On the locally self-similar solution of the surface quasi-geostrophic equation with decaying or non-decaying profiles

Motivated by the numerical simulation and the study on several 1D models, we consider the locally self-similar singular solutions for the surface quasi-geostrophic equation with decaying or non-decaying blowup profiles. Based on a suitable local LpLp-inequality in terms of the profile and the bootstrapping method, we show some exclusion results and derive the asymptotic behavior of the possible blowup profiles.