2022/12/22 by Thomas Y. Hou, Hou, Thomas Y., Shumao Zhang +1 · 1 citation
Mathematics · Engineering · #Navier-Stokes equation solutions #Fluid Dynamics and Turbulent Flows #Computational Fluid Dynamics and Aerodynamics
paper · pdf · doi:10.48550/arxiv.2212.11924
In Part II of this sequence to our previous paper for the 3-dimensional Euler equations \citezhang2022potential, we investigate potential singularity of the n-diemnsional axisymmetric Euler equations with Cα 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 \citezhang2022potential, the scaling behavior and regularity properties of our initial data are quite different from those of the initial data considered by Elgindi in \citeelgindi2021finite.