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Steady-State Navier–Stokes Problem Past a Rotating Body: Geometric-Functional Properties and Related Questions

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Topics in Mathematical Fluid Mechanics

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2073))

Abstract

As is well known, the three-dimensional steady motion of a viscous, incompressible (Navier–Stokes) liquid around a rigid body, \(\mathcal{B}\), is among the fundamental and most studied questions in fluid dynamics; see e.g. [4].

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Notes

  1. 1.

    \({}^{\ddag }\)Partially supported by NSF grant DMS-1062381

  2. 2.

    Or, equivalently, the body is at rest and the liquid tends to a constant uniform flow at large distances from the body.

  3. 3.

    A reasonably complete list of reference would be too long to be included here. A good source of information is provided by the articles collected in [26] and the bibliography quoted therein.

  4. 4.

    See the third term on the left-hand side of (1).

  5. 5.

    As a matter of fact, even in absence of rotation, this type of properties have been addressed only very recently; see [17, 18, 24, 25].

  6. 6.

    See e.g. [20, p. 10].

  7. 7.

    This assumption also implies, of course, that both \(\boldsymbol{\eta }\) and \(\boldsymbol{\omega }\) are non-zero. While the case \(\boldsymbol{\omega } = \boldsymbol{0}\) can be still treated by the methods described in these Notes to obtain very similar results [18], the same methods do not work if \(\boldsymbol{\eta } = \boldsymbol{0}\). We refer the reader to [20, Chap. XI] for the known results in this latter case.

  8. 8.

    As customary, \(\mathbb{R}_{+}\) denotes the set of all positive real numbers.

  9. 9.

    More precisely, Y is isomorphic to \(\mathcal{D}_{0}^{1,2}(\Omega )\).

  10. 10.

    This assumption is certainly verified if \(\boldsymbol{f}\) satisfies further summability hypothesis.

  11. 11.

    Let S be any space of real functions. As a rule, we shall use the same symbol S to denote the corresponding space of vector and tensor-valued functions.

  12. 12.

    Throughout these Notes, “linear” functional on a Banach space X, means a map \(F : X\mapsto \mathbb{R}\) (or \(\mathbb{C}\)) with D(F) = X, that is bounded and distributive. See Definition 9.

  13. 13.

    Sometimes in the literature, our definition of coerciveness is also referred to as weak coerciveness.

  14. 14.

    See Remark 34.

  15. 15.

    However, problem (33) is well-posed for \(\boldsymbol{u}\) and \(\boldsymbol{f}\) in suitable Lorentz spaces; see [30].

  16. 16.

    We recall that a subset B of X is said to be a Banach manifold of class C k if for any x ∈ B there is an open neighborhood U(x) in X such that U(x) ∩ B is C k -diffeomorphic to an open set in a Banach space X x.

  17. 17.

    The definition of degree can be suitably extended to the case when D(M) ⊊X. However, such a circumstance will not happen in the applications we have in mind. For this more general case, we refer the reader to, e.g., [5, p. 263 and ff.].

  18. 18.

    See (47) for notation.

  19. 19.

    Formally, (58) is obtained by first writing, in (58), \(\boldsymbol{v} = \boldsymbol{u} + \boldsymbol{V }\), then by taking the scalar product of both sides of the resulting equation by \(\boldsymbol{\varphi }\), and, finally, by integrating by parts over Ω.

  20. 20.

    In fact, one can easily construct examples proving the invalidity of (60), if \(\boldsymbol{u}\) only belongs to \(\mathcal{D}_{0}^{1,2}(\Omega )\).

  21. 21.

    See (??) below with q = 2.

  22. 22.

    Interpolation inequalities for positive anisotropic Sobolev spaces are well-known; see, e.g., [6].

  23. 23.

    For future reference, we remark that, in view of (79)2, the redefined pressure satisfies the asymptotic condition \(p_{k} - {p}^{(0)} = O(\vert x{\vert }^{-2})\) as | x | → ∞.

  24. 24.

    See footnote 22.

  25. 25.

    Notice that from (117)1, (112) and (119) it follows \(\boldsymbol{w} \in {L}^{2}({\mathbb{R}}^{3})\).

  26. 26.

    For simplicity, in what follows, we suppress the dependence on \(\boldsymbol{\mathfrak{p}}\).

  27. 27.

    A detailed analysis of the spectrum of the lienearized operator is given in [9].

  28. 28.

    This may depend on the particular non-dimensionalization of the Navier–Stokes equations and on the special form of the family of solutions \(\boldsymbol{u}_{0}\). In fact, there are several interesting problems formulated in exterior domains where this circumstance takes place, like, for example, the problem of steady bifurcation considered in the previous section and the one studied in [23, Sect. 6].

  29. 29.

    Namely, that \((\boldsymbol{w}_{k},\boldsymbol{\Phi })\) can be uniquely extended to an element of \(\mathcal{D}_{0}^{-1,2}({\Omega }^{R})\) with preservation of the norm.

  30. 30.

    In fact, setting ∫ −  = \(\frac{1}{2\pi}\int^{2\pi}_{0}\), and recalling (115)2 and (114) in Sect. 3, we find

    $$\begin{array}{rl} \boldsymbol{w}_{0}({\boldsymbol{Q}}^{\top }(t) \cdot \boldsymbol{ y}) & =\displaystyle\int -\boldsymbol{Q}(\tau ) \cdot \boldsymbol{v}({\boldsymbol{Q}}^{\top }(\tau ) \cdot {\boldsymbol{Q}}^{\top }(t) \cdot \boldsymbol{ y})d\tau =\displaystyle\int -\boldsymbol{Q}(\tau ) \cdot \boldsymbol{v}({\boldsymbol{Q}}^{\top }(\tau + t) \cdot \boldsymbol{ y})d\tau \\ & =\displaystyle\int -\boldsymbol{Q}(\tau - t) \cdot \boldsymbol{v}({\boldsymbol{Q}}^{\top }(\tau ) \cdot \boldsymbol{ y})d\tau = \boldsymbol{Q}(-t) \cdot \displaystyle\int -\boldsymbol{Q}(\tau ) \cdot \boldsymbol{v}({\boldsymbol{Q}}^{\top }(\tau ) \cdot \boldsymbol{ y})d\tau \\ & ={ \boldsymbol{Q}}^{\top }(t) \cdot \displaystyle\int -\boldsymbol{w}(y,\tau )d\tau ={ \boldsymbol{Q}}^{\top }(t) \cdot \boldsymbol{w}_{0}(y) \end{array}$$
  31. 31.

    Of course, the assumption \(\boldsymbol{u}_{0} \in L_{\mathrm{loc}}^{4}(\overline{\Omega })\) is redundant if \(\boldsymbol{u}_{0} \in X(\Omega )\).

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Authors and Affiliations

  1. Department of Mechanical Engineering and Material Science, Pittsburgh, USA

    Giovanni P. Galdi

  2. Department of Mathematics, University of Pittsburgh, Pittsburgh, USA

    Giovanni P. Galdi

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Galdi, G.P. (2013). Steady-State Navier–Stokes Problem Past a Rotating Body: Geometric-Functional Properties and Related Questions. In: Topics in Mathematical Fluid Mechanics. Lecture Notes in Mathematics(), vol 2073. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36297-2_3

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