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Peter Wittwer University of Geneva (peter.wittwer@unige.ch)

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

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Peter Wittwer University of Geneva (peter.wittwer@unige.ch)

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  1. Stationary and time periodic solutionsof the Navier-Stokes equations inexterior domains: a new approachto open problems Review of some open problems New approach for solving such problems Importance of results for modeling Peter WittwerUniversity of Geneva(peter.wittwer@unige.ch)

  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. Guillaume van Baalenand P.W. Time periodic problem (d=3): Dept. of Mathematics and StatisticsBoston University 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.

  7. Today’s case (d=2): y x 1

  8. Associated exterior problem y x 2

  9. Connection between and 2 1 y x 2 1

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

  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 initial data convert stationary (or time periodic) equations into evolution systems

  13. Reduction to an evolution system I

  14. Reduction to an evolution system II

  15. Heuristic aspects y x x

  16. Decomposition

  17. Fourier transform

  18. Integral equations I

  19. Integral equations II

  20. Functional framework I

  21. Functional framework II Existence by contraction mapping principle

  22. Typical asymptotic result

  23. Adaptive boundary conditions

  24. Precision Results for Forces V. Heuvelineet al. 2005, 2007, 2008

  25. Importance of results for modeling References: Institute of Thermal-Fluid DynamicsRoma, Italy. F. Takemura, J. MagnaudetThe transverse force on clean and contaminated bubbles rising near a vertical wall at moderate Reynoldsnumber Journal of Fluid Mechanics 495, pp 235-253, 2003.

  26. THANK YOU !

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