Abstract
Consider the Navier–Stokes flow in 3-dimensional exterior domains, where a rigid body is translating with prescribed translational velocity \(-\,h(t)u_\infty \) with constant vector \(u_\infty \in {\mathbb {R}}^3{\setminus }\{0\}\). Finn raised the question whether his steady solutions are attainable as limits for \(t\rightarrow \infty \) of unsteady solutions starting from motionless state when \(h(t)=1\) after some finite time and \(h(0)=0\) (starting problem). This was affirmatively solved by Galdi et al. (Arch Ration Mech Anal 138:307–318, 1997) for small \(u_\infty \). We study some generalized situation in which unsteady solutions start from large motions being in \(L^3\). We then conclude that the steady solutions for small \(u_\infty \) are still attainable as limits of evolution of those fluid motions which are found as a sort of weak solutions. The opposite situation, in which \(h(t)=0\) after some finite time and \(h(0)=1\) (landing problem), is also discussed. In this latter case, the rest state is attainable no matter how large \(u_\infty \) is.
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Hishida, T., Maremonti, P. Navier–Stokes Flow Past a Rigid Body: Attainability of Steady Solutions as Limits of Unsteady Weak Solutions, Starting and Landing Cases. J. Math. Fluid Mech. 20, 771–800 (2018). https://doi.org/10.1007/s00021-017-0344-3
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DOI: https://doi.org/10.1007/s00021-017-0344-3
Keywords
- Navier–Stokes flow
- exterior domain
- starting problem
- landing problem
- steady flow
- attainability
- Oseen semigroup