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 Schrödinger equation (4NLS) (i∂t+εΔ+Δ2)u=Pm((∂xαu)|α|≤2,(∂xαu¯)|α|≤2),m≥3on Rd (d≥2) with random initial data. Here, Pm 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 Hs(Rd) with some range of s<sc, where sc is the scaling critical regularity, i.e., sc≔d∕2−2∕(m−1). Our results contain the cases s∈(sc−1∕2,sc] when d≥3 and m≥5. Similar supercritical well-posedness results also hold for d=2,m≥4 and d≥3,3≤m<5.