.957. |
[33] |
E. J. Villamizar-Roa, M. A. Rodríguez-Bellido, M. A and Ro jas-Medar,
The Boussinesq system with mixed nonsmooth boundary data, C. R. Math. Acad. Sci. Paris, 343 (2006), 191-196.
doi: 10.1016/j.crma.2006.06.011. |
[34] |
Y. Zeng,
Steady states of Hall-MHD system, J. Math. Anal. Appl., 451 (2017), 757-793.
doi: 10.1016/j.jmaa.2017.02.023. |
[35] |
C. S. Zhao,
Initial boundary value problem for the evolution system of MHD type describing geophysical flow in three-dimensional domains, Math. Methods Appl. Sci., 26 (2003), 759-781.
doi: 10.1002/mma.394. |
show all references
References:
[1] |
G. V. Alekseev,
Solvability of control problems for stationary equations of the magnetohydrodynamics of a viscous fluid., Siberian Math. J., 45 (2004), 197-213.
doi: 10.1023/B:SIMJ.0000021277.82617.3b. |
[2] |
K. A. Ames and J. C. Song,
Decay bounds for magnetohydrodynamic geophysical flow, Nonlinear Anal., 65 (2006), 1318-1333.
doi: 10.1016/j.na.2005.10.013. |
[3] |
C. Amrouche and V. Girault,
Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czechoslov. Math. J., 44 (1994), 109-140.
|
[4] |
C. Amrouche and U. Razafison,
Weighted Sobolev spaces for a scalar model of the stationary Oseen equations in $\Bbb R^3$, J. Math. Fluid Mech., 9 (2007), 181-210.
doi: 10.1007/s00021-005-0195-1. |
[5] |
C. Amrouche, $ \rm\check{S} $. Ne$ \rm\check{c} $asová and Y. Raudin,
Very weak, generalized and strong solutions to the Stokes system in the half-space, J. Differ. Equ., 244 (2008), 887-915.
doi: 10.1016/j.jde.2007.10.007. |
[6] |
C. Amrouche and M. Á. Rodríguez-Bellido,
Very weak solutions for the stationary Stokes equations, C. R. Math. Acad. Sci. Paris, 348 (2010), 223-228.
doi: 10.1016/j.crma.2009.12.020. |
[7] |
C. Amrouche and M. Á. Rodríguez-Bellido,
Very weak solutions for the stationary Oseen and Navier-Stokes equations, C. R. Math. Acad. Sci. Paris, 348 (2010), 335-339.
doi: 10.1016/j.crma.2009.12.021. |
[8] |
C. Amrouche and M. Á. Rodríguez-Bellido,
Stationary Stokes, Oseen and Navier-Stokes equations with singular data, Arch. Ration. Mech. Anal., 199 (2011), 597-651.
doi: 10.1007/s00205-010-0340-8. |
[9] |
F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, 183. Springer, New York, 2003.
doi: 10.1007/978-1-4614-5975-0. |
[10] |
M. Bulí$ \rm\check{c} $ek, J. Burczak and S. Schwarzacher,
A unified theory for some non Newtonian fluids under singular forcing, SIAM J. Math. Anal., 48 (2016), 4241-4267.
doi: 10.1137/16M1073881. |
[11] |
C. Cao and J. Wu,
Two regularity criteria for the 3D MHD equations, J. Diff. Equations, 248 (2010), 2263-2274.
doi: 10.1016/j.jde.2009.09.020. |
[12] |
L. Cattabriga,
Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Sem. Mat. Univ. Padova, 31 (1961), 308-340.
|
[13] |
E. V. Chizhonkov,
A system of equations of magnetohydrodynamic type, Dokl. Akad. Nauk SSSR, 278 (1984), 1074-1077.
|
[14] |
R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Volume 3 Spectral Theory and Applications, Springer-Verlag Berlin Heidelberg, 1990. |
[15] |
G. Duvaut and J.-L. Lions,
Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Ration. Mech. Anal., 46 (1972), 241-279.
doi: 10.1007/BF00250512. |
[16] |
G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems, Second Edition, Springer, New York, 2011. |
[17] |
G. P. Galdi, C. G. Simader and H. Sohr,
A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in $W^{-1/q,q}$, Math. Ann., 331 (2005), 41-74.
