March  2021, 29(1): 1783-1801. doi: 10.3934/era.2020091

Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations

1. 

Departamento de Matemática, Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas, Campinas, SP, Brazil

2. 

Departamento de Matemática, Universidad de La Serena, La Serena, Chile

3. 

Departamento de Matemática, Universidad de Tarapacá, Casilla 7D, Arica, Chile

* Corresponding author: Marko A. Rojas-Medar

Received  January 2020 Revised  June 2020 Published  September 2020

We prove some results on the stability of slow stationary solutions of the MHD equations in two- and three-dimensional bounded domains for external force fields that are asymptotically autonomous. Our results show that weak solutions are asymptotically stable in time in the $ L^2 $-norm. Further, assuming certain regularity hypotheses on the problem data, strong solutions are asymptotically stable in the $ H^1 $ and $ H^2 $-norms.

Citation: José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091
References:
[1]

C. Amrouche and V. Girault, On the existence and regularity of the solutions of Stokes problem in arbitrary dimension, Proc. Japan Acad. Ser. A Math. Sci., 67 (1991), 171-175.  doi: 10.3792/pjaa.67.171.  Google Scholar

[2]

H. Beirão da Veiga, Existence and asymptotic behavior for strong solutions of the Navier-Stokes equations in the whole space, Indiana Univ. Math. J., 36 (1987), 149-166.  doi: 10.1512/iumj.1987.36.36008.  Google Scholar

[3]

H. Beirão da Veiga and P. Secchi, $L^p$-stability for the strong solutions of the Navier-Stokes equations n the whole space, Arch. Rational Mech. Anal., 98 (1987), 65-69.  doi: 10.1007/BF00279962.  Google Scholar

[4]

J. L. Boldrini and M. Rojas-Medar, On a system of evolution equations of magnetohydrodynamic type, Mat. Contemp., 8 (1995), 1-19.   Google Scholar

[5]

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

[6]

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

[7]

P. D. Damázio and M. A. Rojas-Medar, On some questions of the weak solutions of evolution equations for magnetohydrodynamic type, Proyecciones, 16 (1997), 83-97.  doi: 10.22199/S07160917.1997.0002.00001.  Google Scholar

[8]

Y. He and K. Li, Asymptotic behavior and time discretization analysis for the nonstationary Navier-Stokes problem, Num. Math., 98 (2004), 647-673.  doi: 10.1007/s00211-004-0532-y.  Google Scholar

[9]

J. G. Heywood, An error estimate uniform in time for spectral Galerkin approximations of the Navier-Stokes problem, Pacific J. Math., 98 (1982), 333-345.  doi: 10.2140/pjm.1982.98.333.  Google Scholar

[10]

J. G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem. Ⅰ. Regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal., 19 (1982), 275-311.  doi: 10.1137/0719018.  Google Scholar

[11]

J. G. Heywood and R. Rannacher, Finite element approximations of the nonstationary Navier-Stokes equations. Ⅱ. Stability of solutions and error estimates uniform in time, SIAM J. Numer. Anal., 23 (1986), 750-77.  doi: 10.1137/0723049.  Google Scholar

[12]

I. Kondrashuk, E. Notte-Cuello, M. Poblete-Cantellano and M. A. Rojas-Medar, Periodic solution for the magnetohydrodynamic equations with inhomogeneous boundary condition, Axioms, 8 (2019), 44, http://dx.doi.org/10.3390/axioms8020044. Google Scholar

[13]

G. Lassner, Über ein Rand-Anfangswertproblem der Magnetohydrodynamik, Arch. Rational Mech. Anal., 25 (1967), 388-405.  doi: 10.1007/BF00291938.  Google Scholar

[14]

E. A. Notte-Cuello and M. A. Rojas-Medar, On a system of evolution equations of magnetohydrodynamic type: An iterational approach, Proyecciones, 17 (1998), 133-165.  doi: 10.22199/S07160917.1998.0002.00001.  Google Scholar

[15]

S. B. Pikelner, Grundlagen der Kosmischen Elektrodynamik [Russ.], Moskau, 1966. Google Scholar

[16]

C. S. Qu and P. Wang, $L^p$ exponential stability for the equilibrium solutions of the Navier-Stokes equations, J. Math. Anal. Appl., 190 (1995), 419-427.  doi: 10.1006/jmaa.1995.1085.  Google Scholar

[17]

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

[18]

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

[19]

A. Schlüter, Dynamik des Plasma, Iund II, Z. Naturforsch., 5a (1950), 72–78; 6a (1951), 73–79. Google Scholar

[20]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Third edition, North-Holland Publishing Co., Amsterdam, 1984.  Google Scholar

[21]

