SYSTEMATIC SEARCH FOR EXTREME BEHAVIOR IN 3D NAVIER-STOKES FLOWS BASED ON THE LADYZHENSKAYA-PRODI-SERRIN CONDITIONS

Abstract

One of the most famous problems in theoretical fluid mechanics concerns the question whether the 3D Navier-Stokes equations always produce smooth solutions. More specifically, it is not known whether all sufficiently smooth initial data lead to the existence of regular solutions for all time or if singularities may form in finite time. One approach to study this problem is based on the so called “conditional regularity results”; if such statements are shown to hold, this would imply that the corresponding flows are regular and satisfy the Navier-Stokes system in the classical sense. Arguably, the best known result of this type is the enstrophy condition. It has inspired recent works attempting to search for initial conditions that maximize the enstrophy over a certain time window to identify the worst-case scenarios that could result in singularity formation in finite time. Motivated by these studies, in this investigation we conduct a systematic computational search for potential singularities in three-dimensional Navier-Stokes flows using the Ladyzhenskaya-Prodi-Serrin conditions. They assert that for a solution u(t) of the Navier-Stokes system to be regular on an interval [0,T], the integral int_{0}^{T}∥u(t)∥^p_{L^q} dt, where 2/p + 3/q = 1, q > 3, must be bounded. Our main contribution is to conduct a systematic search for flows that might become singular and violate this condition, by solving a family of variational PDE optimization problems on a periodic domain Ω. In these problems, we identify initial conditions u0 that locally maximizes the integral int_{0}^{T} ∥u(t)∥^p_{L^q} dt for a range of different values of q and p, different time windows T and several sizes ∥u_0∥_{L^q} of the initial data. Such local maximizers are found numerically with a state-of-the-art adjoint-based Riemannian gradient method. Four formulations are considered with optimal solutions sought in Hilbert-Sobolev and Lebesgue function spaces. This is the first time the worst-case behavior of Navier-Stokes flows is thoroughly investigated through the lens of the Ladyzhenskaya-Prodi-Serrin conditions. In order to study how a hypothetical singularity could develop, we analyze the rate of growth of ∥u(t)∥_{L^q} and of the enstrophy in the extreme flows obtained by solving theoptimization problems. We derive and analyze explicit bounds on the rate of growth for the L^q norm of Navier-Stokes flows for which singularity formation is impossible. By combining them with existing bounds on the rate of growth of ∥u(t)∥_{L^q}, we identify specific regimes such that if the corresponding rate of growth is sustained, this would lead to singularity formation in finite time in Navier-Stokes flows. Although we did not find any evidence for blow-up, these relations allow us to quantify how “close” the extreme flows arising in such worst-case scenarios come to producing a singularity.