Bartosz Protas: Systematic Search For Singularities in 3D Navier-Stokes Flows

Abstract

This investigation concerns a systematic computational search for potentially singular behavior in 3D Navier-Stokes flows. Enstrophy \(\mathcal{E}(t)\) serves as a convenient indicator of the regularity of solutions to the Navier Stokes equation --- as long as this quantity remains finite, the solutions are guaranteed to be smooth and satisfy the equations in the classical (pointwise) sense. Another well-known conditional regularity result are the Ladyzhenskaya-Prodi-Serrin conditions asserting that the quantity \(\mathcal{L}_{q,p}:=\int_0^T \| \mathbf{u}(t) \|_{L^q(\Omega)}^p \, dt\), where \(2/p+3/q \le 1\), \(q>3\), must remain bounded if the solution \(\mathbf{u}(t)\) is smooth on the interval \([0,T]\). However, there are no finite a priori bounds available for these quantities and hence the regularity problem for the 3D Navier-Stokes system remains open. To quantify the maximum possible growth of \(\mathcal{E}(t)\) and \(\mathcal{L}_{q,p}\), we consider families of PDE optimization problems in which initial conditions are sought subject to certain constraints so that these quantities in the resulting Navier-Stokes flows are maximized. These problems are solved computationally using a large-scale adjoint-based gradient approach. By solving these problems for a broad range of parameter values we demonstrate that the maximum growth of \(\mathcal{E}(t)\) and \(\mathcal{L}_{q,p}\) appears finite and follows well-defined power-law relations in terms of the size of the initial data. Thus, in the worst-case scenarios the two quantities remain bounded for all times and there is no evidence for singularity formation in finite time. We will also review earlier results where a similar approach allowed us to probe the sharpness of a priori bounds on the growth of enstrophy and palinstrophy in 1D Burgers and 2D Navier-Stokes flows.