Hostname: page-component-745bb68f8f-b6zl4 Total loading time: 0 Render date: 2025-01-11T04:11:07.820Z Has data issue: false hasContentIssue false
  • English
  • Français

The Hele-Shaw flow and moduli of holomorphic discs

Part of: Curves Miscellaneous topics of analysis in the complex domain Differential operators in several variables Incompressible viscous fluids

Published online by Cambridge University Press:  18 August 2015

Julius Ross  and
David Witt Nyström
Julius Ross
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, UK email [email protected]
David Witt Nyström
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, UK email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We present a new connection between the Hele-Shaw flow, also known as two-dimensional Laplacian growth, and the theory of holomorphic discs with boundary contained in a totally real submanifold. Using this, we prove short-time existence and uniqueness of the Hele-Shaw flow with varying permeability both when starting from a single point and also when starting from a smooth Jordan domain. Applying the same ideas, we prove that the moduli space of smooth quadrature domains is a smooth manifold whose dimension we also calculate, and we give a local existence theorem for the inverse potential problem in the plane.

Keywords

Hele-Shaw flowLaplacian growthmoduli of holomorphic discshomogeneous complex Monge–Ampère equations

MSC classification

Primary: 76D27: Other free-boundary flows; Hele-Shaw flows 30E05: Moment problems, interpolation problems 14H15: Families, moduli (analytic) 32W20: Complex Monge-Ampère operators
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 12 , December 2015 , pp. 2301 - 2328
Copyright
© The Authors 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aharanov, D. and Shapiro, H. S., Domains in which analytic functions satisfy quadrature identities, J. Anal. Math. 30 (1976), 39–73.CrossRefGoogle Scholar
Antontsev, S. N., Gonçalves, C. R. and Meirmanov, A. M., Local existence of classical solutions to the well-posed Hele-Shaw problem, Port. Math. (N.S.) 59 (2002), 435–452.Google Scholar
Bedford, E. and Kalka, M., Foliations and complex Monge–Ampère equations, Comm. Pure Appl. Math. 30 (1977), 543–571.CrossRefGoogle Scholar
Bishop, E., Differentiable manifolds in complex Euclidean space, Duke Math. J. 32 (1965), 1–22.CrossRefGoogle Scholar
Cherednichenko, V. G., Inverse logarithmic potential problem, Inverse and Ill-posed Problems Series, vol. 5 (VSP, Utrecht, 1996).CrossRefGoogle Scholar
Cho, S. and Pai, S. R., On the regularity of the Riemann mapping function in the plane, Pusan Kyongnam Math. J. 12 (1996), 203–211.Google Scholar
Donaldson, S. K., Holomorphic discs and the complex Monge–Ampère equation, J. Symplectic Geom. 1 (2002), 171–196.CrossRefGoogle Scholar
Elliott, C. M. and Janovsky, V., A variational inequality approach to Hele-Shaw flow with a moving boundary, Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), 93–107.CrossRefGoogle Scholar
Escher, J. and Simonett, G., On Hele-Shaw models with surface tension, Math. Res. Lett. 3 (1996), 467–474.CrossRefGoogle Scholar
Escher, J. and Simonett, G., Classical solutions for Hele-Shaw models with surface tension, Adv. Differential Equations 2 (1997), 619–642.CrossRefGoogle Scholar
Escher, J. and Simonett, G., Classical solutions of multidimensional Hele-Shaw models, SIAM J. Math. Anal. 28 (1997), 1028–1047.CrossRefGoogle Scholar
Forstnerič, F., Analytic disks with boundaries in a maximal real submanifold of ℂ2, Ann. Inst. Fourier (Grenoble) 37 (1987), 1–44.CrossRefGoogle Scholar
Gakhov, F. D., Boundary value problems (Pergamon Press, Oxford–New York, 1966).CrossRefGoogle Scholar
Gromov, M., Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307–347.CrossRefGoogle Scholar
