?. ??-n and ?. D. , sont des courbesrégulì eres localement rectilignes en s 1 et s 2 , ? lorsque ? est polygonal, les angles aux sommets autres que s 1 et s 2

. En-effet and . Dans-le, premier cas la régularité H 2 est obtenue en tout point de ??\{s 1 , s 2 }. Dans le second cas, les singularités apparaissant aux sommets tels que ? < ? < 2? sont de même nature que les singularités apparaissant en s ? lorsque 0 < ? ? < ?. Dans ces deux cas

. Probì-eme-de-stokes-leibenson-pour-unécoulementunécoulement-de-hele-shaw, Dans ce chapitre sont présentés les aspects théoriques de l'´ etude réalisée dans les travaux [20, 21]. 4.1 Introduction Les premiers travaux sur le sujet remontentàremontentà 1898 : Hele-Shaw présentà a Londres lors de la session de printemps de la Marine Engineering Society un montage expérimental destinédestinéà montrer les lignes de courant d'un fluide en mouvement autour de divers obstacles. La même année, Stokes prouva que l'´ ecoulement produit par ce montagé etait une solution hydrodynamique auprobì eme de la construction des

@. Lohéac and J. P. , Advective estimates, Mathematical Methods in the Applied Sciences, vol.13, issue.6, pp.453-465, 1990.
DOI : 10.1002/mma.1670130602

@. Lohéac and J. P. , An artifical boundary condition for an advection-diffusion equation, Math. methods appl. sci, vol.1464, pp.155-175, 1991.

@. Lohéac and J. P. , Artifical boundary conditions for advection-diffusion equations. ICIAM/GAMM 95, ZAMM, Progress in partial differential equations: elliptic and parabolic problems, Pitman Res. Notes Math. Ser, vol.26665, pp.207-219, 1992.

@. Lohéac and J. P. , In-and out-flow artificial boundary conditions for advection-diffusion equations, ICIAM/GAMM 95, ZAMM, pp.309-310, 1996.

@. Castro, C. Lohéac, and J. P. , 2002 A characteristic cone formula for the linear wave equation and application to observability. En préparation

F. Alabau and V. Komornik, Observabilit??, contr??labilit?? et stabilisation fronti??re du syst??me d?????lasticit?? lin??aire, Comptes Rendus de l'Acad??mie des Sciences - Series I - Mathematics, vol.324, issue.5, pp.519-524, 1999.
DOI : 10.1016/S0764-4442(99)80383-8

F. Alabau and V. Komornik, Boundary Observability, Controllability, and Stabilization of Linear Elastodynamic Systems, SIAM Journal on Control and Optimization, vol.37, issue.2, pp.521-542, 1999.
DOI : 10.1137/S0363012996313835

R. Almgren, Crystalline Saffman???Taylor Fingers, SIAM Journal on Applied Mathematics, vol.55, issue.6, pp.1511-1535, 1995.
DOI : 10.1137/S0036139993259908

M. Asch and G. Lebeau, Geometrical aspects of exact boundary controllability for the wave equation ? a numerical study. ESAIM: Control, Optimization and Calculus of Variations, pp.163-212, 1998.

C. Bardos, G. Lebeau, and J. Rauch, Sharp Sufficient Conditions for the Observation, Control, and Stabilization of Waves from the Boundary, SIAM Journal on Control and Optimization, vol.30, issue.5, pp.1024-1065, 1992.
DOI : 10.1137/0330055

R. Bey, A. Heminna, and J. P. Lohéac, Stabilisation fronti??re du syst??me de l'??lasticit??. Nouvelle approche, Comptes Rendus de l'Acad??mie des Sciences - Series I - Mathematics, vol.330, issue.7, pp.563-566, 2000.
DOI : 10.1016/S0764-4442(00)00198-1

R. Bey, A. Heminna, and J. P. Lohéac, Stabilisation fronti??re du syst??me de l'??lasticit?? lin??aire anisotrope, Comptes Rendus de l'Acad??mie des Sciences - Series I - Mathematics, vol.333, issue.12, pp.1083-1088, 2001.
DOI : 10.1016/S0764-4442(01)02194-2

R. Bey, A. Heminna, and J. P. Lohéac, Boundary stabilization of the linear elastodynamic system with variable coefficients, Electron. J. differential equations, vol.78, pp.1-23, 2001.

