Local and global well-posedness for a quadratic Schrödinger system on Zoll manifolds

Abstract

We consider the initial value problem (IVP) associated to a quadratic Schrödinger system{i∂tv±Δgv−v=ϵ1uv¯,t∈R,x∈M,iσ∂tu±Δgu−αu=ϵ22v2,σ>0,α∈R,ϵi∈C(i=1,2),(v(0),u(0))=(v0,u0), posed on a d-dimensional compact Zoll manifold M. Considering σ=θβ with θ,β∈{n2:n∈Z} we derive a bilinear Strichartz type estimate and use it to prove the local well-posedness results for given data (v0,u0)∈Hs(M)×Hs(M) whenever s>14 when d=2 and s>d−22 when d≥3. Moreover, in dimensions 2 and 3, we use a Gagliardo-Nirenberg type inequality and conservation laws to prove that the local solution can be extended globally in time whenever s≥1.