Probing fundamental bounds in hydrodynamics using variational optimisation methods
Seminar Room 1, Newton Institute
This work demonstrates how the modern methods of PDE-constrained optimization can be used to assess sharpness of a class of fundamental functional estimates in fluid mechanics. These estimates concern bounds on the instantaneous rate of growth and finite-time growth of quadratic quantities such as the enstrophy and palinstrophy in viscous incompressible flows. Sharpness of such estimates is inherently related to the problem of singularity formation in the 3D Navier-Stokes system. In our presentation we will first review earlier results of Lu & Doering (2008) and Ayala & Protas (2011) concerning the maximum growth of enstrophy the 1D Burgers equation. We will then present several new results regarding the maximum growth of palinstrophy in 2D flows and will discuss some questions concerning sharpness of the corresponding analytical estimates. While it is well known that solutions of 1D Burgers equations and 2D Navier-Stokes equation evolving from smooth initial data remain smooth for all times, the question whether the best available estimates for the maximum growth of enstrophy and palinstrophy are sharp is both interesting and relevant. One reason is that such estimates are derived using similar mathematical techniques as in the 3D case where blow-up cannot be ruled out. We will show how new insights regarding these problems can be obtained by formulating them as variational PDE optimization problems which can be solved computationally using suitable adjoint--based gradient descent (or ascent) methods. In particular, we will discuss certain topological features of the families of vorticity fields maximizing the instantaneous rate of growth of palinstrophy in 2D. In offering a systematic approach to finding flow solutions closest to saturating a given analytical bound, the proposed approach provides a bridge between theory and computation.