Abstract We consider the initial value problem (IVP) associated to a quadratic Schrodinger system { i ∂ t v ± Δ g v − v = ϵ 1 u v ¯ , t ∈ R , x ∈ M , i σ ∂ t u ± Δ g u − α u = ϵ 2 2 v 2 , σ > 0 , α ∈ R , ϵ i ∈ C ( i = 1 , 2 ) , ( v ( 0 ) , u ( 0 ) ) = ( v 0 , u 0 ) , posed on a d-dimensional compact Zoll manifold M. Considering σ = θ β with θ , β ∈ { n 2 : n ∈ Z } we derive a bilinear Strichartz type estimate and use it to prove the local well-posedness results for given data ( v 0 , u 0 ) ∈ H s ( M ) × H s ( M ) whenever s > 1 4 when d = 2 and s > d − 2 2 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 .

中文翻译：

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Zoll 流形上二次薛定谔系统的局部和全局适定性

摘要 我们考虑与二次薛定谔系统相关的初值问题 (IVP) { i ∂ tv ± Δ gv − v = ϵ 1 uv ¯ , t ∈ R , x ∈ M , i σ ∂ tu ± Δ gu − α u = ϵ 2 2 v 2 , σ > 0 , α ∈ R , ϵ i ∈ C ( i = 1 , 2 ) , ( v ( 0 ) , u ( 0 ) ) = ( v 0 , u 0 ) , 放在 a d 上维紧致 Zoll 流形 M. 考虑 σ = θ β 和 θ , β ∈ { n 2 : n ∈ Z } 我们推导出双线性 Strichartz 类型估计并用它来证明给定数据 ( v 0 , u 0 ) ∈ H s ( M ) × H s ( M ) 当 d = 2 时 s > 1 4 并且当 d ≥ 3 时 s > d − 2 2 。此外，在维度 2 和维度 3 中，我们使用 Gagliardo-Nirenberg 型不等式和守恒定律来证明当 s ≥ 1 时，局部解可以在时间上全局扩展。