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

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.

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.

Main idea, cut problem into two

Problems in half planes

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

y x 1 Today’s case (d=2):

Associated exterior problem y x 2

Connection between and y x

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

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

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

Reduction to an evolution system I

Reduction to an evolution system II

y x Heuristic aspects x

Decomposition

Fourier transform

Integral equations I

Integral equations II

Integral equations III

Functional framework I

Functional framework II Existence by contraction mapping principle

Properties of α-solutions I Bootstrap:

Properties of α-solutions II

Uniqueness of α-solutions I

Uniqueness of α-solutions II

Typical asymptotic result

Adaptive boundary conditions

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

cut I cut II Scaled velocity components

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

Comparison with Experiment

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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