Consider the Navier–Stokes flow in 3-dimensional exterior domains, where a rigid body is translating with prescribed translational velocity $$-\,h(t)u_\infty $$ - h ( t ) u ∞ with constant vector $$u_\infty \in {\mathbb {R}}^3{\setminus }\{0\}$$ u ∞ ∈ R 3 \ { 0 } . Finn raised the question whether his steady solutions are attainable as limits for $$t\rightarrow \infty $$ t → ∞ of unsteady solutions starting from motionless state when $$h(t)=1$$ h ( t ) = 1 after some finite time and $$h(0)=0$$ h ( 0 ) = 0 (starting problem). This was affirmatively solved by Galdi et al. (Arch Ration Mech Anal 138:307–318, 1997) for small $$u_\infty $$ u ∞ . We study some generalized situation in which unsteady solutions start from large motions being in $$L^3$$ L 3 . We then conclude that the steady solutions for small $$u_\infty $$ u ∞ 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 $$h(t)=0$$ h ( t ) = 0 after some finite time and $$h(0)=1$$ h ( 0 ) = 1 (landing problem), is also discussed. In this latter case, the rest state is attainable no matter how large $$u_\infty $$ u ∞ is.
Journal of Mathematical Fluid Mechanics – Springer Journals
Published: Nov 8, 2017
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