zbMATH — the first resource for mathematics

The Hele–Shaw problem and area-preserving curve-shortening motions. (English) Zbl 0780.35117
The author studies the following Hele-Shaw problem: Given a simply connected curve in a smooth bounded domain, find the motion of the curve such that its normal velocity equals the normal derivative jump of a function which is harmonic in the complement of the curve in the domain and whose boundary value on the curve equals its curvature. It is shown that a weak solution exists locally in time; if the initial curve is close to a circle, global existence follows. In the latter case, it is proved that the global solution converges to a circle exponentially fast.
The main new idea in the existence proof is to regularize the boundary condition involving the normal velocity of the curve by a term involving the second derivative of the curvature with respect to the arc length. For the regularized system existence can be shown using Schauder’s fixed point theorem. Then a priori estimates and compactness arguments yield the existence of a sequence of regularized solutions that converge to a solution of the original Hele-Shaw problem.
Reviewer: J.Sprekels (Essen)

35R35 Free boundary problems for PDEs
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
35R30 Inverse problems for PDEs
35B45 A priori estimates in context of PDEs
Full Text: DOI
[1] N. Alikakos &G. Fusco,Slow dynamics of spherical fronts for the Cahn-Hilliard equation, preprint. · Zbl 0851.35065
[2] F. J. Almgren &L. Wang,Mathematical Existence of crystal growth with Gibbs-Thomson curvature effects, Informal Lecture notes for the workshop on Evolving Phase Boundaries, at Carnegie Mellon University, March 1991.
[3] G. Caginalp,Stefan and Hele-Shaw type models as asymptotic limits of the phase field equations, Phys. Rev. A39 (1989), 5887-5896. · Zbl 1027.80505
[4] Y. G. Chen, Y. Giga &S. Goto,Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Diff. Geom.33 (1991), 749-786. · Zbl 0696.35087
[5] J. Duchon &R. Robert,Évolution d’une interface par capillarité et diffusion de volume I. Existence locale en temps, Ann. Inst. H. Poincaré, Analyse non linéaire1 (1984), 361-378. · Zbl 0572.35051
[6] C. M. Elliott &J. Ockendon,Weak and Variational Methods for Moving Boundary Problems, Pitman, Boston, 1982. · Zbl 0476.35080
[7] L. C. Evans &J. Spruck,Motion of level set by mean curvature I, J. Diff. Geom.33 (1991), 635-681. · Zbl 0726.53029
[8] M. Gage &R. Hamilton,The shrinking of convex plane curves by heat equation, J. Diff. Geom.23 (1986), 69-96. · Zbl 0621.53001
[9] G. Gilbarg &N. S. Trudinger,Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, 1983. · Zbl 0562.35001
[10] M. Grayson,The heat equation shrinks embedded plane curves to points, J. Diff. Geom.26 (1987), 285-314. · Zbl 0667.53001
[11] R. S. Hamilton,Three manifolds with positive Ricci curvature, J. Diff. Geom.17 (1982), 255-306. · Zbl 0504.53034
[12] G. Huisken,Flow by mean curvature of convex surface into point. J. Diff. Geom.20 (1984), 237-266. · Zbl 0556.53001
[13] D. S. Jerison &C. E. Kenig,The Neumann problem on Lipschitz domains, Bull. Amer. Math. Soc.4 (1981), 203-207. · Zbl 0471.35026
[14] C. E. Kenig,Elliptic boundary value problems on Lipschitz domains, in Beijing Lectures in Harmonic Analysis (E. M. Stein, ed.) Princeton, New Jersey, 1986, 131-184.
[15] J. L. Lions &E. Magenes,Non-homogeneous Boundary Value Problems and Applications, Vol. II, Springer-Verlag, 1972.
[16] S. Luckhaus,Solutions for the two-phase Stefan problem with Gibbs-Thomson law for the melting temperature, Euro. J. Appl. Math.1 (1990), 101-111. · Zbl 0734.35159
[17] L. E. Payne &H. F. Weinberger,New bounds in harmonic and biharmonic problems, J. Math. Phys.33 (1954), 291-307. · Zbl 0064.09903
[18] R. L. Pego,Front migration in the nonlinear Cahn-Hilliard equation, Proc. R. Soc. Lond. A422 (1989), 261-278. · Zbl 0701.35159
[19] E. Radkevitch,The Gibbs-Thomson correction and conditions for the classical solution of the modified Stefan problem, Sov. Math. Doklady43 (1991), 274-278.
[20] H. M. Soner,Motion of a set by the curvature of its boundary, Preprint. · Zbl 0769.35070
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.