American Institute of Mathematical Sciences

Advanced
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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Sem. Mat. Univ. Padova, 31 (1961), 308-340.   Google Scholar

[13]

E. V. Chizhonkov, A system of equations of magnetohydrodynamic type, Dokl. Akad. Nauk SSSR, 278 (1984), 1074-1077.   Google Scholar

[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.  Google Scholar

[15]

G. Duvaut and J.-L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Ration. Mech. Anal., 46 (1972), 241-279.  doi: 10.1007/BF00250512.  Google Scholar

[16]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems, Second Edition, Springer, New York, 2011. Google Scholar

[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.  Google Scholar

[18]

C. Gerhardt, Stationary solutions to the Navier-Stokes equations in dimension four, Math. Z., 165 (1979), 193-197.  doi: 10.1007/BF01182469.  Google Scholar

[19]

D. Glibarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

R. Hide, On planetary atmospheres and interiors, In Mathematical Problems in the Gcophysical Sciences Ⅰ. Amer. Math. Soc., Providence, RI, (1971), 229-353. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Sem. Mat. Univ. Padova, 31 (1961), 308-340.   Google Scholar

[13]

E. V. Chizhonkov, A system of equations of magnetohydrodynamic type, Dokl. Akad. Nauk SSSR, 278 (1984), 1074-1077.   Google Scholar

[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.  Google Scholar

[15]

G. Duvaut and J.-L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Ration. Mech. Anal., 46 (1972), 241-279.  doi: 10.1007/BF00250512.  Google Scholar

[16]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems, Second Edition, Springer, New York, 2011. Google Scholar

[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.  Google Scholar

[18]

C. Gerhardt, Stationary solutions to the Navier-Stokes equations in dimension four, Math. Z., 165 (1979), 193-197.  doi: 10.1007/BF01182469.  Google Scholar

[19]

D. Glibarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

R. Hide, On planetary atmospheres and interiors, In Mathematical Problems in the Gcophysical Sciences Ⅰ. Amer. Math. Soc., Providence, RI, (1971), 229-353. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Elder Jesús Villamizar-Roa, Henry Lamos-Díaz, Gilberto Arenas-Díaz. Very weak solutions for the magnetohydrodynamic type equations. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 957-972. doi: 10.3934/dcdsb.2008.10.957
[2]

Peiying Chen. Existence and uniqueness of weak solutions for a class of nonlinear parabolic equations. Electronic Research Announcements, 2017, 24: 38-52. doi: 10.3934/era.2017.24.005
[3]

Shihui Zhu. Existence and uniqueness of global weak solutions of the Camassa-Holm equation with a forcing. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5201-5221. doi: 10.3934/dcds.2016026
[4]

Verena Bögelein, Frank Duzaar, Ugo Gianazza. Very weak solutions of singular porous medium equations with measure data. Communications on Pure & Applied Analysis, 2015, 14 (1) : 23-49. doi: 10.3934/cpaa.2015.14.23
[5]

Hongbin Chen, Yi Li. Existence, uniqueness, and stability of periodic solutions of an equation of duffing type. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 793-807. doi: 10.3934/dcds.2007.18.793
[6]

Huajun Gong, Jinkai Li. Global existence of strong solutions to incompressible MHD. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1553-1561. doi: 10.3934/cpaa.2014.13.1553
[7]

Huajun Gong, Jinkai Li. Global existence of strong solutions to incompressible MHD. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1337-1345. doi: 10.3934/cpaa.2014.13.1337
[8]

Igor Chueshov, Irena Lasiecka. Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 777-809. doi: 10.3934/dcds.2006.15.777
[9]

Jiahong Wu. Regularity results for weak solutions of the 3D MHD equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 543-556. doi: 10.3934/dcds.2004.10.543
[10]

Allen Montz, Hamid Bellout, Frederick Bloom. Existence and uniqueness of steady flows of nonlinear bipolar viscous fluids in a cylinder. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2107-2128. doi: 10.3934/dcdsb.2015.20.2107
[11]

Jesus Idelfonso Díaz, Jean Michel Rakotoson. On very weak solutions of semi-linear elliptic equations in the framework of weighted spaces with respect to the distance to the boundary. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1037-1058. doi: 10.3934/dcds.2010.27.1037
[12]

Alexander Quaas, Aliang Xia. Existence and uniqueness of positive solutions for a class of logistic type elliptic equations in $\mathbb{R}^N$ involving fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2653-2668. doi: 10.3934/dcds.2017113
[13]

Hua Nie, Wenhao Xie, Jianhua Wu. Uniqueness of positive steady state solutions to the unstirred chemostat model with external inhibitor. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1279-1297. doi: 10.3934/cpaa.2013.12.1279
[14]

Emeric Bouin, Jean Dolbeault, Christian Schmeiser. Diffusion and kinetic transport with very weak confinement. Kinetic & Related Models, 2020, 13 (2) : 345-371. doi: 10.3934/krm.2020012
[15]

Andrea L. Bertozzi, Dejan Slepcev. Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1617-1637. doi: 10.3934/cpaa.2010.9.1617
[16]

Xin Lai, Xinfu Chen, Mingxin Wang, Cong Qin, Yajing Zhang. Existence, uniqueness, and stability of bubble solutions of a chemotaxis model. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 805-832. doi: 10.3934/dcds.2016.36.805
[17]

Peter Markowich, Jesús Sierra. Non-uniqueness of weak solutions of the Quantum-Hydrodynamic system. Kinetic & Related Models, 2019, 12 (2) : 347-356. doi: 10.3934/krm.2019015
[18]

Carlos J. Garcia-Cervera, Xiao-Ping Wang. Spin-polarized transport: Existence of weak solutions. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 87-100. doi: 10.3934/dcdsb.2007.7.87
[19]

Changchun Liu, Jingxue Yin, Juan Zhou. Existence of weak solutions for a generalized thin film equation. Communications on Pure & Applied Analysis, 2007, 6 (2) : 465-480. doi: 10.3934/cpaa.2007.6.465
[20]

Chérif Amrouche, María Ángeles Rodríguez-Bellido. On the very weak solution for the Oseen and Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 159-183. doi: 10.3934/dcdss.2010.3.159

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (148)
  • HTML views (151)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences