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Register a new userThe local structure of the energy landscape in multiphase mean curvature flow: Weak-strong uniqueness and stability of evolutions
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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 that one has reached a global minimum in the energy landscape, the existence of a gradient flow calibration ensures that the route of steepest descent in the energy landscape is unique and stable.
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We derive a weak-strong uniqueness principle for BV solutions to multiphase mean curvature flow of triple line clusters in three dimensions. Our proof is based on the explicit construction of a gradient-flow calibration in the sense of the recent work of Fischer et al. [arXiv:2003.05478] for any such cluster. This extends the two-dimensional construction to the three-dimensional case of surfaces meeting along triple junctions.
We establish weak-strong uniqueness and stability properties of renormalised solutions to a class of energy-reaction-diffusion systems, which genuinely feature cross-diffusion effects. The systems considered are motivated by thermodynamically consistent models, and their formal entropy structure allows us to use as a key tool a suitably adjusted relative entropy method. Weak-strong uniqueness is obtained for general entropy-dissipating reactions without growth restrictions, and certain models with a non-integrable diffusive flux. The results also apply to a class of (isoenergetic) reaction-cross-diffusion systems.
We consider a variational scheme for the anisotropic (including crystalline) mean curvature flow of sets with strictly positive anisotropic mean curvature. We show that such condition is preserved by the scheme, and we prove the strict convergence in BV of the time-integrated perimeters of the approximating evolutions, extending a recent result of De Philippis and Laux to the anisotropic setting. We also prove uniqueness of the flat flow obtained in the limit.
We will give a new proof of the existence of hypercylinder expander of the inverse mean curvature flow which is a radially symmetric homothetic soliton of the inverse mean curvature flow in $mathbb{R}^ntimes mathbb{R}$, $nge 2$, of the form $(r,y(r))$ or $(r(y),y)$ where $r=|x|$, $xinmathbb{R}^n$, is the radially symmetric coordinate and $yin mathbb{R}$. More precisely for any $lambda>frac{1}{n-1}$ and $mu>0$, we will give a new proof of the existence of a unique even solution $r(y)$ of the equation $frac{r(y)}{1+r(y)^2}=frac{n-1}{r(y)}-frac{1+r(y)^2}{lambda(r(y)-yr(y))}$ in $mathbb{R}$ which satisfies $r(0)=mu$, $r(0)=0$ and $r(y)>yr(y)>0$ for any $yinmathbb{R}$. We will prove that $lim_{ytoinfty}r(y)=infty$ and $a_1:=lim_{ytoinfty}r(y)$ exists with $0le a_1<infty$. We will also give a new proof of the existence of a constant $y_1>0$ such that $r(y_1)=0$, $r(y)>0$ for any $0<y<y_1$ and $r(y)<0$ for any $y>y_1$.
An existence and uniqueness result, up to fattening, for crystalline mean curvature flows with forcing and arbitrary (convex) mobilities, is proven. This is achieved by introducing a new notion of solution to the corresponding level set formulation. Such a solution satisfies the comparison principle and a stability property with respect to the approximation by suitably regularized problems. The results are valid in any dimension and for arbitrary, possibly unbounded, initial closed sets. The approach accounts for the possible presence of a time-dependent bounded forcing term, with spatial Lipschitz continuity. As a by-product of the analysis, the problem of the convergence of the Almgren-Taylor-Wang minimizing movements scheme to a unique (up to fattening) flat flow in the case of general, possibly crystalline, anisotropies is settled.
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