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

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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.

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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.

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Main idea, cut problem into two

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Problems in half planes

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

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y x 1 Today’s case (d=2):

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Associated exterior problem y x 2

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Connection between and y x

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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) TOULOUSE Cedex 09, FRANCE

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

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

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Reduction to an evolution system I

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Reduction to an evolution system II

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y x Heuristic aspects x

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Decomposition

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Fourier transform

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Integral equations I

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Integral equations II

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Integral equations III

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Functional framework I

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Functional framework II Existence by contraction mapping principle

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Properties of α-solutions I Bootstrap:

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Properties of α-solutions II

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Uniqueness of α-solutions I

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Uniqueness of α-solutions II

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Typical asymptotic result

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Adaptive boundary conditions

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

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cut I cut II Scaled velocity components

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

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Comparison with Experiment

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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 , 2003.

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THANK YOU !

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