Analysis and Applied Mathematics Seminar
Bartosz Protas
McMaster University
PROBING FUNDAMENTAL BOUNDS IN FLUID MECHANICS USING VARIATIONAL OPTIMIZATION METHODS
Abstract: Rigorous mathematical analysis of the equations governing the
motion fluids leads to various a priori bounds expressing fundamental
limitations on the forms of extreme behavior possible in fluid flows.
In relation to turbulence, such bounds concern, for example, the
maximum production of enstrophy and the maximum energy or enstrophy
dissipation realizable in Navier-Stokes flows under different
constraints. While by virtue of how they are obtained such a priori
bounds account for all possible flow evolutions, they are often
conservative and hence amenable to improvement. We will present a
framework allowing one to systematically test the sharpness of such
bounds by solving a family of suitably-defined PDE optimization
problems. They are solved computationally using an adjoint-based
Riemannian gradient method. This approach will be illustrated with two
classical problems. First, we consider the question of (the absence of)
the dissipation anomaly in 2D Navier-Stokes flows. After recalling some
rigorous priori estimates describing the vanishing of the enstrophy
dissipation in the inviscid limit, we solve a family of PDE
optimization problems aimed at maximizing this quantity with respect to
the initial data. These results show that the extreme behavior found in
this way saturates an estimate due to Ciampa, Crippa & Spirito (2021),
thereby demonstrating the sharpness of this bound. The second problem
we discuss is motivated by the question about the possibility of
finite-time singularity formation in 3D Navier-Stokes flows. The
mathematical analysis of this problem revolves around conditional
regularity results which provide bounds that must be satisfied by all
smooth (classical) solutions, such that violation of these bounds
signals formation of a singularity. Our optimization-based approach
allows one to systematically search for the most singular behavior
possible in Navier-Stokes flows. However, no evidence for singularity
formation was detected in extreme flows realizing such worst-case
scenarios.
Joint work with D. Kang, P. Matharu, E. Ramirez and T. Yoneda
Monday January 27, 2025 at 4:00 PM in 636 SEO