Navier-Stokes equations under slip boundary conditions:Lower bounds to the minimal amplitude of possible time-discontinuities of solutions with two components in L^(∞)(L^(3))

The main purpose of this paper is to extend the result obtained by Beirao da Veiga(2000)from the whole-space case to slip boundary cases.Denote by a two components of the velocity u.To fix ideas setū=(u_(1),u_(2),0)(the half-space)orū=?_(1)ê_(1)+?_(2)ê_(2)(the general boundary case(see(7.1))).We show that there exists a constant K,which enjoys very simple and significant expressions such that if at some timeτ∈(0,T)one has lim sup_(t→^(τ)-0)‖ū(t)‖_(L^(3)(Ω))^(3)<‖ū(τ)‖_(L^(3)(Ω))^(3)+K,then u is continuous atτwith values in L^(3)(Ω).Roughly speaking,the above norm-discontinuity of merely two components of the velocity cannot occur for steps'amplitudes smaller than K.In particular,if the above condition holds at eachτ∈(0,T),the solution is smooth in(0,T)×Ω.Note that here there is no limitation on the width of the norms‖ū(t)‖_(L^(3)(Ω))^(3)·So K is independent of these quantities.Many other related results are discussed and compared among them.