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Behavior Laws

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Abstract

Behavior laws of fluid mechanics. Force–velocity relations.

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Notes

  1. 1.

    See, for instance, [Germ], [D-L19, ch. IA.2.6].

  2. 2.

    This notation anticipates the fact that ϕ∗, being a strict l.s.c. convex function, is equal to the conjugate function of ϕ.

  3. 3.

    Note that the vector bundle S2T∗M is naturally equipped with a fibre metric, since M is a Riemannian (even Euclidean) manifold, so that L2(M, S2M) is a Hilbert space.

  4. 4.

    The viscous coefficients can be x-dependent.

  5. 5.

    This operator is similar to the capacity operator (often called Dirichlet-to-Neumann operator) in potential theory (see [D-L19]), or to the Calderón operator (or impedance operator) in electromagnetism (see [Ces]).

  6. 6.

    Beware of confusing the notation uΓ with uΓ(x), x ∈ Γ for the velocities: uΓ is not necessarily tangent to Γ, but is in Tx M = Tx( Γ) ⊕ Tx( Γ)⊥. But we keep the notation fΓ for the forces in \(T^{*}_{x}M\). But in some cases, the more natural notation uΓ is used instead of uΓ.

  7. 7.

    For the properties of these spaces, see [Lio-Mag].

  8. 8.

    See (4.9), and [Aub, ch. I.3.5, p. 46].

  9. 9.

    See [Eke-Tem, ch. VII.4], [Aub, ch. 1.5.2; 5.3].

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Authors and Affiliations

  1. Institut Henri Poincaré, Société de Mathématiques Appliquées et Industrielles, Paris Cedex 05, Paris, France

    Michel Cessenat

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  1. Michel Cessenat
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Cessenat, M. (2018). Behavior Laws. In: Mathematical Modelling of Physical Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-94758-7_4

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