doi: 10.1007/s00208-004-0573-7. |
[18] |
C. Gerhardt,
Stationary solutions to the Navier-Stokes equations in dimension four, Math. Z., 165 (1979), 193-197.
doi: 10.1007/BF01182469. |
[19] |
D. Glibarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. |
[20] |
F. Guillénâ€"González, M. A. Rodríguez-Bellido and M. A. Rojas-Medar,
Hydrostatic Stokes equations with non-smooth data for mixed boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 807-826.
doi: 10.1016/j.anihpc.2003.11.002. |
[21] |
M. D. Gunzburger, A. J. Meir and J. S. Peterson,
On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics, Math. Comp., 56 (1991), 523-563.
doi: 10.1090/S0025-5718-1991-1066834-0. |
[22] |
C. He and Z. Xin,
On the regularity of weak solutions to the magnetohydrodynamic equations, J. Diff. Equations, 213 (2005), 235-254.
doi: 10.1016/j.jde.2004.07.002. |
[23] |
R. Hide, On planetary atmospheres and interiors, In Mathematical Problems in the Gcophysical Sciences Ⅰ. Amer. Math. Soc., Providence, RI, (1971), 229-353. |
[24] |
H. Kim,
Existence and regularity of very weak solutions of the stationary Navier-Stokes equations, Arch. Ration. Mech. Anal., 193 (2009), 117-152.
doi: 10.1007/s00205-008-0168-7. |
[25] |
I. Kondrashuk, E. A. Notte-Cuello and M. A. Rojas-Medar,
Magnetohydrodynamics's type equations over Clifford algebras, J. Nonlinear Math. Phys., 17 (2010), 337-347.
doi: 10.1142/S1402925110000933. |
[26] |
H. Li and C. Lin,
Spatial decay bounds for time dependent magnetohydrodynamic geophysical flow, Nonlinear Anal. Real World Appl., 11 (2010), 665-682.
doi: 10.1016/j.nonrwa.2009.01.013. |
[27] |
Y. Li and C. Zhao,
Existence, uniqueness and decay properties of strong solutions to an evolutionary system of MHD type in $\Bbb R^3$, J. Dynam. Differential Equations, 18 (2006), 393-426.
doi: 10.1007/s10884-006-9012-7. |
[28] |
E. Maru$ \rm\check{s} $ić-Paloka,
Solvability of the Navier-Stokes system with $L^2$ boundary data, Appl. Math. Optim., 41 (2000), 365-375.
doi: 10.1007/s002459911018. |
[29] |
M. A. Rojas-Medar and J. L. Boldrini,
The weak solutions and reproductive property for a system of evolution equations of magnetohydrodynamic type, Proyecciones, 13 (1994), 85-97.
doi: 10.22199/S07160917.1994.0002.00002. |
[30] |
M. A. Rojas-Medar and J. L. Boldrini,
Global strong solutions of equations of magnetohydrodynamic type, J. Austral. Math. Soc., Ser. B, 38 (1997), 291-306.
doi: 10.1017/S0334270000000680. |
[31] |
M. Sermange and R. Temam,
Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664.
doi: 10.1002/cpa.3160360506. |
[32] |
E. J. Villamizar-Roa, H. Lamos-Díaz and G. Arenas-Díaz,
Very weak solutions for the magnetohydrodynamic type equations, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 957-972.
doi: 10.3934/dcdsb.2008.10.957. |
[33] |
E. J. Villamizar-Roa, M. A. Rodríguez-Bellido, M. A and Ro jas-Medar,
The Boussinesq system with mixed nonsmooth boundary data, C. R. Math. Acad. Sci. Paris, 343 (2006), 191-196.
doi: 10.1016/j.crma.2006.06.011. |
[34] |
Y. Zeng,
Steady states of Hall-MHD system, J. Math. Anal. Appl., 451 (2017), 757-793.
doi: 10.1016/j.jmaa.2017.02.023. |
[35] |
C. S. Zhao,
Initial boundary value problem for the evolution system of MHD type describing geophysical flow in three-dimensional domains, Math. Methods Appl. Sci., 26 (2003), 759-781.
doi: 10.1002/mma.394. |
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