Eulerian uniqueness of the $α$-SQG patch problem

Abstract

We consider the patch problem of the $α$-SQG equation with $α=0$ being the 2D Euler and $α= \frac{1}{2}$ the SQG equations respectively. In the Eulerian setting, we prove the uniqueness of patch solutions of regularity $W^{2, \frac{1}{1-2α} +} $ when $0<α< \frac{1}{2}$ and $C^{1, 4α+ }$ when $0<α< \frac{1}{4} $. The proof is intrinsic to the modified Biot-Savart law and independent of the local existence of patch solutions.

Figures (1)

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Theorems & Definitions (20)

• Definition 1.1
• Theorem 1.2: $W^{2, \frac{1}{1 -2\alpha } + }$ Uniqueness
• Theorem 1.3: $C^{1,4\alpha + }$ Uniqueness
• Lemma 2.1
• proof
• Definition 3.1
• Theorem 3.2: asqg
• Lemma 3.3
• proof
• Lemma 3.4