scholarly journals Existence of weak solution for mean curvature flow with transport term and forcing term

2020 ◽  
Vol 19 (5) ◽  
pp. 2655-2677
Author(s):  
Keisuke Takasao ◽  
Keyword(s):  
Weak Solution ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Forcing Term ◽  
Transport Term

NUMERICAL EVIDENCE OF FATTENING FOR THE MEAN CURVATURE FLOW

1996 ◽  
Vol 06 (06) ◽  
pp. 793-813 ◽  
Author(s):  
FRANCESCA FIERRO ◽  
MAURIZIO PAOLINI
Keyword(s):  
Numerical Simulations ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Normal Direction ◽  
Compact Surface ◽  
Principal Curvatures ◽  
Forcing Term ◽  
The Mean ◽  
Motion By Mean Curvature

In this paper we describe some numerical simulations in the context of mean curvature flow. We recover a few different approaches in modeling the evolution of an interface Σ which evolves according to the law: V=κ+g where V is the velocity in the inward normal direction, κ is the sum of the principal curvatures and g is a given forcing term. We will discuss about the phenomenon of fattening or nonuniqueness of the solution, recalling what is known about this subject. Finally we show some interesting numerical simulations that suggest evidence of fattening starting from different initial interfaces. Of particular interest is the result obtained for a torus in ℝ4 which would be a first example of a regular and compact surface showing evidence of fattening in the case of pure motion by mean curvature (no forcing term).

scholarly journals On the existence of mean curvature flow with transport term

2010 ◽  
pp. 251-277 ◽  
Author(s):  
Chun Liu ◽  
Norifumi Sato ◽  
Yoshihiro Tonegawa
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Transport Term

scholarly journals Stationary sets of the mean curvature flow with a forcing term

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Vesa Julin ◽  
Joonas Niinikoski
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Higher Dimensions ◽  
Curvature Flow ◽  
Planar Case ◽  
Forcing Term ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
The Mean ◽  
Flat Flow

Abstract We consider the flat flow solution to the mean curvature equation with forcing in ℝ n {\mathbb{R}^{n}} . Our main result states that tangential balls in ℝ n {\mathbb{R}^{n}} under a flat flow with a bounded forcing term will experience fattening, which generalizes the result in [N. Fusco, V. Julin and M. Morini, Stationary sets and asymptotic behavior of the mean curvature flow with forcing in the plane, preprint 2020, https://arxiv.org/abs/2004.07734] from the planar case to higher dimensions. Then, as in the planar case, we characterize stationary sets in ℝ n {\mathbb{R}^{n}} for a constant forcing term as finite unions of equisize balls with mutually positive distance.

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