PreprintRegularisation effects of nonlinear semigroupsThierry Coulhon and Daniel HauerAbstractWe introduce natural and simple methods to deduce
\(L^{s}\)\(L^{\infty}\)regularisation estimates for \(1\le s <
\infty\) of nonlinear semigroups holding uniformly for all time
with sharp exponents from natural GagliardoNirenberg
inequalities. From \(L^{q}\)\(L^{r}\) GagliardoNirenberg
inequalities, \(1\le q, r\le \infty\), 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 \(L^{\tilde{q}}\)\(L^{\infty}\) estimates for some
\(\tilde{q}\), \(1\le \tilde{q} < \infty\). Finally one is able
to extrapolate to \(L^{s}\)\(L^{\infty}\) estimates for \(1\le
s < q\). 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
AMS Subject Classification: Primary 47H06,47H20,35K55,46B70,35B65. This paper is available as a pdf (1300kB) file. This paper is also on the arXiv: arxiv.org/abs/1604.08737.
