Gradient estimates for degenerate quasi-linear parabolic equations

TL;DR

This paper establishes gradient estimates and Lipschitz continuity for solutions to a broad class of degenerate quasi-linear parabolic equations, advancing understanding of their regularity properties.

Contribution

It provides new $L^q$-estimates for gradients and Lipschitz regularity results for solutions under minimal assumptions on coefficients.

Findings

$L^q$-estimates for solution gradients

Lipschitz continuity of solutions

Applicable to general degenerate quasi-linear parabolic equations

Abstract

For a general class of divergence type quasi-linear degenerate parabolic equations with differentiable structure and lower order coefficients form bounded with respect to the Laplacian we obtain $L^q$-estimates for the gradients of solutions, and for the lower order coefficients from a Kato-type class we show that the solutions are Lipschitz continuous with respect to the space variable.