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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.
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.
This paper focuses on the global existence and time-decay rates of the strong solution for the Boussinesq system with full viscosity in R n for n ≥ 3 . Under the initial assumption of θ 0 , u 0 ∈ L n / 3 × L n with a small norm, and n > 3 or n = 3 and θ 0 ∈ L r 0 for some r 0 > 1 , global existence and uniqueness of the strong solution θ , u for the Boussinesq system is established. This solution is proven to obey the following estimates: θ t r ≤ C t − 3 − n / p / 2 for n / 3 ≤ p < ∞ , u t p ≤ C t − 1 − n / q / 2 for n ≤ q ≤ ∞ , ∇ θ t p ≤ C t − 3 − n / p / 2 − 1 / 2 and ∇ 2 θ t p = O t − n 1 / r − 1 / p / 2 − 1 as t ⟶ ∞ for r ≤ p < n / 2 , and ∇ u t q ≤ C t − 1 − n / q / 2 − 1 / 2 and ∇ 2 u t q = O t − n 1 / r − 1 / q / 2 − 1 as t ⟶ ∞ for n ≤ q < 2 n , where r = n / 3 if n > 3 and 1 < r < min r 0 , n / 2 if n = 3 .
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