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$0 \leq \nu < 2$.