1. Stationary and time periodic solutions of the Navier-Stokes equations in exterior domains: a new approach to open problems Peter Wittwer University of Geneva (peter.wittwer@unige.ch) 1. Review of some open problems 2. New approach for solving such problems 3. Importance of results for modeling

2. Main open problem (d=2): G. P. Galdi. Handbook of differential equations, stationary partial differential equations, Vol. 1, M. Chipot, P. Quittner ed., Elsevier 2004.

3. Less difficult problem (d=2): G. P. Galdi. Handbook of differential equations, stationary partial differential equations, Vol. 1, M. Chipot, P. Quittner ed., Elsevier 2004.

4. Main idea, cut problem into two

5. Problems in half planes

6. Time periodic problem (d=3): Associated exterior problem H. F. Weinberger. On the steady fall of a body in a Navier- Stokes fluid, 1978. G. P. Galdi and A.L. Silvestre. The steady motion of a Navier- Stokes liquid around a rigid body, 2007, 2008. Guillaume van Baalen and P.W. Dept. of Mathematics and Statistics Boston University

7. y x 1 Today’s case (d=2):

8. Associated exterior problem y x 2

9. Connection between and y x 1 2 21

10. 1. Show existence of weak solutions for (2) 2. Provides weak solutions for (1) 3. Show existence of strong solutions for (1) (for small data) 4. Show a weak-strong uniqueness result for (1) (for small data) Strategy : Matthieu Hillairet and P.W. 2007, 2008, 2009 Laboratoire MIP UMR CNRS 5640 Université Paul Sabatier (Toulouse 3) 31062 TOULOUSE Cedex 09, FRANCE

11. Result for today's case Theorem For all sufficiently small there exists a solution The solution is unique in

12. Method of proof: y = time convert stationary (or time periodic) equations into evolution systems initial data

13. Reduction to an evolution system I

14. Reduction to an evolution system II

15. y x Heuristic aspects x

16. Decomposition

17. Fourier transform

18. Integral equations I

19. Integral equations II

20. Integral equations III

21. Functional framework I

22. Functional framework II Existence by contraction mapping principle

23. Properties of α-solutions I Bootstrap:

24. Properties of α-solutions II

25. Uniqueness of α-solutions I

26. Uniqueness of α-solutions II

27. Typical asymptotic result

28. Adaptive boundary conditions

29. Streamlines V. Heuveline et al. 2005, 2007, 2008

30. cut I cut II Scaled velocity components

31. Precision Results for Forces V. Heuveline et al. 2005, 2007, 2008

32. Comparison with Experiment

33. Importance of results for modeling References: Institute of Thermal-Fluid Dynamics Roma, Italy. F. Takemura, J. Magnaudet The transverse force on clean and contaminated bubbles rising near a vertical wall at moderate Reynolds number Journal of Fluid Mechanics 495, pp 235-253, 2003.

34. THANK YOU !