文摘
We obtain lower bounds on blow-up of solutions for the 3D magneto-micropolar equations. More precisely, we establish some estimates for the solution \((\mathbf{u},\mathbf{w},\mathbf{b}) (t)\) in its maximal interval \([0,T^{*})\) provided that \(T^{*}<\infty\), which show for \(\delta\in(0,1)\) that \(\|(\mathbf{u},\mathbf{w},\mathbf{b})(t)\|_{\dot{H}^{s}}\) is at least of the order \((T^{*}-t)^{-(\delta s)/(1+2\delta)}\) for \(s\geq1/2+\delta\). In particular, by choosing a suitable \(\delta\), one concludes that \(\|(\mathbf{u},\mathbf{w},\mathbf{b})(t)\|_{\dot{H}^{s}}\) is at least of the order \((T^{*}-t)^{-s/4}\), and \((T^{*}-t)^{1/4-s/2}\) for \(s\geq1\), and \(1/2< s<3/2\), respectively. We also show that \((T^{*}-t)^{-s/3}\) is a lower rate for \(\|(\mathbf{u},\mathbf{w},\mathbf{b})(t)\|_{\dot{H}^{s}}\) if \(s>3/2\).