The 3D Nematic Liquid Crystal Equations With Blow-up Criteria In Terms Of Pressure

In this paper,we concern the 3D nematic liquid crystal equations and prove three almost Serrin-type blow-up criteria for the breakdown of local in time smooth solutions in terms of pressure and gradient of the orientation field.More precisely,let[0,T*)be the maximal time of the local smooth solution,then T*<+∞ if and only if∫0T*‖‖‖p(·,t)‖Lx1p‖Lx2q‖Lx3rβ+‖▽d(·,t)‖L48dt=∞,with 2/β+1/p+1/q+1/r= 3 and 1 ≤ p,q,r ≤ ∞,1-(1/2p + 1/2r)>0,and∫0T*‖‖‖▽p(·,t)‖Lx1p‖Lx2q‖Lx3rβ+‖▽d(·,t)‖L48dt=∞,with 2/β+ 1/γ +2/α=κ ∈[2,3)and 3/л ≤ γ ≤α<1/л-2.and∫0T*‖‖(?)3P(·,t)‖Lx3γ‖Lx1x2αβ + ‖▽d(·,t)‖L48dt = ∞,with 2/β+ 1/γ +2/α=κ ∈[2,3)and 3/κ ≤ γ ≤α<1/κ-2.