Pulsating wave for mean curvature flow in inhomogeneous medium. (English) Zbl 1185.53076

In the present study the authors deal with the mean curvature flow of a hypersurface in a periodic inhomogeneous medium. More precisely, they consider the evolution \(\{\Gamma(t)\subset \mathbb{R}^{n+1}\mid t\geq 0\}\) of an \(n\)-dimensional surface with its motion law given by \[ V_N(p)=H(p)+\delta f(p),\;p\in\Gamma(t),\tag{1} \] where \(V_N\) and \(H\) are the normal velocity and mean curvature of \(\Gamma (t)\), and \(\delta\) is a positive number which measures the strength of the spatial inhomogeneity, represented by \(f:\mathbb{R}^{n+1}\to \mathbb{R}\). Under rather weak assumptions on the data of (1), the authors are able to show for any direction \(\vec\nu\) the existence of a unique speed \(c_\nu\) and a number \(D<\infty\) such that the solution of (1) starting from a plane with normal \(\vec\nu\) stays as a graph over the same plane for all times, and moreover, this graph lies within a distance \(D\) from a plane which has normal \(\vec\nu\) and moves with normal velocity \(c_\nu\). Furthermore, if \(c_\nu\neq 0\), the authors show that pulsating waves exist.


53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations
Full Text: DOI


[1] DOI: 10.1016/j.anihpc.2004.10.009 · Zbl 1135.35092
[2] DOI: 10.1007/s000300050029 · Zbl 0887.35070
[3] Huisken, J. Diff. Geom. 20 pp 237– (1984)
[4] Friedman, Pacific J. Math. 8 pp 201– (1958) · Zbl 0103.06403
[5] Evans, J. Geom. Anal. 2 pp 121– (1992) · Zbl 0768.53003
[6] DOI: 10.1007/978-3-7643-7719-9_13
[7] Evans, J. Differential Geom. 33 pp 635– (1991)
[8] DOI: 10.1090/S0273-0979-1992-00266-5 · Zbl 0755.35015
[9] DOI: 10.1007/BF01232278 · Zbl 0707.53008
[10] Chen, J. Diff. Geom. 33 pp 749– (1991)
[11] DOI: 10.2307/1971452 · Zbl 0696.53036
[12] Dirr, Interfaces Free Bound. 8 pp 79– (2006)
[13] Cahn, SIAM J. Appl. Math. 59 pp 455– (1999)
[14] Dirr, Interfaces Free Bound. 8 pp 47– (2006)
[15] DOI: 10.1002/cpa.10008 · Zbl 1036.49040
[16] Bhattacharya, Interfaces Free Bound. 6 pp 151– (2004)
[17] Bhattacharya, Proc. Royal. Soc. Edin. 133A pp 773– (2003)
[18] Allen, Acta Metall. 27 pp 1084– (1979)
[19] DOI: 10.1002/cpa.20046 · Zbl 1065.49011
[20] Phillips, Crystals, Defects and Microstructures (2001)
[21] Korevaar, Proc. Sympos. Pure Math. Part 2, Amer. Math. Soc. 45 pp 81– (1986)
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.