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Advances in Mathematical Fluid Mechanics pp 473–511Cite as

On Stokes' Problem

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Abstract

We consider the Stokes problem of viscous hydrodynamics in bounded and exterior Lipschitz domains O of with boundary datum in. We show that this problem has a unique very weak solution in bounded domains. As far as exterior domains are concerned, we prove that a very weak solution exists such that at infinity, with pk a Stokes's polynomial of degree k, if and only if the data satisfy a suitable compatibility condition. In particular, we derive the well-known Stokes'paradox of hydrodynamics for very weak solutions. We use this results to prove the existence of a very weak solution to the Navier-Stokes problem in bounded and exterior Lipschitz domains of by requiring that the boundary datum belongs to.

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  1. Dipartimento di Matematica, Seconda Universit`a di Napoli, 81100, Caserta, Italy

    Remigio Russo

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  1. Remigio Russo
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  1. Interdisziplinäres Zentrum für, Universität Heidelberg, Im Neuenheimer Feld 368, Heidelberg, 69120, Germany

    Rolf Rannacher

  2. Depto. Mathemática, Istituto Superior Técnico, Av. Rovisco Pais, Lisboa, 1049-001, Portugal

    Adélia Sequeira

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Russo, R. (2010). On Stokes' Problem. In: Rannacher, R., Sequeira, A. (eds) Advances in Mathematical Fluid Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04068-9_28

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