Sobolev multipliers, maximal functions and parabolic equations with a quadratic nonlinearity

We develop a general framework to describe global mild solutions to a Cauchy problem with small initial values concerning a general class of semilinear parabolic equations with a quadratic nonlinearity. This class includes the Navier-Stokes equations, the subcritical dissipative quasi-geostrophic equation and the parabolic-elliptic Keller-Segel system.