Corpus ID: 212675755

The local structure of the energy landscape in multiphase mean curvature flow: Weak-strong uniqueness and stability of evolutions

@article{Fischer2020TheLS,
  title={The local structure of the energy landscape in multiphase mean curvature flow: Weak-strong uniqueness and stability of evolutions},
  author={Julian Fischer and Sebastian Hensel and Tim Laux and Thilo M. Simon},
  journal={arXiv: Analysis of PDEs},
  year={2020}
}
We prove that in the absence of topological changes, the notion of BV solutions to planar multiphase mean curvature flow does not allow for a mechanism for (unphysical) non-uniqueness. Our approach is based on the local structure of the energy landscape near a classical evolution by mean curvature. Mean curvature flow being the gradient flow of the surface energy functional, we develop a gradient-flow analogue of the notion of calibrations. Just like the existence of a calibration guarantees… Expand
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References

SHOWING 1-10 OF 50 REFERENCES
Evolution of networks with multiple junctions
We consider the motion by curvature of a network of curves in the plane and we discuss existence, uniqueness, singularity formation and asymptotic behavior of the flow.
The motion of a surface by its mean curvature
TLDR
The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press to preserve the original texts of these important books while presenting them in durable paperback editions. Expand
The grain boundary mobility tensor
TLDR
It is argued that the mobility is, in general, a tensor (classically, it is a scalar) and determine all of its components and demonstrated that stress generation during GB migration necessarily slows grain growth and reduces GB mobility in polycrystals. Expand
Weak-strong uniqueness for multiphase mean curvature flow of double bubbles
  • in preparation,
  • 2020
Calibrations and null-Lagrangians for nonlocal perimeters and an application to the viscosity theory
For nonnegative even kernels K , we consider the K -nonlocal perimeter functional acting on sets. Assuming the existence of a foliation of space made of solutions of the associated K -nonlocal meanExpand
Halfspaces minimise nonlocal perimeter: a proof via calibrations
We consider a nonlocal functional $J_K$ that may be regarded as a nonlocal version of the total variation. More precisely, for any measurable function $u\colon \mathbb{R}^d \to \mathbb{R}$, we defineExpand
On short time existence for the planar network flow
We prove the existence of the flow by curvature of regular planar networks starting from an initial network which is non-regular. The proof relies on a monotonicity formula for expanding solutionsExpand
Weak–Strong Uniqueness for the Navier–Stokes Equation for Two Fluids with Surface Tension
In the present work, we consider the evolution of two fluids separated by a sharp interface in the presence of surface tension—like, for example, the evolution of oil bubbles in water. Our mainExpand
Grain-boundary kinetics: A unified approach
Grain boundaries (GBs) are central defects for describing polycrystalline materials, and playing major role in a wide-range of physical properties of polycrystals. Control over GB kinetics providesExpand
Brakke’s inequality for the thresholding scheme
We continue our analysis of the thresholding scheme from the variational viewpoint and prove a conditional convergence result towards Brakke’s notion of mean curvature flow. Our proof is based on aExpand
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