Peter Wittwer University of Geneva ([email protected])
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Stationary and time periodic solutionsof the Navier-Stokes equations inexterior domains: a new approach
to open problems
Peter WittwerUniversity of Geneva
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.
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