A revisit of patch solutions for the 2D Loglog-Euler type equation

In this paper, we revisit the patch solutions for a class of inviscid whole-space active scalar equations that interpolate between the 2D Euler equation and the $\alpha$-SQG equation. Compared with the 2D Euler equation in vorticity form, there is an additional Fourier multiplier $m(\Lambda)$ ($\Lambda = (-\Delta)^{1/2}$) in the Biot-Savart law. If the symbol $m$ satisfies the Osgood-type condition $$\int_2^{+\infty} \frac{1}{r (\log r) m(r)} dr= +\infty$$ and certain mild assumptions, the system is referred to as the 2D Loglog-Euler type equation. First, we prove a Yudovich-type theorem establishing the existence and uniqueness of a global weak solution for the Loglog-Euler type equation associated with bounded and integrable initial data. This result directly applies to patch solutions, which are weak solutions corresponding to patch initial data given by characteristic functions of disjoint, regular, bounded domains. Next, we revisit the seminal result by Elgindi ( Arch. Ration. Mech. Anal. 211(3) 965-990, 2014 ) and provide a different proof under explicit assumptions on $m$, showing that for the 2D Loglog-Euler type equation with $C^{1,\mu}$ ($0<\mu<1$) single-patch initial data, the evolved patch boundary globally preserves the $C^{1,\mu-\varepsilon}$ regularity for any $\varepsilon \in (0,\mu)$. In contrast to the frequency-space argument in Elgindi's result, we develop an entirely physical-space-based approach that avoids the Littlewood-Paley theory and offers advantages for potential extensions to the half-plane or bounded smooth domains. Furthermore, we investigate the global propagation of higher-order $C^{n,\mu}$ boundary regularity for patch solutions with any $n \in \mathbb{N}^\star$, and analyze the evolution of multiple patches.