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

Dirr, N.; Karali, G.; Yip, N. K.

Eur. J. Appl. Math. 19, No. 6, 661-699 (2008).

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.

Reviewer: Messoud A. Efendiev (Berlin)

MSC:

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

Keywords:

mean curvature flow; inhomogeneous medians; pulsating wave

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