Communications in Mathematical Sciences

Volume 18 (2020)

Number 6

Two-front solutions of the SQG equation and its generalizations

Pages: 1685 – 1741



John K. Hunter (Department of Mathematics, University of California, Davis, Cal., U.S.A.)

Jingyang Shu (Department of Mathematics, Temple University, Philadelphia, Pennsylvania, U.S.A.)

Qingtian Zhang (Department of Mathematics, West Virginia University, Morgantown, W.V., U.S.A.)


The generalized surface quasi-geostrophic (GSQG) equations are transport equations for an active scalar that depend on a parameter $0 \lt \alpha \leq 2$. Special cases are the two-dimensional incompressible Euler equations $(\alpha = 2)$ and the surface quasi-geostrophic (SQG) equations $(\alpha = 1)$. We derive contour-dynamics equations for a class of two-front solutions of the GSQG equations when the fronts are a graph. Scalar reductions of these equations include ones that describe a single front in the presence of a rigid, flat boundary. We use the contour dynamics equations to determine the linearized stability of the GSQG shear flows that correspond to two flat fronts. We also prove local-intime existence and uniqueness for large, smooth solutions of the two-front equations in the parameter regime $1 \lt \alpha \leq 2$, and small, smooth solutions in the parameter regime $0 \lt \alpha \leq 1$.


surface quasi-geostrophic equation, contour dynamics, fronts, stability, well-posedness

2010 Mathematics Subject Classification

35Q35, 35Q86, 76B03, 86A10

J.K.H. was supported by the NSF under grant numbers DMS-1616988 and DMS-1908947.

J.S. would like to thank Javier Gómez-Serrano for discussions in the “MathFluids” Workshop held in Mathematical Institute of University of Seville, Seville, Spain, June 12–15, 2018.

Received 15 August 2019

Accepted 1 April 2020

Published 4 November 2020