In this paper we study a mathematical model for the motion of a Stokesian fluid in a Hele-Shaw cell surrounded by a gas at uniform pressure. The model is based on a non-Newtonian version of Darcy’s law for the bulk fluid, as suggested in [9,12]. Besides a general existence and uniqueness result for classical solutions, it is also shown that classical solutions exist globally and tend to circles exponentially fast, provided the initial data is sufficiently close to a circle. Finally, our analysis discloses the influence of surface tension and the effective viscosity on the rate of convergence.
@article{ASNSP_2010_5_9_2_325_0,
author = {Escher, Joachim and Matioc, Anca-Voichita and Matioc, Bogdan-Vasile},
title = {Classical solutions and stability results for {Stokesian} {Hele-Shaw} flows},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {325--349},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 9},
number = {2},
year = {2010},
mrnumber = {2731159},
zbl = {1202.35028},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2010_5_9_2_325_0/}
}
TY - JOUR AU - Escher, Joachim AU - Matioc, Anca-Voichita AU - Matioc, Bogdan-Vasile TI - Classical solutions and stability results for Stokesian Hele-Shaw flows JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2010 SP - 325 EP - 349 VL - 9 IS - 2 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2010_5_9_2_325_0/ LA - en ID - ASNSP_2010_5_9_2_325_0 ER -
%0 Journal Article %A Escher, Joachim %A Matioc, Anca-Voichita %A Matioc, Bogdan-Vasile %T Classical solutions and stability results for Stokesian Hele-Shaw flows %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2010 %P 325-349 %V 9 %N 2 %I Scuola Normale Superiore, Pisa %U http://www.numdam.org/item/ASNSP_2010_5_9_2_325_0/ %G en %F ASNSP_2010_5_9_2_325_0
Escher, Joachim; Matioc, Anca-Voichita; Matioc, Bogdan-Vasile. Classical solutions and stability results for Stokesian Hele-Shaw flows. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 2, pp. 325-349. http://www.numdam.org/item/ASNSP_2010_5_9_2_325_0/
[1] , “Linear and Quasilinear Parabolic Problems", Volume I, Birkhäuser, Basel, 1995. | MR
[2] – , Operator-valued Fourier multipliers on periodic Besov spaces and applications, Proc. Edinb. Math. Soc. (2) 47 (2004), 15–33. | MR | Zbl
[3] and , -Theory for a class of non-Newtonian fluids, SIAM J. Math. Anal. 39 (2007), 379-421. | MR | Zbl
[4] and , Stability, instability, and center manifold theorem for fully nonlinear autonomous parabolic equations in Banach space, Arch. Ration. Mech. Anal. 101 (1988), 115–141. | MR | Zbl
[5] and , A moving boundary problem for periodic Stokesian Hele-Shaw flows, Interfaces Free Bound. 11 (2009), 119–137. | MR | Zbl
[6] and , Multidimensional Hele-Shaw flows modeling Stokesian fluids, Math. Methods Appl. Sci. 32 (2009), 577–593. | MR | Zbl
[7] and , Analyticity of the interface in a free boundary problem, Math. Ann. 305 (1996), 435–459. | EuDML | MR | Zbl
[8] and , A center manifold analysis for the Mullins-Sekerka model, J. Differential Equations 143 (1998), 267–292. | MR | Zbl
[9] , , and , Pattern formation in non-Newtonian Hele-Shaw flow, Phys. Fluids 13 (2001), 1191–1212. | MR | Zbl
[10] and , “Elliptic Partial Differential Equations of Second Order", Springer-Verlag, New York, 2001. | MR | Zbl
[11] , “Perturbation Theory for Linear Operators", Springer-Verlag, Berlin Heidelberg, 1995. | MR | Zbl
[12] , and , Models of non-Newtonian Hele-Shaw flow, Phys. Rev. E (5) 54 (1996), R4536–R4539.
[13] , “The Mathematical Theory of Viscous Incompressible Flow", Gordon and Beach, New York, 1969. | MR | Zbl
[14] and ,“Linear and Quasilinear Elliptic Equations”, Academic Press, New York, 1968. | MR
[15] , “Analytic Semigroups and Optimal Regularity in Parabolic Problems", Birkhäuser, Basel, 1995. | MR | Zbl
[16] and , “Topics in Fourier Analysis and Function Spaces", John Wiley and Sons Limited, New York, 1987. | MR | Zbl
[17] , and , “Introduction to Fluid Mechanics", Oxford University Press, New York, 2005.
[18] , and , Hele-Shaw flow and pattern formation in a time-dependent gap, Nonlinearity 10 (1997), 1471–1495. | MR | Zbl
[19] , Center manifolds for quasilinear reaction-diffusion systems, Differential Integral Equations (4) 8 (1995), 753–796. | MR | Zbl
[20] , On the abstract Cauchy problem of parabolic type in spaces of continuous functions, J. Math. Anal. Appl. 107 (1985), 16–66. | MR | Zbl
[21] , “Nonlinear Differential Equations and Dynamical Systems", Springer-Verlag, Berlin Heidelberg, 1990. | MR | Zbl
