In the presentation we will discuss our research program concerning the study of extreme vortex events in viscous incompressible flows. These vortex states arise as the flows saturating certain fundamental mathematical estimates, such as the bounds on the maximum enstrophy growth in 3D (Lu \& Doering, 2008). They are therefore intimately related to the question of 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 which for given initial enstrophy and time window produces the largest possible growth of enstrophy at the end of the time window. This allows us then to demonstrate that when such extreme initial data is used, the maximum enstrophy attained during the flow evolution is bounded and exhibits a power-law dependence on the initial enstrophy. 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.