Resolvent Estimates in L^p for the Stokes Operator in Lipschitz Domains

TL;DR

This paper proves $L^p$ resolvent estimates for the Stokes operator in Lipschitz domains, showing it generates a bounded analytic semigroup in certain $L^p$ ranges, confirming a conjecture by M. Taylor.

Contribution

It establishes new $L^p$ resolvent estimates for the Stokes operator in Lipschitz domains, extending the understanding of its analytic semigroup generation.

Findings

Resolves the conjecture of M. Taylor on Stokes operator in Lipschitz domains.

Shows the Stokes operator generates a bounded analytic semigroup in specified $L^p$ ranges.

Provides $L^p$ resolvent estimates for $d eq 2$ in Lipschitz domains.

Abstract

We establish the $L^p$ resolvent estimates for the Stokes operator in Lipschitz domains in $R^d$, $d\ge 3$ for $|\frac{1}{p}-1/2|< \frac{1}{2d} +\epsilon$. The result, in particular, implies that the Stokes operator in a three-dimensional Lipschitz domain generates a bounded analytic semigroup in $L^p$ for $(3/2)-\varep < p< 3+\epsilon$. This gives an affirmative answer to a conjecture of M. Taylor.