Oseen resolvent estimates with small resolvent parameter

We consider the Oseen system with resolvent term in an exterior domain in R3, supplemented by homogeneous Dirichlet boundary conditions. Under the assumption that the resolvent parameter λ is close to zero and ℜλ≥0, λ≠0, we estimate the Lp-norm of the velocity against the Lp-norm of the right-hand side, times a factor C|λ|−2, with C>0 independent of λ. Such an estimate cannot hold for this range of λ if |λ|−2 is replaced by |λ|−κ with κ<3/2, and there are indications that κ∈[3/2,2) cannot be admitted either. We present various other Lp-estimates of Oseen resolvent flows for the same range of λ. Our article is complementary to the work by T. Kobayashi and Y. Shibata (1998) [20], where Oseen resolvent estimates are derived under the assumption that |λ|≥c0, for some arbitrary but fixed c0>0, with the constant in the resolvent estimate depending on c0.