Two-front solutions of the SQG equation and its generalizations

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 $0 \lt \alpha \leq 1$.

Keywords

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

2010 Mathematics Subject Classification

35Q35, 35Q86, 76B03, 86A10

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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