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Attainability of a stationary Navier-Stokes flow around a moving rigid body

髙橋, 知希 名古屋大学

2022.05.19

概要

We consider the large time behavior of a viscous incompressible flow around a moving rigid body O ⊂ R n (n ≥ 3). Specifically, if the rigid body O ⊂ R 3 translates with a prescribed constant velocity, we then expect from the physical point of view that solutions to the NavierStokes equation possess anisotropic decay structure at spatial infinity. In a series of his celebrated papers, see for instance [17–20], Finn succeeded in constructing a stationary solution, termed by him physically reasonable solution, that exhibits a paraboloidal wake region behind the body, and developed the theory of stationary Navier-Stokes problem in exterior domains. For even better understanding, in [18, Section 6], he raised a question relating to the convergence of nonstationary solutions to stationary solutions. This is well known as Finn’s starting problem; to be precise, suppose both a rigid body and the fluid filling the outside of the body are initially at rest and the body starts to translate with a velocity which gradually increases and is maintained after a certain finite time, does a nonstationary flow then converge to a stationary solution corresponding to a terminal velocity of the body as time goes to infinity? If the answer is affirmative, the stationary solution is said to be attainable by following Heywood [29, Section 6], who first studied Finn’s starting problem but gave a partial answer merely in a special situation that the net force exerted by the fluid on the body is identically zero, yielding the square summability of stationary solutions. Since stationary solutions do not belong to L 2 space in general, energy methods are far from enough to analyze the starting problem and we do need L q framework. Thus the problem had remained open until Kobayashi and Shibata [39] developed the L q theory of the linearized problem, which is called the Oseen problem. By making use of estimates established in [39] via Kato’s approach [37], Finn’s starting problem was affirmatively solved by Galdi, Heywood and Shibata [27] for small terminal velocity of the body. Nevertheless, convergence rates deduced by them were the same as those in stability analysis (Shibata [45]), and they can be improved as we will clarify in this thesis. The concept of attainability is somewhat similar to, however, different from stability. Analysis of attainability is more difficult because the equation is non-autonomous and because one has to deal with several delicate terms. Since the other rigid motion is rotation, it should be even more challenging to study Finn’s starting problem above in which translation is replaced by rotation. This problem was proposed by Hishida [31], but it has remained open since there seems to be no chance to avoid the non-autonomous character unlike the translational case. If a stationary solution is attainable, then it is interesting to compare convergence rates with those in the translational regime. This issue is closely related to asymptotic structure of stationary solutions at infinity.

The objective of this thesis is two-fold. The first one is to develop further analysis of Finn’s starting problem with translation to derive new convergence rates which improve [27].

Moreover, we extend this result to the case of higher dimensions. Such generalization is never obvious because our knowledge about stationary solutions in higher dimensions is quite less than in 3D case. Therefore, analysis of the stationary problem must be an important step. The second objective is to prove attainability of a stationary solution around a rigid body rotating from rest.

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参考文献

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