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Preprints, Working Papers, ... Year : 2017

Oseen resolvent estimates with small resolvent parameter

Abstract

We consider the Oseen system with resolvent term in an exterior domain in R^3 , supplemented by homogeneous Dirichlet boundary conditions. Under the assumption that the resolvent parameter λ is close to zero and Re λ ≥ 0, λ different from 0, we estimate the L^p-norm of the velocity against the L^p-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 L^p-estimates of Oseen resolvent flows for the same range of λ. Our article is complementary to the work by T. Kobayashi and Y. Shibata, "On the Oseen equation in the three dimensional exterior domains", Math. Ann. 310 (1998), 1–45, where Oseen resolvent estimates are derived under the assumption that |λ| ≥ c, for some arbitrary but fixed c>0, with the constant in the resolvent estimate depending on c .
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Dates and versions

hal-01534690 , version 1 (08-06-2017)

Identifiers

  • HAL Id : hal-01534690 , version 1

Cite

Paul Deuring. Oseen resolvent estimates with small resolvent parameter. 2017. ⟨hal-01534690⟩

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