scholarly journals Remarks on large time behavior of level-set mean curvature flow equations with driving and source terms

2020 ◽  
Vol 25 (10) ◽  
pp. 3983-3999
Author(s):  
Yoshikazu Giga ◽  
◽  
Hiroyoshi Mitake ◽  
Hung V. Tran ◽  
Keyword(s):  
Level Set ◽  
Mean Curvature ◽  
Large Time ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Large Time Behavior ◽  
Time Behavior ◽  
Source Terms ◽  
Flow Equations

scholarly journals Existence of martingale solutions and large-time behavior for a stochastic mean curvature flow of graphs

2020 ◽  
Author(s):  
Nils Dabrock ◽  
Martina Hofmanová ◽  
Matthias Röger
Keyword(s):  
Mean Curvature ◽  
Large Time ◽  
Mean Curvature Flow ◽  
Stochastic Perturbation ◽  
Curvature Flow ◽  
Large Time Behavior ◽  
Time Behavior ◽  
Viscosity Sense ◽  
Stochastic Mean Curvature Flow ◽  
Martingale Solutions

Abstract We are concerned with a stochastic mean curvature flow of graphs over a periodic domain of any space dimension. For the first time, we are able to construct martingale solutions which satisfy the equation pointwise and not only in a generalized (distributional or viscosity) sense. Moreover, we study their large-time behavior. Our analysis is based on a viscous approximation and new global bounds, namely, an $$L^{\infty }_{\omega ,x,t}$$ L ω , x , t ∞ estimate for the gradient and an $$L^{2}_{\omega ,x,t}$$ L ω , x , t 2 bound for the Hessian. The proof makes essential use of the delicate interplay between the deterministic mean curvature part and the stochastic perturbation, which permits to show that certain gradient-dependent energies are supermartingales. Our energy bounds in particular imply that solutions become asymptotically spatially homogeneous and approach a Brownian motion perturbed by a random constant.

scholarly journals On obstacle problem for mean curvature flow with driving force

2019 ◽  
Vol 4 (1) ◽  
pp. 9-29
Author(s):  
Yoshikazu Giga ◽  
Hung V. Tran ◽  
Longjie Zhang
Keyword(s):  
Mean Curvature ◽  
Driving Force ◽  
Obstacle Problem ◽  
Mean Curvature Flow ◽  
Boundary Regularity ◽  
Curvature Flow ◽  
Convergence Result ◽  
Large Time Behavior ◽  
Time Behavior ◽  
Full Characterization

Abstract In this paper, we study an obstacle problem associated with the mean curvature flow with constant driving force. Our first main result concerns interior and boundary regularity of the solution. We then study in details the large time behavior of the solution and obtain the convergence result. In particular, we give full characterization of the limiting profiles in the radially symmetric setting.

Computation of geometric partial differential equations and mean curvature flow

Acta Numerica ◽  
2005 ◽  
Vol 14 ◽  
pp. 139-232 ◽  
Author(s):  
Klaus Deckelnick ◽  
Gerhard Dziuk ◽  
Charles M. Elliott
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Level Set ◽  
Phase Field ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Partial Differential ◽  
Geometric Partial Differential Equations ◽  
Geometric Pdes

This review concerns the computation of curvature-dependent interface motion governed by geometric partial differential equations. The canonical problem of mean curvature flow is that of finding a surface which evolves so that, at every point on the surface, the normal velocity is given by the mean curvature. In recent years the interest in geometric PDEs involving curvature has burgeoned. Examples of applications are, amongst others, the motion of grain boundaries in alloys, phase transitions and image processing. The methods of analysis, discretization and numerical analysis depend on how the surface is represented. The simplest approach is when the surface is a graph over a base domain. This is an example of a sharp interface approach which, in the general parametric approach, involves seeking a parametrization of the surface over a base surface, such as a sphere. On the other hand an interface can be represented implicitly as a level surface of a function, and this idea gives rise to the so-called level set method. Another implicit approach is the phase field method, which approximates the interface by a zero level set of a phase field satisfying a PDE depending on a new parameter. Each approach has its own advantages and disadvantages. In the article we describe the mathematical formulations of these approaches and their discretizations. Algorithms are set out for each approach, convergence results are given and are supported by computational results and numerous graphical figures. Besides mean curvature flow, the topics of anisotropy and the higher order geometric PDEs for Willmore flow and surface diffusion are covered.

scholarly journals Planar Waves of the Buffered Bistable System

2013 ◽  
Vol 2013 ◽  
pp. 1-13
Author(s):  
Xiaohuan Wang ◽  
Guangying Lv
Keyword(s):  
Large Time ◽  
Mean Curvature Flow ◽  
Large Time Behavior ◽  
Bistable System ◽  
Time Behavior ◽  
Almost Periodic ◽  
Drift Term ◽  
Planar Front ◽  
Special Cases ◽  
The Mean

This paper is concerned with the large time behavior of disturbed planar fronts in the buffered bistable system inℝn(n≥2). We first show that the large time behavior of the disturbed fronts can be approximated by that of the mean curvature flow with a drift term for all large time up tot=+∞. And then we prove that the planar front is asymptotically stable inL∞(ℝn)under ergodic perturbations, which include quasiperiodic and almost periodic ones as special cases.

scholarly journals Anisotropic mean curvature flow of Lipschitz graphs and convergence to self-similar solutions

2021 ◽  
Author(s):  
Annalisa Cesaroni ◽  
Heiko Kröner ◽  
Matteo Novaga
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Time Behavior ◽  
Long Time ◽  
Self Similar ◽  
The Stability ◽  
Lipschitz Graphs ◽  
Similar Solutions ◽  
Anisotropic Mean Curvature

We consider the anisotropic mean curvature flow of entire Lipschitz graphs. We prove existence and uniqueness of expanding self-similar solutions which are asymptotic to a prescribed cone, and we characterize the long time behavior of solutions, after suitable rescaling, when the initial datum is a sublinear perturbation of a cone. In the case of regular anisotropies, we prove the stability of self-similar solutions asymptotic to strictly mean convex cones, with respect to perturbations vanishing at infinity. We also show the stability of hyperplanes, with a proof which is novel also for the isotropic mean curvature flow.

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