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 $$\mathrm {L}^2_{\sigma } ({\mathbb {R}}^d)$$ L σ 2 ( 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 $$\mathrm {L}^{2 , \nu } ({\mathbb {R}}^d)$$ L 2 , ν ( R d ) with $$0 \le \nu < 2$$ 0 ≤ ν < 2 .