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

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 $0 \leq \nu < 2$.