On the exponential type explosion of Navier–Stokes equations

The classical results on the explosion of the maximal solution of incompressible Navier–Stokes equations are of type c(T∗−t)−σ0c(T∗−t)−σ0 for some σ0>0σ0>0. Inspired by the works Benameur and Selmi (2012)  [15], Chemin (2004)  [16], we use the Sobolev–Gevrey spaces to get better explosion results, precisely if ea|D|1/σu0∈Hs(R3)ea|D|1/σu0∈Hs(R3), then |ea|D|1/σu(t)|Hs|ea|D|1/σu(t)|Hs is at least of the order (T∗−t)−σ1exp(c(T∗−t)−σ2)(T∗−t)−σ1exp(c(T∗−t)−σ2) for some σ1>0σ1>0 and σ2>0σ2>0. Fourier analysis and standard techniques are used.