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October  2020, 40(10): 5617-5638. doi: 10.3934/dcds.2020240

Existence and uniqueness of very weak solution of the MHD type system

Yong Zeng

School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China

Received  October 2018 Revised  April 2020 Published  June 2020

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.

Keywords: Steady MHD type, existence, uniqueness, very weak solutions.
Mathematics Subject Classification: 35A01, 35A02, 35D99, 35Q35, 35Q60.
Citation: Yong Zeng. Existence and uniqueness of very weak solution of the MHD type system. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 5617-5638. doi: 10.3934/dcds.2020240
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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