• Previous Article
    A conservative semi-Lagrangian finite difference WENO scheme based on exponential integrator for one-dimensional scalar nonlinear hyperbolic equations
  • ERA Home
  • This Issue
  • Next Article
    Regular random attractors for non-autonomous stochastic reaction-diffusion equations on thin domains
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, 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

[1]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

[2]

Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016

[3]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[4]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[5]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118

[6]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[7]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020268

[8]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074

[9]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

[10]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364

[11]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454

[12]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[13]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276

[14]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119

[15]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383

[16]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[17]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[18]

Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020274

[19]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[20]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

 Impact Factor: 0.263

Article outline

[Back to Top]