In the presentation we will discuss our research program concerning the search for the most singular behaviors possible in viscous incompressible flows. These events are characterized by extremal growth of

various quantities, such as the enstrophy, which control the regularity of the solution. They are therefore intimately related to the question of possible singularity formation in the 3D Navier-Stokes system, known as the hydrodynamic blow-up problem. We demonstrate how new insights concerning such questions can be obtained by formulating them as variational PDE optimization problems which can be solved computationally using suitable discrete gradient flows. More specifically, such an optimization formulation allows one to identify

"extreme" initial data which, subject to certain constraints, leads to the most singular flow evolution. In offering a systematic approach to finding flow solutions which may saturate known estimates, the proposed paradigm provides a bridge between mathematical analysis and scientific computation. In particular, it makes it possible to determine whether or not certain mathematical estimates are "sharp", in the sense that they can be realized by actual vector fields, or if these estimates may still be improved. In the presentation we will review a number of results concerning 1D and 2D flows characterized by the maximum possible growth of different Sobolev norms of the solutions. As regards 3D flows, we focus on the enstrophy which is a well-known indicator of the regularity of the solution. We find a family of initial data with fixed enstrophy which leads to the largest possible growth of this quantity at some prescribed final time. Since even with such worst-case initial data the enstrophy remains finite, this indicates that the 3D Navier-Stokes system reveals no tendency for singularity formation in finite time.

[joint work with Dongfang Yun and Bartosz Protas]