In this paper, we prove several convergence theorems for the mean curvature flow of n-dimensional closed submanifolds in the unit sphere \(\mathbb{S}^{n+k}\) under integral curvature pinching conditions. In particular, we prove that if the Ln-norm of the second fundamental form of the initial submanifold is small enough, then the mean curvature flow either shrinks to a round point in finite time, or converges to a totally geodesic submanifold as the time tends to infinity. As a consequence of the smooth convergence theorems, we obtain several differentiable sphere theorems for certain submanifolds in \(\mathbb{S}^{n+k}\) .
