Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions

Toshiaki Hishida ()
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Toshiaki Hishida: Nagoya University, Graduate School of Mathematics

Chapter 6 in Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, 2018, pp 299-339 from Springer

Abstract: Abstract Consider the stationary Navier-Stokes flow in 3D exterior domains with zero velocity at infinity. What is of particular interest is the spatial behavior of the flow at infinity, especially optimal decay (summability) observed in general and the asymptotic structure. When the obstacle is translating, the answer is found in some classic literature by Finn; in fact, the optimal summability is L q with q > 2 and the leading profile is the Oseen fundamental solution. This presentation is devoted to the other cases developed in the last decade, mainly the case where the obstacle is at rest, together with several remarks even on the challenging case where the obstacle is rotating. The optimal summability for those cases is L3,∞ (weak-L3) and the leading term of small solutions being in this class is the homogeneous Navier-Stokes flow of degree (−1), which is called the Landau solution. In any case, the total net force is closely related to the asymptotic structure of the flow. An insight into the homogeneous Navier-Stokes flow of degree (−1), due to Šverák, plays an important role. It would be also worthwhile finding a class of the external force, as large as possible, which ensures the asymptotic expansion of the flow at infinity.

Date: 2018
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DOI: 10.1007/978-3-319-13344-7_6

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