Abstract: 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]
