On the blow-up criterion of periodic solutions for micropolar equations in homogeneous Sobolev spaces

We prove lower estimates for space periodic solutions (u,w)(t) of the micropolar equations in their maximal interval [0,T∗)[0,T∗) provided that T∗<∞T∗<∞. For example, we show for 0<δ<10<δ<1 that ‖(u,w)(t)‖Ḣs(T3) is at least of the order (T∗−t)−(δs)/(1+2δ)(T∗−t)−(δs)/(1+2δ) for s≥1/2+δs≥1/2+δ. Moreover, we prove the inequality ‖(û,ŵ)(t)‖l1(Z3)≥C(T∗−t)−1/2, which yields the blow-up rate (T∗−t)−s/3(T∗−t)−s/3 for ‖(u,w)(t)‖Ḣs(T3) for s>3/2s>3/2.