Georgia Institute of Technology College of Sciences

We first discuss blow-up rates and the blow-up profiles of possible asymptotically self-similar singularities of the 3D Euler equations, where the sense of convergence and self-similarity are considered in various sense. We extend much further, in particular, the previous nonexistence results of self-similar/asymptotically self-similar singularities. In the second part of the talk we discuss some observations on the Euler equations with symmetries, which shows that the point-wise behavior of the pressure along the flows is closely related to the blow-up of of solutions.