Global solutions for a family of GSQG front equations

Abstract

We prove the global existence of solutions with small and smooth initial data of a nonlinear dispersive equation for the motion of generalized surface quasi-geostrophic (GSQG) fronts in a parameter regime $1<\alpha <2$, where $\alpha =1$ corresponds to the SQG equation and $\alpha =2$ corresponds to the incompressible Euler equations. This result completes previous global well-posedness results for $0<\alpha \le 1$. We also use contour dynamics to derive the GSQG front equations for $1<\alpha <2$.

References in corpus (7)

• Some Liouville theorems for the fractional Laplacian
• A self-similar cascade of instabilities in the surface quasigeostrophic system
• Global Solutions of a Surface Quasi-Geostrophic Front Equation
• Symmetry in stationary and uniformly-rotating solutions of active scalar equations
• Regularized and Approximate Equations for Sharp Fronts in the Surface Quasi-Geostrophic Equation and its Generalizations
• Contour Dynamics for Surface Quasi-Geostrophic Fronts
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Cited by in corpus (2)

• On the approximation of vorticity fronts by the Burgers-Hilbert equation
• The lifespan of classical solutions for the inviscid Surface Quasi-geostrophic equation