S. Zhang, Asymptotic behavior for strong solutions of the Navier-Stokes equations with external forces, Nonlinear Analysis, Ser. A: Theory Methods, 45 (2001), 435-446.  doi: 10.1016/S0362-546X(99)00402-2.  Google Scholar

[22]

C. Zhao and K. Li, On the existence, uniqueness and $L^r$-exponential stability for stationary solutions to the MHD equations in three-dimensional domains, ANZIAM J., 46 (2004), 95-109.  doi: 10.1017/S1446181100013705.  Google Scholar

show all references

References:
[1]

C. Amrouche and V. Girault, On the existence and regularity of the solutions of Stokes problem in arbitrary dimension, Proc. Japan Acad. Ser. A Math. Sci., 67 (1991), 171-175.  doi: 10.3792/pjaa.67.171.  Google Scholar

[2]

H. Beirão da Veiga, Existence and asymptotic behavior for strong solutions of the Navier-Stokes equations in the whole space, Indiana Univ. Math. J., 36 (1987), 149-166.  doi: 10.1512/iumj.1987.36.36008.  Google Scholar

[3]

H. Beirão da Veiga and P. Secchi, $L^p$-stability for the strong solutions of the Navier-Stokes equations n the whole space, Arch. Rational Mech. Anal., 98 (1987), 65-69.  doi: 10.1007/BF00279962.  Google Scholar

[4]

J. L. Boldrini and M. Rojas-Medar, On a system of evolution equations of magnetohydrodynamic type, Mat. Contemp., 8 (1995), 1-19.   Google Scholar

[5]

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

[6]

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

[7]

P. D. Damázio and M. A. Rojas-Medar, On some questions of the weak solutions of evolution equations for magnetohydrodynamic type, Proyecciones, 16 (1997), 83-97.  doi: 10.22199/S07160917.1997.0002.00001.  Google Scholar

[8]

Y. He and K. Li, Asymptotic behavior and time discretization analysis for the nonstationary Navier-Stokes problem, Num. Math., 98 (2004), 647-673.  doi: 10.1007/s00211-004-0532-y.  Google Scholar

[9]

J. G. Heywood, An error estimate uniform in time for spectral Galerkin approximations of the Navier-Stokes problem, Pacific J. Math., 98 (1982), 333-345.  doi: 10.2140/pjm.1982.98.333.  Google Scholar

[10]

J. G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem. Ⅰ. Regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal., 19 (1982), 275-311.  doi: 10.1137/0719018.  Google Scholar

[11]

J. G. Heywood and R. Rannacher, Finite element approximations of the nonstationary Navier-Stokes equations. Ⅱ. Stability of solutions and error estimates uniform in time, SIAM J. Numer. Anal., 23 (1986), 750-77.  doi: 10.1137/0723049.  Google Scholar

[12]

I. Kondrashuk, E. Notte-Cuello, M. Poblete-Cantellano and M. A. Rojas-Medar, Periodic solution for the magnetohydrodynamic equations with inhomogeneous boundary condition, Axioms, 8 (2019), 44, http://dx.doi.org/10.3390/axioms8020044. Google Scholar

[13]

G. Lassner, Über ein Rand-Anfangswertproblem der Magnetohydrodynamik, Arch. Rational Mech. Anal., 25 (1967), 388-405.  doi: 10.1007/BF00291938.  Google Scholar

[14]

E. A. Notte-Cuello and M. A. Rojas-Medar, On a system of evolution equations of magnetohydrodynamic type: An iterational approach, Proyecciones, 17 (1998), 133-165.  doi: 10.22199/S07160917.1998.0002.00001.  Google Scholar

[15]

S. B. Pikelner, Grundlagen der Kosmischen Elektrodynamik [Russ.], Moskau, 1966. Google Scholar

[16]

C. S. Qu and P. Wang, $L^p$ exponential stability for the equilibrium solutions of the Navier-Stokes equations, J. Math. Anal. Appl., 190 (1995), 419-427.  doi: 10.1006/jmaa.1995.1085.  Google Scholar

[17]

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

[18]

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

[19]

A. Schlüter, Dynamik des Plasma, Iund II, Z. Naturforsch., 5a (1950), 72–78; 6a (1951), 73–79. Google Scholar

[20]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Third edition, North-Holland Publishing Co., Amsterdam, 1984.  Google Scholar

[21]

S. Zhang, Asymptotic behavior for strong solutions of the Navier-Stokes equations with external forces, Nonlinear Analysis, Ser. A: Theory Methods, 45 (2001), 435-446.  doi: 10.1016/S0362-546X(99)00402-2.  Google Scholar

[22]

C. Zhao and K. Li, On the existence, uniqueness and $L^r$-exponential stability for stationary solutions to the MHD equations in three-dimensional domains, ANZIAM J., 46 (2004), 95-109.  doi: 10.1017/S1446181100013705.  Google Scholar

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