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

TL;DR

This paper studies patch solutions for a class of 2D active scalar equations that interpolate between Euler and SQG, proving global existence, uniqueness, and boundary regularity preservation using physical-space methods.

Contribution

It provides a new physical-space proof for boundary regularity preservation in Loglog-Euler patches and extends analysis to multiple patches and higher regularity.

Findings

Global weak solutions exist and are unique for bounded initial data.

Patch boundaries maintain $C^{1, u}$ regularity over time.

Higher-order boundary regularity propagates globally.

Abstract

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.