Potential Singularity of the Axisymmetric Euler Equations with C^α Initial Vorticity for A Large Range of α. Part I: the 3-Dimensional Case
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In Part I of our sequence of 2 papers, we provide numerical evidence for a potential finite-time self-similar singularity of the 3D axisymmetric Euler equations with no swirl and with C^α initial vorticity for a large range of α. We employ an adaptive mesh method using a highly effective mesh to resolve the potential singularity sufficiently close to the potential blow-up time. Resolution study shows that our numerical method is at least second-order accurate. Scaling analysis and the dynamic rescaling formulation are presented to quantitatively study the scaling properties of the potential singularity. We demonstrate that this potential blow-up is stable with respect to the perturbation of initial data. Our study shows that the 3D Euler equations with our initial data develop finite-time blow-up when the Hölder exponent α is smaller than some critical value α^*. By properly rescaling the initial data in the z-axis, this upper bound for potential blow-up α^* can asymptotically approach 1/3. Compared with Elgindi's blow-up result in a similar setting <cit.>, our potential blow-up scenario has a different Hölder continuity property in the initial data and the scaling properties of the two initial data are also quite different.
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