On a degenerate parabolic system describing the mean curvature flow of rotationally symmetric closed surfaces

Garcke, Harald; Matioc, Bogdan–Vasile

doi:10.1007/s00028-020-00575-0

Cited by 4 publications

(3 citation statements)

References 32 publications

(63 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To get a further impression of the evolution prescribed by our system of equations (1.1), we study the radial symmetric case as in [14]. The usual mean curvature flow forces a convex, closed surface to shrink to a round point in finite time.…”

Section: Theorem 33 (Decrease Of Surface Area)mentioning

confidence: 99%

Qualitative Properties for a System Coupling Scaled Mean Curvature Flow and Diffusion

Abels¹,

Bürger²,

Garcke³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

We consider a system consisting of a geometric evolution equation for a hypersurface and a parabolic equation on this evolving hypersurface. More precisely, we discuss mean curvature flow scaled with a term that depends on a quantity defined on the surface coupled to a diffusion equation for that quantity. Several properties of solutions are analyzed. Emphasis is placed on to what extent the surface in our setting qualitatively evolves similar as for the usual mean curvature flow. To this end, we show that the surface area is strictly decreasing but give an example of a surface that exists for infinite times nevertheless. Moreover, mean convexity is conserved whereas convexity is not. Finally, we construct an embedded hypersurface that develops a self-intersection in the course of time. Additionally, a formal explanation of how our equations can be interpreted as a gradient flow is included.

show abstract

Section: Theorem 33 (Decrease Of Surface Area)mentioning

confidence: 99%

Qualitative Properties for a System Coupling Scaled Mean Curvature Flow and Diffusion

Abels¹,

Bürger²,

Garcke³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…It is beyond the scope of this paper to prove the existence of a solution to (1.6) with the above regularity. We note, however, that the well-posedness of the corresponding problem, in the case that the curves \Gamma (t) can be written as a graph, was recently studied in [14].…”

Section: Finite Difference Discretizationmentioning

confidence: 99%

Error Analysis for a Finite Difference Scheme for Axisymmetric Mean Curvature Flow of Genus-0 Surfaces

Deckelnick¹,

Nürnberg²

2021

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

We consider a finite difference approximation of mean curvature flow for axisymmetric surfaces of genus zero. A careful treatment of the degeneracy at the axis of rotation for the onedimensional partial differential equation for a parameterization of the generating curve allows us to prove error bounds with respect to discrete L 2 -and H 1 -norms for a fully discrete approximation. The theoretical results are confirmed with the help of numerical convergence experiments. We also present numerical simulations for some genus-0 surfaces, including for a nonembedded self-shrinker for mean curvature flow.

show abstract

“…It is beyond the scope of this paper to prove the existence of a solution to (1.7) with the above regularity. We note, however, that the well-posedness of the corresponding problem, in the case that the curves Γ(t) can be written as a graph, was recently studied in [14].…”

Section: Introductionmentioning

confidence: 99%

Error analysis for a finite difference scheme for axisymmetric mean curvature flow of genus-0 surfaces

Deckelnick,

Nürnberg

2020

Preprint

View full text Add to dashboard Cite

We consider a finite difference approximation of mean curvature flow for axisymmetric surfaces of genus zero. A careful treatment of the degeneracy at the axis of rotation for the one dimensional partial differential equation for a parameterization of the generating curve allows us to prove error bounds with respect to discrete L 2 -and H 1 -norms for a fully discrete approximation. The theoretical results are confirmed with the help of numerical convergence experiments. We also present numerical simulations for some genus-0 surfaces, including for a non-embedded self-shrinker for mean curvature flow.

show abstract

On a degenerate parabolic system describing the mean curvature flow of rotationally symmetric closed surfaces

Cited by 4 publications

References 32 publications

Qualitative Properties for a System Coupling Scaled Mean Curvature Flow and Diffusion

Qualitative Properties for a System Coupling Scaled Mean Curvature Flow and Diffusion

Error Analysis for a Finite Difference Scheme for Axisymmetric Mean Curvature Flow of Genus-0 Surfaces

Error analysis for a finite difference scheme for axisymmetric mean curvature flow of genus-0 surfaces

Contact Info

Product

Resources

About