Lower Bounds on Blow-up of Solutions for Magneto-Micropolar Fluid Systems in Homogeneous Sobolev Spaces

摘要

We obtain lower bounds on blow-up of solutions for the 3D magneto-micropolar equations. More precisely, we establish some estimates for the solution (u, w, b)(t) in its maximal interval [0, T*) provided that T* < infinity, which show for delta is an element of (0, 1) that vertical bar vertical bar(u, w, b)(t)vertical bar vertical bar ((H) over dots) is at least of the order (T* - t)(-(delta s)/(1 + 2 delta)) for s >= 1/2 + delta. In particular, by choosing a suitable d, one concludes that vertical bar vertical bar(u, w, b)(t)vertical bar vertical bar ((H) over dots) is at least of the order (T* - t)(-s/4), and (T* - t)(1/4-s/2) for s >= 1, and 1/2 < s < 3/2, respectively. We also show that (T* - t)(-s/3) is a lower rate for vertical bar vertical bar(u, w, b)(t)vertical bar vertical bar ((H) over dots) if s > 3/2.