Well‐Posedness for Fourth‐Order Cubic Schrödinger Equation on Spheres

ABSTRACT In this article, we investigate the local and global well‐posedness for the cubic fourth‐order defocusing nonlinear Schrödinger equation (4NLS) posed on ‐dimensional spheres with . First, we establish the bilinear estimates for the fourth‐order Schrödinger operator in Bourgain spaces based on the sharp bilinear eigenfunction estimate on spheres . As an application, we prove the (4NLS) is locally well‐posed in with , where Furthermore, combining bilinear estimates with the I‐method adapting to non‐flat settings and the multilinear eigenfunction estimate on spheres, we prove that the (4NLS) is globally well‐posed in with , where Since the spectrum corresponding to the is discrete, we cannot obtain improved bilinear estimates for as we would in Euclidean space. Therefore, we find that does not affect the local well‐posedness results. In proving the global well‐posedness results, we need bilinear estimates in a short time interval. However, if we employ the manifold rescaling techniques in our cases, the bilinear estimate we obtain will include a large constant corresponding to the rescaling scale, which is unacceptable. Therefore, we will establish a bilinear estimate in a short time interval via the regularity transformation of Bourgain's spaces and obtain the global well‐posedness results.