We consider the well-posedness of the generalized surface quasigeostrophic (gSQG)
front equation. By using the null structure of the equation via a paradifferential
normal form analysis, we obtain balanced energy estimates, which allow us to prove
the local well-posedness of the nonperiodic gSQG front equation at a low level of
regularity (in the SQG case, at only one-half derivatives above scaling). In addition,
we establish global well-posedness for small and localized rough initial data, as well
as modified scattering, by using the testing by wave packet approach of Ifrim and
Tataru.
Keywords
gSQG front equation, low regularity, normal forms,
paralinearization, modified energies, frequency envelopes,
wave packet testing
