The Algebraic Decay Behavior of Weak Solutions to the Magnetohydrodynamic Equations in Unbounded Domains

This paper investigates the long-term asymptotic behavior of solutions to the initial-boundary value problem for the three-dimensional incompressible viscous magnetohydrodynamic (MHD) equations in general unbounded domains. Addressing the difficulty that traditional analytical methods (such as Fourier separation techniques and semigroup estimates for the Stokes operator) fail in unbounded domains, we introduce the operator regularization technique to construct a sequence of approximate solutions. By combining spectral analysis skills and the theory of analytic semigroups, a unified estimation method applicable to the nonlinear terms in the system is proposed. Through energy estimates and the theory of weak convergence, the existence of global weak solutions is proven, and the algebraic decay rate of the solutions is further derived. The results show that the decay behavior of the weak solutions is mainly dominated by the corresponding linear part (i.e., the semigroup solution of the Stokes equations). The estimation method established in this paper is applicable to general smooth unbounded domains, which generalizes the existing results that were only applicable to special domains.