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In Part I of our sequence of 2 papers, we provide numerical evidence for apotential finite-time self-similar singularity of the 3D axisymmetric Eulerequations with no swirl and with $C^\alpha$ initial vorticity for a large rangeof $\alpha$. We employ an adaptive mesh method using a highly effective mesh toresolve the potential singularity sufficiently close to the potential blow-uptime. Resolution study shows that our numerical method is at least second-orderaccurate. Scaling analysis and the dynamic rescaling formulation are presentedto quantitatively study the scaling properties of the potential singularity. Wedemonstrate that this potential blow-up is stable with respect to theperturbation of initial data. Our study shows that the 3D Euler equations withour initial data develop finite-time blow-up when the H\"older exponent$\alpha$ is smaller than some critical value $\alpha^*$. By properly rescalingthe initial data in the $z$-axis, this upper bound for potential blow-up$\alpha^*$ can asymptotically approach $1/3$. Compared with Elgindi's blow-upresult in a similar setting \cite{elgindi2021finite}, our potential blow-upscenario has a different H\"older continuity property in the initial data andthe scaling properties of the two initial data are also quite different.

Potential Singularity of the Axisymmetric Euler Equations with  $C^\alpha$ Initial Vorticity for A Large Range of $\alpha$. Part I: the  $3$-Dimensional Case