Regularisation effects of nonlinear semigroups

TL;DR

This paper develops methods to derive regularisation estimates for nonlinear semigroups using Gagliardo-Nirenberg inequalities, enabling analysis of minimal regularity solutions in nonlinear parabolic problems.

Contribution

It introduces new nonlinear interpolation techniques and sharp regularisation estimates, broadening the understanding of nonlinear semigroup regularity under minimal assumptions.

Findings

Derived $L^{s}$-$L^{ {0.5}}$-regularisation estimates for nonlinear semigroups.

Established $L^{q}$-$L^{r}$ estimates from Gagliardo-Nirenberg inequalities.

Introduced nonlinear interpolation methods for extrapolating estimates.

Abstract

One introduces natural and simple methods to deduce $L^{s}$-$L^{\infty}$-re\-gularisation estimates for $1\le s< \infty$ of nonlinear semigroups holding uniformly for all time with sharp exponents from natural Gagliardo-Nirenberg inequalities. From $L^{q}$-$L^{r}$ Gagliardo-Nirenberg inequalities, $1\le q, r\le \infty$, one deduces $L^{q}$-$L^{r}$ estimates for the semigroup. New nonlinear interpolation techniques of independent interest are introduced in order to extrapolate such estimates to $L^{\tilde{q}}$-$L^{\infty}$ estimates for some $\tilde{q}$, $1\le \tilde{q}<\infty$. Finally one is able to extrapolate to $L^{s}$-$L^{\infty}$ estimates for $1\le s<q$. The theory developed in this monograph allows to work with minimal regularity assumptions on solutions of nonlinear parabolic boundary value problems as illustrated in a plethora of examples including nonlocal diffusion processes.