ON WELL-POSEDNESS OF α-SQG EQUATIONS IN THE HALF-PLANE - Jeong, In Jee; Kim, Junha; Yao, Yao (Transactions of the American Mathematical Society, 2025)
We investigate the well-posedness of α-surface quasi-geostrophic (α-SQG) equations in the half-plane, where α = 0 and α = 1 correspond to the 2D Euler and SQG equations respectively. For 0 < α ≤ 1/2, we prove local well-posedness in certain weighted anisotropic Hölder spaces. We also show that such a well-posedness result is sharp: for any 0 < α ≤ 1, we prove nonexistence of Hölder regular solutions (with the Hölder regularity depending on α) for initial data smooth up to the boundary.
Keywords: fluid dynamics; Ill-posedness; norm inflation; surface quasi-geostrophic equation