The magneto-micropolar equations with periodic boundary conditions: Solution properties at potential blow-up times

We examine some estimates involving strong solutions, at potential blow-up times, for the magneto-micropolar equations with periodic boundary conditions. More precisely, if T⁎<∞T⁎<∞ is the first blow-up instant of a solution (u,w,b)(t)(u,w,b)(t) defined in the interval [0,T⁎)[0,T⁎) and s≥1/2+δs≥1/2+δ, with δ∈(0,1)δ∈(0,1), then it holds the inequality ‖(u,w,b)(t)‖H˙s(T3)≥C(T⁎−t)−(δs)/(1+2δ). In addition, we prove that ‖(uˆ,wˆ,bˆ)(t)‖l1(Z3)≥C(T⁎−t)−1/2 in order to obtain as a result the estimate ‖(u,w,b)(t)‖H˙s(T3)≥C(T⁎−t)−s/3 for s>3/2s>3/2.