Gustafsson, B., Quadrature identities and the Schottky double, Acta Appl. Math. 1 (1983), 209–240.CrossRefGoogle Scholar
Gustafsson, B., On a differential equation arising in a Hele-Shaw flow moving boundary problem, Ark. Mat. 22 (1984), 251–268.CrossRefGoogle Scholar
Gustafsson, B., Applications of variational inequalities to a moving boundary problem for Hele-Shaw flows, SIAM J. Math. Anal. 16 (1985), 279–300.CrossRefGoogle Scholar
Gustafsson, B. and Putinar, M., Selected topics on quadrature domains, Physica D 235 (2007), 90–100.CrossRefGoogle Scholar
Gustafsson, B. and Vasil’ev, A., Conformal and potential analysis in Hele-Shaw cells, Advances in Mathematical Fluid Mechanics (Birkhäuser, Basel, 2006).Google Scholar
Hanzawa, E., Classical solutions of the Stefan problem, Tohoku Math. J. (2) 33 (1981), 297–335.CrossRefGoogle Scholar
Hedenmalm, H. and Olofsson, A., Hele-Shaw flow on weakly hyperbolic surfaces, Indiana Univ. Math. J. 54 (2005), 1161–1180.CrossRefGoogle Scholar
Hedenmalm, H. and Shimorin, S., Hele-Shaw flow on hyperbolic surfaces, J. Math. Pures Appl. (9) 81 (2002), 187–222.CrossRefGoogle Scholar
Isakov, V., Inverse source problems, Mathematical Surveys and Monographs, vol. 34 (American Mathematical Society, Providence, RI, 1990).CrossRefGoogle Scholar
Khavinson, D., Mineev-Weinstein, M. and Putinar, M., Planar elliptic growth, Complex Anal. Oper. Theory 3 (2009), 425–451.CrossRefGoogle Scholar
LeBrun, C., Twistors, holomorphic discs, and Riemann surfaces with boundary, in Perspectives in Riemannian geometry, CRM Proceedings and Lecture Notes, vol. 40 (American Mathematical Society, Providence, RI, 2006), 209–221.CrossRefGoogle Scholar
LeBrun, C. and Mason, L. J., Zoll manifolds and complex surfaces, J. Differential Geom. 61 (2002), 453–535.CrossRefGoogle Scholar
Lin, Y.-L., Perturbation theorems for Hele-Shaw flows and their applications, Ark. Mat. 49 (2011), 357–382.CrossRefGoogle Scholar
McDuff, D. and Salamon, D., J-holomorphic curves and symplectic topology, American Mathematical Society Colloquium Publications, vol. 52 (American Mathematical Society, Providence, RI, 2004).CrossRefGoogle Scholar
Reissig, M. and von Wolfersdorf, L., A simplified proof for a moving boundary problem for Hele-Shaw flows in the plane, Ark. Mat. 31 (1993), 101–116.CrossRefGoogle Scholar
Richardson, S., Hele-Shaw flows with a free boundary produced by the injection of fluid into a narrow channel, J. Fluid Mech. 56 (1972), 609–618.CrossRefGoogle Scholar
Sakai, M., Quadrature domains, Lecture Notes in Mathematics, vol. 934 (Springer, New York, 1982).CrossRefGoogle Scholar
Sakai, M., Regularity of boundaries having a Schwarz function, Acta Math. 166 (1991), 263–297.CrossRefGoogle Scholar
Sakai, M., Regularities of boundaries in two dimensions, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 20 (1993), 323–339.Google Scholar
Sakai, M., Application of variational inequalities to the existence theorem on quadrature domains, Trans. Amer. Math. Soc. 276 (1993), 267–279.CrossRefGoogle Scholar
Semmes, S., Complex Monge–Ampère and symplectic manifolds, Amer. J. Math. 114 (1992), 495–550.CrossRefGoogle Scholar
Shapiro, H. S., The Schwarz function and its generalization to higher dimensions, University of Arkansas Lecture Notes in the Mathematical Sciences, vol. 9 (John Wiley, New York, 1992).Google Scholar
Tian, F. R., A Cauchy integral approach to Hele-Shaw problems with a free boundary, Arch. Ration. Mech. Anal. 135 (1996), 175–196.CrossRefGoogle Scholar
Tian, F. R., Hele-Shaw problems in multidimensional spaces, J. Nonlinear Sci. 10 (2000), 275–290.CrossRefGoogle Scholar
Vinogradov, Y. P. and Kufarev, P. P., On a problem of filtration, Akad. Nauk SSSR Prikl. Mat. Meh. 12 (1948), 181–198; (in Russian).Google Scholar
Wiegmann, P. B. and Zabrodin, A., Conformal maps and dispersionless integrable hierarchies, Preprint (1999), arXiv:hep-th/9909147.Google Scholar