We prove the existence of unique solutions for the 3D incompressible Navier-Stokes equations in an exterior domain with small boundary data which do not necessarily decay in time. As a corollary, the existence of unique small time-periodic solutions is shown. We next show that the spatial asymptotics of the periodic solution is given by the same Landau solution at all times. Lastly we show that if the boundary datum is time-periodic and the initial datum is asymptotically self-similar, then the solution converges to the sum of a time-periodic vector field and a forward self-similar vector field as time goes to infinity.
Bibliographical noteFunding Information:
We thank Professor C.-C. Chen for providing the references [1, 34] for (5.36), and Professors Y. Giga and Y. Maekawa for discussing Lemma 2.2. We also thank the referees for several valuable suggestions. Part of this work was done when all of us visited the National Center for Theoretical Sciences (Taipei Office) and the Department of Mathematics, National Taiwan University, when both Kang and Miura visited the University of British Columbia, and when Tsai visited Sungkyunkwan University and Taida Institute for Mathematical Sciences. We would like to thank the kind hospitality of these institutions and Professors D. Chae, C.-C. Chen, S. Gustafson, J. Lee, C.-S. Lin and C.-L. Wang. The research of Kang is partly supported by KRF-2008-331-C00024 and R01-2008-000-11008-0. The research of Miura was partly supported by the JSPS grant no. 191437. The research of Tsai is partly supported by Natural Sciences and Engineering Research Council of Canada, grant no. 261356-08.
All Science Journal Classification (ASJC) codes
- Applied Mathematics