We survey our recent works for a class of nonlinear evolution equations including Navier-Stokes, NLKG, nonlocal NLS and semilinear heat equation with initial data in super-critical spaces \(E^{\sigma ,s}\) for which the norm is defined as \( \Vert u_0\Vert _{E^{\sigma ,s}} = \Vert \langle \xi \rangle ^\sigma 2^{s|\xi |} \widehat{u}_0\Vert _{L^2(\mathbb {R}^d)}, \ s<0. \) If \(s<0\) , then \(H^\kappa \subset E^{\sigma ,s}\) for any \(\kappa ,\sigma \in \mathbb {R}\) . The global existence and uniqueness of solutions are obtained for the initial data \(u_0 \in E^{\sigma ,s}\) and \(\widehat{u}_0\) is supported in the first octant, where \(s<0\) , \(\sigma \) is the (sub-)critical index for the nonlinear evolution equations.