Local estimates on two linear parabolic equations with singular coefficients
Qi Zhang · Pacific Journal of Mathematics · 2006
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.p loc ޒ( n ), p > n, weak solutions to (1) are locally bounded and Hölder-continuous.This condition is sharp in general.Here is an example, taken from [Han and Lin 1997, p. 108].The function u = ln ln |x| -1 -ln ln R -1 is an unbounded weak solution of u + b∇u = 0 MSC2000: 35K40, 76D05.