Global and local existence for the dissipative critical SQG equation with small oscillations

Abstract

This article is devoted to the study of the critical dissipative surface quasi-geostrophic $(SQG)$ equation in ${\mathbb{R}}^2$. For any initial data ${\theta }_0$ belonging to the space ${\Lambda }^s(H_{uloc}^s({\mathbb{R}}^2))\cap L^{\infty }({\mathbb{R}}^2)$, we show that the critical (SQG) equation has at least one global weak solution in time for all $1/4\le s\le 1/2$ and at least one local weak solution in time for all $0<s<1/4$. The proof for the global existence is based on a new energy inequality which improves the one obtain in \cite{Laz} whereas the local existence uses more refined energy estimates based on Besov space techniques.

arXiv comment17 pages. To appear in JMFM

References in corpus (2)

• On the global well-posedness of the critical quasi-geostrophic equation
• Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation

Cited by in corpus (2)

• Global existence for some transport equations with nonlocal velocity
• Suppression of blow up by a logistic source in $2$D Keller-Segel system with fractional dissipation