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\noindent
\textbf{SPEAKER}:
\begin{center}
%\textbf{SPEAKER}:
Patrick Tolksdorf,  Johannes Gutenberg-Universit\"at Mainz \\
 Institut f\"ur Mathematik 
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\noindent
{\bf TITLE}:
\begin{center}
$\mathrm{L}^p$-extrapolation of the generalized Stokes operator
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\noindent
{\bf ABSTRACT:}\\
 In this talk, we discuss the Stokes operator with bounded measurable coefficients $\mu$, formally given by
\begin{align}
\label{Eq: Stokes}
 A u := - \mathrm{div} (\mu \nabla u) + \nabla \phi, \quad \mathrm{div} (u) = 0 \quad \text{in } \mathbb{R}^d.
\end{align}
This operator satisfies $\mathrm{L}^2$-resolvent estimates of the form
\begin{align*}
 \| \lambda (\lambda + A)^{-1} f \|_{\mathrm{L}^2} \leq C \| f \|_{\mathrm{L}^2} \qquad (f \in \mathrm{L}^2_{\sigma} (\mathbb{R}^d))
\end{align*}
for $\lambda$ in some complex sector $\{ z \in \mathbb{C} \setminus \{ 0 \} : \lvert \arg(z) \rvert < \theta\}$. We describe how an analogue of such a resolvent estimate can be established in $\mathrm{L}^p$ by virtue of certain non-local Caccioppoli inequalities combined with an extrapolation argument of Shen.  
Such estimates build the foundation for many important functional analytic properties of these operators like maximal $\mathrm{L}^q$-regularity and the boundedness of its $\mathrm{H}^{\infty}$-calculus. \par
More precisely, we establish resolvent estimates in $\mathrm{L}^p$ for $p$ satisfying
\begin{align}
\label{Eq: Interval}
 \Big\lvert \frac{1}{p} - \frac{1}{2} \Big\rvert < \frac{1}{d}\cdotp
\end{align}
This resembles a well-known situation for elliptic systems in divergence form with $\mathrm{L}^{\infty}$-coefficients. Here, important estimates like Gaussian upper bounds for the semigroup cease to exist and the $\mathrm{L}^p$-extrapolation has be concluded by other means. In particular, for elliptic systems one can establish resolvent bounds for numbers $p$ that satisfy~\eqref{Eq: Interval}. Moreover, if $d \geq 3$, Davies constructed examples which show that corresponding resolvent bounds do not generally hold in $\mathrm{L}^p$ for numbers $1 < p < \infty$ that satisfy
\begin{align*}
 \Big\lvert \frac{1}{p} - \frac{1}{2} \Big\rvert > \frac{1}{d}\cdotp
\end{align*}
These elliptic results give an indication that the corresponding result for the Stokes operator with $\mathrm{L}^{\infty}$-coefficients is optimal.




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