R. Bey, A. Heminna, and J. P. Lohéac, Boundary stabilization of the linear elastodynamic system by a Lyapunov-type method, Revista Matem??tica Complutense, vol.16, issue.2, 2002.
DOI : 10.5209/rev_REMA.2003.v16.n2.16815

R. Bey, J. P. Lohéac, and M. Moussaoui, Singularities of the solution of a mixed problem for a general second order elliptic equation and boundary stabilizationof the wave equation, Journal de Math??matiques Pures et Appliqu??es, vol.78, issue.10, pp.1043-1067, 1999.
DOI : 10.1016/S0021-7824(99)00028-8

R. Bey, J. P. Lohéac, and M. Moussaoui, Stabilisation non linéaire de l'´ equation des ondes, 1998.

R. Bey, J. P. Lohéac, and M. Moussaoui, Nonlinear boundary stabilization of the wave equation . Partial differential equations, theory and numerical solution, CRC Res. Notes Math, vol.406, pp.45-48, 1999.

C. Castro and J. P. Lohéac, 2002 A characteristic cone formula for the linear wave equation and application to observability

X. Chen, The Hele-Shaw problem and area-preserving curve-shortening motions, Archive for Rational Mechanics and Analysis, vol.43, issue.2, pp.117-151, 1993.
DOI : 10.1007/BF00695274

F. Conrad, F. Rao, and B. , Decay of solutions of the wawe equation in a star-shaped domain with non-linear boundary feedback, Asymptotic Analysis, vol.7, pp.159-177, 1993.

A. S. Demidov, A polygonal model for the Hele-Shaw flow, Uspekhi Mat. Nauk, vol.4, pp.195-196, 1998.

A. S. Demidov, Some applications of the Helmholtz-Kirchhoff method (equilibrium plasma in tokamaks, Hele-Shaw flow, and high-frequency asymptotics), Russian Journal of Mathematical Physics, vol.7, issue.2, pp.166-186, 2000.

A. S. Demidov and J. P. Lohéac, 2001 A quasi-contour model of the Stokes-Leibenson problem for Hele-Shaw flows. CNRS UMR 5585 MAPLY. Prépublication 328, 2001.

A. S. Demidov and J. P. Lohéac, Some attractive submanifold in a space of contours for Stokes- Leibenson problem, 2002.

A. S. Demidov and O. A. Vasilieva, The finite point model of the Stokes-Leibenson problem for the Hele-Shaw flow, Fundament. Priklad. Mat, vol.1, pp.67-84, 1999.

V. M. Entov, P. L. Etingof, D. Kleinbock, and . Ya, On nonlinear interface dynamics in Hele-Shaw flows, European Journal of Applied Mathematics, vol.35, issue.05, pp.399-420, 1995.
DOI : 10.1017/S0022112081002632

J. Escher and G. Simonett, Classical Solutions of Multidimensional Hele--Shaw Models, SIAM Journal on Mathematical Analysis, vol.28, issue.5, pp.1028-1047, 1997.
DOI : 10.1137/S0036141095291919

L. Galin, A. 1945 Unsteady filtration with a free surface, Dokl. Akad. Nauk SSSR, vol.47, pp.250-253
URL : https://hal.archives-ouvertes.fr/in2p3-00384381

P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol.24
DOI : 10.1137/1.9781611972030

P. Grisvard and P. , Contrôlabilité exacte des solutions de l'´ equation des ondes en présence de singularités, J. Math. pures et appli, vol.68, pp.215-259, 1989.

A. Guesmia, ON LINEAR ELASTICITY SYSTEMS WITH VARIABLE COEFFICIENTS, Kyushu Journal of Mathematics, vol.52, issue.1, pp.227-248, 1998.
DOI : 10.2206/kyushujm.52.227
URL : https://hal.archives-ouvertes.fr/hal-01282803

A. Guesmia, 1998 Observability, controllability and boundary stabilization of some linear elasticity systems, Acta Scientiarum Mathematicarum, vol.64, pp.109-120

A. Guesmia, Existence globale et stabilisationfrontì ere non lináire d'un système d'´ elasticité, Portugaliae Mathematica, vol.56, pp.361-379, 1999.

A. Guesmia, On the decay estimates for elasticity systems with some localized dissipations, Asymptotic Analysis, vol.22, pp.1-14, 2000.

B. Gustafsson, Applications of Variational Inequalities to a Moving Boundary Problem for Hele Shaw Flows, SIAM Journal on Mathematical Analysis, vol.16, issue.2, pp.279-300, 1985.
DOI : 10.1137/0516021

A. Heibig and M. Moussaoui, Exact controllability of the wave equation for a domain with slits and for mixed boundary conditions, Discrete and Continuous Dynamical Systems, pp.367-386, 1996.

