On off-diagonal decay properties of the generalized Stokes semigroup with bounded measurable coefficients

TL;DR

This paper studies how the generalized Stokes semigroup with bounded measurable coefficients exhibits off-diagonal decay properties, extending known heat kernel bounds to more general elliptic systems.

Contribution

It establishes resolvent estimates in Morrey spaces for the generalized Stokes semigroup, revealing new off-diagonal decay behaviors for systems with measurable coefficients.

Findings

Proves off-diagonal decay estimates for the Stokes semigroup.

Establishes resolvent estimates in Morrey spaces.

Extends classical heat kernel bounds to more general elliptic systems.

Abstract

We investigate off-diagonal decay properties of the generalized Stokes semigroup with bounded measurable coefficients on $\mathrm{L}^2_{\sigma} (\mathbb{R}^d)$. Such estimates are well-known for elliptic equations in the form of pointwise heat kernel bounds and for elliptic systems in the form of integrated off-diagonal estimates. On our way to unveil this off-diagonal behavior we prove resolvent estimates in Morrey spaces $\mathrm{L}^{2 , \nu} (\mathbb{R}^d)$ with $0 \leq \nu < 2$.