$L^2$ decay estimates of weak solutions to 3D fractional MHD equations in exterior domains

TL;DR

This work analyzes the 3D fractional MHD system in an exterior domain with Dirichlet boundary conditions, proving the global existence of weak solutions and establishing algebraic $L^2$ decay rates dictated by the linear fractional dissipations $\mathcal{A}^\alpha$ and $\mathcal{A}^\beta$ for $\alpha,\beta \in (\tfrac{3}{4},1]$. The authors construct approximate solutions via mollification, obtain uniform energy bounds, and pass to the limit through compactness to obtain a global weak solution. The decay analysis hinges on the spectral representation of the exterior Stokes operator and a bootstrap argument that connects nonlinear dynamics to the linear semigroup decay, yielding explicit decay rates depending on $\alpha$ and $\beta$. The results extend classical exterior-domain decay for Navier–Stokes to fractional MHD, providing a framework for understanding long-time behavior in exterior domains with fractional dissipation.

Abstract

Consider three-dimensional fractional MHD equations in an exterior domain with the Dirichlet boundary condition assumed. Asymptotic behaviours of weak solutions to the three-dimensional exterior fractional MHD equations are studied. $L^2$ decay estimates of the weak solutions are obtained.

Theorems & Definitions (7)

• Definition 1.1
• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• Theorem 3.1
• Proposition 3.1
• proof