Nonlinear Schrödinger equations on compact Zoll manifolds with odd growth

Local well‐posedness of critical nonlinear Schrödinger equation on Zoll manifolds of odd‐growth

“…As in , we can obtain the Lipschitz continuity of the flow map associated to . In contrast to the results in , we do not have the uniform local well‐posedness of the Cauchy problem in the critical case just depending on the Sobolev norm of the H s ( M ). Because in the critical case, the lifespan of a solution depends not only on the Sobolev norm but also on the profile of its initial data (Section 5).…”

“…From , we have the threshold for well‐posedness of cubic NLS on , that is, from scaling), which means that the uniform well‐posedness holds if and fails if . And this phenomena does not exist in the case of torus and sphere in higher dimensions , which means that the threshold depends on the geometric properties of the manifolds. In and , by developing a new set of multilinear estimates for spherical harmonics, the authors obtained the global existence of strong H 1 solutions of sub‐quintic defocusing NLS on .…”

“…When , the theory of the well‐posedness of with general power‐type nonlinearities is highly developed, which mainly depends on the Strichartz estimates of the operator e i t Δ in . We refer to for the pertinent references.…”

“…Second, we will recall the multilinear spectral projector estimates from , Formula (3.14)] in Lemma and extend the spectral localization of products of eigenfunctions Lemma . Lemma is a replacement of the fine orthogonality in the cases of or .…”

“…For the second case, when d = 2,4 < p < 6 the Strichartz estimates is not as good as other cases. Besides, when d = 2, the spectral projector estimates in Lemma 3.3 from , Formula (3.14)] are valid only when p j ≥4, which also results in the failure of Theorem 1.1.…”