Large time behavior for the incompressible magnetohydrodynamic equations in half-spaces

Abstract

Communicated by Z. XinThe weighted L r -asymptotic behavior of the strong solution and its first-order spacial derivatives to the incompressible magnetohydrodynamic (MHD) equations is established in a half-space. Further, the L 1 -decay rates of the second-order spatial derivatives of the strong solution are derived by using the Stokes solution formula and employing a decomposition for the nonlinear terms in MHD equations.

The exponential decay of solutions to the nonstationary magneto‐hydrodynamic equations

“…Schonbek et al 18 studied the derivation of upper and lower bounds on the decay of the total energy and the magnetic energy of a viscous incompressible electrically conducting resistive fluid in $$$ {\mathbb{R}}^n $$$; their analysis is based on the comparison of the decay rates of the weak solutions of the MHD equations that satisfy the energy inequality with those solutions of a suitable heat system with the same initial data. Although there are many asymptotic results for solutions of (), for example, see other works, 19–22 as far as we know, there are few exponential time decays (including the first spatial derivatives and pointwise estimates) on MHD flows in $$$ {L}^r\left({\mathbb{R}}^2\right),1<r\le \infty $$$.…”

The Green tensor of the nonstationary Stokes system in the half space

“…where f is the coupling with the magnetic field b : R 3 + × (0, ∞) → R 3 in the magnetohydrodynamic equations in the half space R 3 + with boundary conditions b 3 = 0 and (∇ × b) × e 3 = 0 (see [18,15,19,30]), and g is the coupling with the orientation field d : R 3 + × (0, ∞) → S 2 in the nematic liquid crystal flows with boundary conditions ∂ 3 d| Σ = 0 and lim |x|→∞ d = e 3 (see [16]). Pointwise estimates are also useful for the study of the local and asymptotic behavior of the solutions of (NS), see e.g.…”