On rigidity of the steady Ericksen-Leslie system

We study solutions, with scaling-invariant bounds, to the steady simplified Ericksen-Leslie system in R n ∖ { 0 } R^n \{0\} . When n = 2 n=2 , we construct and classify a class of self-similar solutions. When n ≥ 3 n≥ 3 , we establish the rigidity asserting that if ( u , d ) (u,d) satisfies a scaling-invariant bound with a small constant, then u ≡ 0 u≡ 0 and d = d= constant for n ≥ 4 n≥ 4 or u u is a Landau solution and d = d= constant for n = 3 n=3 . Such a smallness condition can be weaken when n = 4 n=4 or the solutions are self-similar.