Potential Singularity of the Axisymmetric Euler Equations with C^α Initial Vorticity for A Large Range of α. Part II: the N-Dimensional Case

Abstract

In Part II of this sequence to our previous paper for the 3-dimensional Euler equations [8], we investigate potential singularity of the n-diemnsional axisymmetric Euler equations with C^α initial vorticity for a large range of α. We use the adaptive mesh method to solve the n-dimensional axisymmetric Euler equations and use the scaling analysis and dynamic rescaling method to examine the potential blow-up and capture its self-similar profile. Our study shows that the n-dimensional axisymmetric Euler equations with our initial data develop finite-time blow-up when the Hölder exponent α < α^∗, and this upper bound α∗ can asymptotically approach 1 − 2/n. Moreover, we introduce a stretching parameter δ along the z-direction. Based on a few assumptions inspired by our numerical experiments, we obtain α^∗ = 1 − 2/n by studying the limiting case of δ→0. For the general case, we propose a relatively simple one-dimensional model and numerically verify its approximation to the n-dimensional Euler equations. This one-dimensional model sheds useful light to our understanding of the blowup mechanism for the n-dimensional Euler equations. As shown in [8], the scaling behavior and regularity properties of our initial data are quite different from those of the initial data considered by Elgindi in [6].