Stationary and time periodic solutionsof the Navier-Stokes equations inexterior domains: a new approach

to open problems

Peter WittwerUniversity of Geneva

(peter.wittwer@unige.ch)

1. Review of some open problems2. New approach for solving such problems3. 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, 1978.G. P. Galdi and A.L. Silvestre. The steady motion of a Navier-Stokes liquid around a rigid body, 2007, 2008.

Guillaume van Baalenand 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 2

2 1

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 MIPUMR CNRS 5640

Université Paul Sabatier (Toulouse 3)

31062 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

Functional framework I

Functional framework II

Existence by contraction mapping principle

Typical asymptotic result

Adaptive boundary conditions

Precision Results for Forces

V. Heuveline et al. 2005, 2007, 2008

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.

THANK YOU !