Local estimates on two linear parabolic equations with singular coefficients

We treat the heat equation with singular drift terms and its generalization: the linearized Navier-Stokes system.In the first case, we obtain boundedness of weak solutions for highly singular, "supercritical" data.In the second case, we obtain regularity results for weak solutions with mildly singular data (those in the Kato class).This not only extends some of the classical regularity theory from the case of elliptic and heat equations to that of linearized Navier-Stokes equations but also proves an unexpected gradient estimate, which extends the recent interesting boundedness result of O'Leary.