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The chapter is devoted to the derivation of the representation formula of the steady Oseen type problem. First a truncated exterior domain \(B_R \backslash {\mathcal {D}}^c\) is considered. Passing with \(R \to \infty \) the representation formula is derived in the case of the exterior domain. In the next parts the volume potential that arises in the problem is investigated. Moreover, the asymptotic profile is shown. Using the representation formula for the Oseen type problem also, such a formula is derived for the Navier–Stokes type problem, see Chap. 11 . The asymptotic profile of the gradient of the velocity field and the second derivatives of the velocity are investigated. Finally, the Leray solution is studied.

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Representation Formula and the Asymptotic Behavior of the Steady Oseen Type Problem

  • 章节

  • 发表日期：

    2025年

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  • Šárka Nečasová,
  • Stanislav Kračmar,
  • Jiří Neustupa,
  • Patrick Penel

摘要

The chapter is devoted to the derivation of the representation formula of the steady Oseen type problem. First a truncated exterior domain \(B_R \backslash {\mathcal {D}}^c\) is considered. Passing with \(R \to \infty \) the representation formula is derived in the case of the exterior domain. In the next parts the volume potential that arises in the problem is investigated. Moreover, the asymptotic profile is shown. Using the representation formula for the Oseen type problem also, such a formula is derived for the Navier–Stokes type problem, see Chap. 11 . The asymptotic profile of the gradient of the velocity field and the second derivatives of the velocity are investigated. Finally, the Leray solution is studied.