L. F. Ho, Observabilitéfrontì ere de l'´ equation des ondes, C.R. Acad. Sci. Paris, série I Math, vol.302, pp.443-446, 1986.

M. A. Horn, Implications of Sharp Trace Regularity Results on Boundary Stabilization of the System of Linear Elasticity, Journal of Mathematical Analysis and Applications, vol.223, issue.1, pp.126-150
DOI : 10.1006/jmaa.1998.5963

S. D. Howison, Cusp Development in Hele???Shaw Flow with a Free Surface, SIAM Journal on Applied Mathematics, vol.46, issue.1, pp.20-26, 1986.
DOI : 10.1137/0146003

M. Kimura, Time local existence of a moving boundary of the Hele-Shaw flow with suction, European Journal of Applied Mathematics, vol.10, issue.6, pp.581-605, 1999.
DOI : 10.1017/S0956792599003897

J. R. King, A. A. Lacey, and J. L. Vasquez, Persistence of corners in free boundaries in Hele-Shaw flow, European Journal of Applied Mathematics, vol.25, issue.05, pp.455-490, 1995.
DOI : 10.1137/0146003

Y. E. Khokhlov and S. D. Howison, On the classification of solutions to the zero-surfacetension model for Hele-Shaw free boundary flow, Quart. Appl. Math, vol.51, pp.777-789, 1993.

Y. E. Khokhlov, S. D. Howison, C. Huntingford, J. R. Ockendon, and A. A. Lacey, 1994 A model for non-smooth free boundaries in Hele-Shaw flows, Qu. J. Mech. Appl. Math, vol.47, pp.107-128

Y. E. Khokhlov, S. D. Howison, and A. A. Lacey, Some open questions on regularization of ill-posed free boundary problems, FBP News, vol.4, pp.11-12, 1994.

V. Komornik, On the nonlinear boundary stabilization of the wave equation, Chin. Ann. of Math, vol.14, issue.2, pp.153-164, 1993.

V. Komornik, Exact controllability and stabilization; The multiplier method, 1994.

V. Komornik, Stabilisation rapide deprobì emes d'´ evolution linéaires, C.R. Acad. Sci. Paris, série I Math, vol.321, pp.581-586, 1995.

V. Komornik and B. Rao, Stabilisationfrontì ere d'un système d'´ equations des ondes, C.R. Acad. Sci. Paris, série I Math, vol.69, pp.833-838, 1995.

V. Komornik and B. Rao, Boundary stabilization of compactly coupled wave equations, Asymptotic Analysis, vol.14, pp.339-359, 1997.

V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. pures et appli, vol.69, pp.33-54, 1990.

J. E. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation, Journal of Differential Equations, vol.50, issue.2, pp.304-318, 1971.
DOI : 10.1016/0022-0396(83)90073-6

J. E. Lagnese, Boundary Stabilization of Linear Elastodynamic Systems, SIAM Journal on Control and Optimization, vol.21, issue.6, pp.968-984, 1983.
DOI : 10.1137/0321059

J. Lagnese, Uniform asymptotic energy estimates for solutions of the equations of dynamic plane elasticity with nonlinear dissipation at the boundary, Nonlinear Analysis: Theory, Methods & Applications, vol.16, issue.1, pp.35-54, 1991.
DOI : 10.1016/0362-546X(91)90129-O

I. Lasiecka and R. Triggiani, Uniform exponential energy decay of wave equations in a bounded region with L2(0, ???; L2 (??))-feedback control in the Dirichlet boundary conditions, Journal of Differential Equations, vol.66, issue.3, pp.340-390, 1987.
DOI : 10.1016/0022-0396(87)90025-8

G. Lebeau, Propagation des ondes dans les vari??t??s ?? coins, Annales Scientifiques de l?????cole Normale Sup??rieure, vol.30, issue.4, pp.429-497, 1997.
DOI : 10.1016/S0012-9593(97)89928-4

G. Lebeau, The wave equation with oscillating density: observability at low frequency ESAIM: Control, Optimization and Calculus of Variations, pp.219-258, 2000.

