  This paper considers a family of active scalar equations which modify the
generalized surface quasi-geostrophic (gSQG) equations through its constitutive
law or dissipative perturbations. These modifications are characteristically
mild in the sense that they are logarithmic. The problem of well-posedness, in
the sense of Hadamard, is then studied in a borderline setting of regularity in
analog to the scaling-critical spaces of the gSQG equations. A novelty of the
system considered is the nuanced form of smoothing provided by the proposed
mild form of dissipation, which is able to support global well-posedness at the
Euler endpoint, but in a setting where the inviscid counterpart is known to be
ill-posed. A novelty of the analysis lies in the simultaneous treatment of
modifications in the constitutive law, dissipative mechanism, and functional
setting, which the existing literature has typically treated separately. A
putatively sharp relation is identified between each of the distinct
system-modifiers that is consistent with previous studies that considered these
modifications in isolation. This unified perspective is afforded by the
introduction of a linear model equation, referred to as the protean system,
that successfully incorporates the more delicate commutator structure
collectively possessed by the gSQG family and upon which each facet of
well-posedness can be reduced to its study.
