Regularisation effects of nonlinear semigroups

Thierry Coulhon and Daniel Hauer

Abstract

We introduce natural and simple methods to deduce $L^s$-$L^{\mathrm{\infty }}$-regularisation estimates for $1\le s<\mathrm{\infty }$ of nonlinear semigroups holding uniformly for all time with sharp exponents from natural Gagliardo-Nirenberg inequalities. From $L^q$-$L^r$ Gagliardo-Nirenberg inequalities, $1\le q,r\le \mathrm{\infty }$, one deduces $L^q$-$L^r$ estimates for the semigroup. We provide a new nonlinear interpolation theorem which might be of independent interest and use this to extrapolate such estimates to $L^{\tilde{q}}$-$L^{\mathrm{\infty }}$ estimates for some $\tilde{q}$, $1\le \tilde{q}<\mathrm{\infty }$. Finally one is able to extrapolate to $L^s$-$L^{\mathrm{\infty }}$ estimates for $1\le s<q$. Our theory developed in this monograph allows to work with minimal regularity assumptions on solutions of nonlinear parabolic boundary value problems, namely with the notion of mild solutions. We illustrate these new tools in a plethora of examples including nonlinear nonlocal diffusion problems. As an application of $L^1$-$L^{\mathrm{\infty }}$-regularisation estimates, we provide an abstract approach to deduce that mild solutions in $L^1$ admit more regularity. They are weak energy solutions.

Keywords: Nonlinear semigroups, $p$-Laplace operator, porous media operator, doubly nonlinear diffusion operator, nonlocal operators, regularity.

AMS Subject Classification: Primary 47H06,47H20,35K55,46B70,35B65.