G. Lebeau and L. Robbiano, Stabilisation de l'´ equation des ondes par le bord Duke Math, J, vol.86, pp.465-491, 1997.

L. S. Leibenson, Oil producing mechanics, Part II, 1934.

K. Lemrabet, 1987Étude1987´1987Étude de diversprobì emes aux limites de Ventcel d'origine physique ou mécanique dans des domaines non réguliers

J. P. Lohéac, Advective estimates, Mathematical Methods in the Applied Sciences, vol.13, issue.6, pp.453-465, 1990.
DOI : 10.1002/mma.1670130602

J. P. Lohéac, Une condition aux limites artificielle pour unprobì eme de convection-diffusion, 1990.

J. P. Lohéac, An artificial boundary condition for an advection-diffusion equation, Mathematical Methods in the Applied Sciences, vol.1, issue.3, pp.155-175, 1991.
DOI : 10.1002/mma.1670140302

J. P. Lohéac, Artifical boundary conditions for advection-diffusion equations. ICIAM/GAMM 95, ZAMM, Progress in partial differential equations: elliptic and parabolic problems, Pitman Res. Notes Math. Ser, vol.266, pp.207-219, 1992.

J. P. Lohéac, In-and out-flow artificial boundary conditions for advection-diffusion equations, ICIAM/GAMM 95, ZAMM, pp.309-310, 1996.

J. P. Lohéac, F. Nataf, and M. Schatzman, Parabolic Approximations of the Convection-Diffusion Equation, Parabolic approximations of the convectiondiffusion equation, pp.515-530, 1993.
DOI : 10.2307/2153100

B. Merouani, Solutionssingulì eres du système de l'´ elasticité dans un polygône pour différentes conditions aux limites, Maghreb Math. Rev, vol.5, pp.1-2, 1996.

M. Moussaoui, Singularités des solutions duprobì eme mêlé, contrôlabilité exacte et stabilisationfrontì ere, ESAIM Proceedings, ´ Elasticité, Viscoélasticité et Contrôle optimal,Huitì emes Entretiens du Centre Jacques Cartier, pp.157-168, 1996.

N. I. Muskhelishvili, Some problems of the theory of elasticity, 1977.

M. Najafi, G. R. Sarhangi, and H. Wang, The study of the stabilizability of the coupled wave equations under various end conditions, [1992] Proceedings of the 31st IEEE Conference on Decision and Control, 1992.
DOI : 10.1109/CDC.1992.371711

M. T. Niane and O. Seck, Contrôlabilité exacte de l'´ equation des ondes avec conditions mêlées, C.R. Acad. Sci. Paris, série I Math, vol.318, pp.945-948, 1974.

S. Nicaise, Exact Controllability of a Pluridimensional Coupled Problem, Revista Matem??tica Complutense, vol.5, issue.1, pp.91-135, 1992.
DOI : 10.5209/rev_REMA.1992.v5.n1.17940

A. Pazy, 1983 Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathmatical Science, vol.44

P. Polubarinova-kochina and . Ya, On the motion of the oil contour, Dokl. Akad. Nauk SSSR, vol.47, pp.254-257, 1945.

P. Polubarinova-kochina and . Ya, Concerning unsteady motions in the theory of filtration, Prik. Mat. Mech, vol.9, pp.79-90, 1945.

J. P. Puel and E. Zuazua, Exact controllability for some models of multidimensional vibrating structures. Mathematics, climate and environment, RMA: Research Notes in Applied Mathematics, vol.27, pp.288-295, 1991.

J. P. Puel and E. Zuazua, Synopsis, Proc.Royal Soc. Edinburgh, 123 A, pp.323-344, 1993.
DOI : 10.1137/0325029

M. Reissig, The existence and uniqueness of analytic solutions for a moving boundary problem for Hele-Shaw flows in the plane, Nonlinear Analysis: Theory, Methods & Applications, vol.23, issue.5, pp.565-576, 1994.
DOI : 10.1016/0362-546X(94)90237-2

S. Richardson, Hele Shaw flows with a free boundary produced by the injection of fluid into a narrow channel, Journal of Fluid Mechanics, vol.5, issue.04, pp.609-618, 1972.
DOI : 10.1007/BF01535177

P. G. Saffman and G. I. Taylor, The Penetration of a Fluid into a Porous Medium or Hele-Shaw Cell Containing a More Viscous Liquid, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.245, issue.1242, pp.312-329, 1958.
DOI : 10.1098/rspa.1958.0085

E. Shamir, Regularization of mixed second-order elliptic problems, Israel Journal of Mathematics, vol.67, issue.2, pp.150-168, 1968.
DOI : 10.1007/BF02760180

Y. P. Vinogradov and P. P. Kufarev, On a filtration problem, Prik. Mat. Mech, vol.12, pp.181-198, 1948.