Random data Cauchy problem for the fourth order Schrödinger equation with the second order derivative nonlinearities. (January 2020) - Chen, Mingjuan; Zhang, Shuai
Abstract: We consider the Cauchy problem of the fourth order nonlinear Schrödinger equation (4NLS) ( i ∂ t + ε 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 .