Abstract
We show, using the spectral Galerkin method together with compactness arguments, the existence and uniqueness of the periodic strong solutions for the magnetohydrodynamic-type equations with inhomogeneous boundary conditions. Furthermore, we study the asymptotic stability for the time periodic solution for this system. In particular, when the magnetic field h ( x , t ) is zero, we obtain the existence, uniqueness, and asymptotic behavior of the strong solutions to the Navier–Stokes equations with inhomogeneous boundary conditions.
Highlights
For many decades, the awareness that the motion of an incompressible electrical conducting fluid can be modeled by the magnetohydrodynamic (MHD) equations, which correspond to the Navier–Stokes (NS) equations coupled to the Maxwell equations, has been consolidated
The awareness that the motion of an incompressible electrical conducting fluid can be modeled by the magnetohydrodynamic (MHD) equations, which correspond to the Navier–Stokes (NS) equations coupled to the Maxwell equations, has been consolidated. This system of equations plays an important role in various applications, for example in phenomena related to the plasma behavior [1], heat conductivity and nematic liquid crystal flows [2,3,4,5], and stochastic dynamics [6]
In the case when the MHD equations have periodic boundary conditions, these equations play an important role in MHD generators [7]
Summary
The awareness that the motion of an incompressible electrical conducting fluid can be modeled by the magnetohydrodynamic (MHD) equations, which correspond to the Navier–Stokes (NS) equations coupled to the Maxwell equations, has been consolidated. U and h are the unknown velocity and magnetic field, respectively; p∗ is an unknown hydrostatic pressure; w is an unknown function related to the heavy ions (in such a way that the density of the electric current, j0 , generated by this motion satisfies the relation rotj0 = −σ ∇w); ρ is the density of the mass of the fluid (assumed to be a positive constant); μ > 0 is a constant magnetic permeability of the medium; σ > 0 is a constant electric conductivity; η > 0 is a constant viscosity of the fluid; f is a given external force field. Following the methodology used by Kato, Notte-Cuello and Rojas-Medar [11] studied the existence and uniqueness of periodic strong solutions with homogeneous boundary conditions for the MHD-type equations. We followed the method used in [23] to perform a study of the asymptotic stability for our system
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