Archive for Rational Mechanics and Analysis· 2026Q1
Unstable Vortices, Sharp Non-Uniqueness with Forcing, and Global Smooth Solutions for the SQG Equation
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- 2026year
Short summary
Researchers constructed smooth, unstable vortices to prove non-uniqueness of weak solutions for the forced $\alpha$-SQG equation in the supercritical regime, a finding applicable to the 2D Euler and Surface Quasi-Geostrophic equations.
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Key points
- Non-uniqueness of weak solutions proven for forced $\alpha$-SQG in supercritical regime ($s < \alpha + 2/p$).
- Construction of smooth, compactly supported, non-linearly unstable vortices is a key step.
- Results apply to 2D Euler ($\alpha=0$) and Surface Quasi-Geostrophic ($\alpha=1$) equations.
- Existence of global smooth, non-rotating, non-traveling solutions shown for the unforced SQG equation.
AI-generated from the title and abstract; the full text is not read.
Abstract
Abstract We prove that non-uniqueness of weak solutions to the forced $$\alpha $$ α -SQG equation with Sobolev regularity $$W^{s,p}$$ W s , p in the supercritical regime $$s < \alpha + \frac{2}{p}$$ s < α + 2 p , covering the 2D Euler equation ( $$\alpha = 0$$ α = 0 ), the Surface Quasi-Geostrophic equation ( $$\alpha = 1$$ α = 1 ), and the intermediate cases. A key step is the construction of smooth, compactly supported vortices that exhibit non-linear instability. As a by-product, we show that existence of global smooth solutions to the (unforced) $$\alpha $$ α -SQG equation that are neither rotating nor traveling.
The authors' abstract, as published at the source. Archive for Rational Mechanics and Analysis, 2026 · DOI ↗
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Field: Applied Mathematics
Applied MathematicsMathematics