Random data Cauchy problem for the fourth order Schrödinger equation with the second order derivative nonlinearities

Abstract We consider the Cauchy problem of the fourth order nonlinear Schrodinger equation (4NLS) ( i ∂ t + e Δ + Δ 2 ) u = P m ( ( ∂ x α u ) | α | ≤ 2 , ( ∂ x α u ¯ ) | α | ≤ 2 ) , m ≥ 3 on R d ( d ≥ 2 ) with random initial data. Here, P m is a homogeneous polynomial of degree m containing the second order derivative. We show the almost sure local well-posedness and small data global existence in H s ( R d ) with some range of s s c , where s c is the scaling critical regularity, i.e., s c ≔ d ∕ 2 − 2 ∕ ( m − 1 ) . Our results contain the cases s ∈ ( s c − 1 ∕ 2 , s c ] when d ≥ 3 and m ≥ 5 . Similar supercritical well-posedness results also hold for d = 2 , m ≥ 4 and d ≥ 3 , 3 ≤ m 5 .