Geometric structures naturally appear in fluid motions. One of the best known examples is Saturn's Hexagon, the huge cloud pattern at the level of Saturn's north pole, remarkable both for the regularity of its shape and its stability during the past decades. In this paper we will address the spontaneous formation of hexagonal structures in planar viscous flows, in the classical setting of Leray's solutions of the Navier–Stokes equations. Our analysis also makes evidence of the isotropic character of the energy density of the fluid for sufficiently localized 2D flows in the far field: it implies, in particular, that fluid particles of such flows are nowhere at rest at large distances.[-]

We study the global existence of the parabolic-parabolic Keller–Segel system in $\mathbb{R}^{d}$. We prove that initial data of arbitrary size give rise to global solutions provided the diffusion parameter $\tau$ is large enough in the equation for the chemoattractant. This fact was observed before in the two-dimensional case by Biler, Guerra and Karch (2015) and Corrias, Escobedo and Matos (2014). Our analysis improves earlier results and extends them to any dimension d ≥ 3. Our size conditions on the initial data for the global existence of solutions seem to be optimal, up to a logarithmic factor in $\tau$ , when $\tau\gg 1$: we illustrate this fact by introducing two toy models, both consisting of systems of two parabolic equations, obtained after a slight modification of the nonlinearity of the usual Keller–Segel system. For these toy models, we establish in a companion paper finite time blowup for a class of large solutions.[-]