The 3D nematic liquid crystal equations with blow-up criteria in terms of pressure

Abstract

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 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 2β+1p+1q+1r=2 and 2≤p,q,r≤∞,1−(1p+1q+1r)≥0,and ∫0T∗‖‖‖∇P(⋅,t)‖Lx1p‖Lx2q‖Lx3rβ+‖∇d(⋅,t)‖L48dt=∞, with 2β+1p+1q+1r=3 and 1≤p,q,r≤∞,1−(12p+12q+12r)≥0,and ∫0T∗‖‖∂3P(⋅,t)‖Lx3γ‖Lx1x2αβ+‖∇d(⋅,t)‖L48dt=∞, with 2β+1γ+2α=k∈[2,3) and 3k≤γ≤α<1k−2.