Local estimates on two linear parabolic equations with singular coefficients

Qi S. Zhang

Vol. 223 (2006), No. 2, 367–396

DOI: 10.2140/pjm.2006.223.367

Abstract

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.

Keywords

heat equation, singular coefficients, Navier–Stokes equation, Kato class

Mathematical Subject Classification 2000

Primary: 35K40, 76D05

Milestones

Received: 19 July 2004

Revised: 15 February 2005

Accepted: 17 May 2005

Published: 1 February 2006

Correction: 15 October 2021