# New gradient estimates for parabolic equations

created by baroni on 28 Sep 2012
modified on 16 Nov 2012

[BibTeX]

Published Paper

Inserted: 28 sep 2012
Last Updated: 16 nov 2012

Journal: Houston J. Math.
Volume: 38
Number: 3
Pages: 855-914
Year: 2012

Abstract:

We prove sharp Lorentz- and Morrey-space estimates for the gradient of solutions $u$ to nonlinear parabolic equations of the type $u_t - {\rm div}\, a(z,Du) = g, \qquad \mbox{ on } \Omega_T = \Omega \times (-T,0),$ where the vector field $a$ is assumed to satisfy classical growth and ellipticity conditions and where the inhomogeneity $g$ is only assumed to be integrable to some power $\gamma > 1$. In particular we investigate the case where $\gamma$ stays below the exponent allowing for weak solutions $u \in L^2(-T,0;W^{1,2}(\Omega))$.