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We consider the family known as modified or generalized surfacequasi-geostrophic equations (mSQG) consisting of the classical inviscid surfacequasi-geostrophic (SQG) equation together with a family of regularized activescalars given by introducing a smoothing operator of nonzero but possiblyarbitrarily small degree. This family naturally interpolates between the 2DEuler equation and the SQG equation. For this family of equations we constructan invariant measure on a rough $L^2$-based Sobolev space and establish theexistence of solutions of arbitrarily large lifespan for initial data in a setof full measure in the rough Sobolev space.

Global Flows with Invariant Measures for the Inviscid Modified SQG Equations