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In \cite{elgindi2022regular}, a family of singular steady states near theBahouri-Chemin patch was introduced. In this paper, we obtain the optimalregularity and convergence of the singular steady states contruced in\cite{elgindi2022regular} to the Bahouri-Chemin patch. We first derive aboundary Harnack principle, and then obtain the optimal convergence resultsusing the singular integral representation based on Green's function.

Optimal H{\"o}lder convergence of a class of singular steady states to the Bahouri-Chemin patch