On the Oseen-type resolvent problem associated with time-periodic flow past a rotating body
- Eiter, Thomas
2020 Mathematics Subject Classification
- 76D07 76U05 47A10 35B10 76D05 35Q30 35A0
- Oseen flow, rotating obstacle, resolvent problem, time-periodic solutions
Consider the time-periodic flow of an incompressible viscous fluid past a body performing a rigid motion with non-zero translational and rotational velocity. We introduce a framework of homogeneous Sobolev spaces that renders the resolvent problem of the associated linear problem well posed on the whole imaginary axis. In contrast to the cases without translation or rotation, the resolvent estimates are merely uniform under additional restrictions, and the existence of time-periodic solutions depends on the ratio of the rotational velocity of the body motion to the angular velocity associated with the time period. Provided that this ratio is a rational number, time-periodic solutions to both the linear and, under suitable smallness conditions, the nonlinear problem can be established. If this ratio is irrational, a counterexample shows that in a special case there is no uniform resolvent estimate and solutions to the time-periodic linear problem do not exist.
- SIAM J. Math. Anal., 54 (2022), pp. 4987--5012, DOI 10.1137/21M1456728 .