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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
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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, 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
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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
1 2
2 1
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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 MIPUMR CNRS 5640
Université Paul Sabatier (Toulouse 3)
31062 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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Functional framework I
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Functional framework II
Existence by contraction mapping principle
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Typical asymptotic result
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Adaptive boundary conditions
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Precision Results for Forces
V. Heuveline et al. 2005, 2007, 2008
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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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THANK YOU !