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Register a new userNeumann heat flow and gradient flow for the entropy on non-convex domains
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For large classes of non-convex subsets $Y$ in ${mathbb R}^n$ or in Riemannian manifolds $(M,g)$ or in RCD-spaces $(X,d,m)$ we prove that the gradient flow for the Boltzmann entropy on the restricted metric measure space $(Y,d_Y,m_Y)$ exists - despite the fact that the entropy is not semiconvex - and coincides with the heat flow on $Y$ with Neumann boundary conditions.
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In this paper we consider the steepest descent $L^2$-gradient flow of the entropy functional. The flow expands convex curves, with the radius of an initial circle growing like the square root of time. Our main result is that, for any initial curve (either immersed locally convex of class $C^2$ or embedded of class $W^{2,2}$ bounding a convex domain), the flow converges smoothly to a round expanding multiply-covered circle.
In this paper we establish new geometric and analytic bounds for Ricci flows, which will form the basis of a compactness, partial regularity and structure theory for Ricci flows in [Bam20a, Bam20b]. The bounds are optimal up to a constant that only depends on the dimension and possibly a lower scalar curvature bound. In the special case in which the flow consists of Einstein metrics, these bounds agree with the optimal bounds for spaces with Ricci curvature bounded from below. Moreover, our bounds are local in the sense that if a bound depends on the collapsedness of the underlying flow, then we are able to quantify this dependence using the pointed Nash entropy based only at the point in question. Among other things, we will show the following bounds: Upper and lower volume bounds for distance balls, dependence of the pointed Nash entropy on its basepoint in space and time, pointwise upper Gaussian bound on the heat kernel and a bound on its derivative and an $L^1$-Poincare inequality. The proofs of these bounds will, in part, rely on a monotonicity formula for a notion, called variance of conjugate heat kernels. We will also derive estimates concerning the dependence of the pointed Nash entropy on its basepoint, which are asymptotically optimal. These will allow us to show that points in spacetime that are nearby in a certain sense have comparable pointed Nash entropy. Hence the pointed Nash entropy is a good quantity to measure local collapsedness of a Ricci flow Our results imply a local $varepsilon$-regularity theorem, improving a result of Hein and Naber. Some of our results also hold for super Ricci flows.
This paper provides sharp Dirichlet heat kernel estimates in inner uniform domains, including bounded inner uniform domains, in the context of certain (possibly non-symmetric) bilinear forms resembling Dirichlet forms. For instance, the results apply to the Dirichlet heat kernel associated with a uniformly elliptic divergence form operator with symmetric second order part and bounded measurable coefficients in inner uniform domains in $mathbb R^n$. The results are applicable to any convex domain, to the complement of any convex domain, and to more exotic examples such as the interior and exterior of the snowflake.
We study the asymptotic behaviour of a gradient system in a regime in which the driving energy becomes singular. For this system gradient-system convergence concepts are ineffective. We characterize the limiting behaviour in a different way, by proving $Gamma$-convergence of the so-called energy-dissipation functional, which combines the gradient-system components of energy and dissipation in a single functional. The $Gamma$-limit of these functionals again characterizes a variational evolution, but this limit functional is not the energy-dissipation functional of any gradient system. The system in question describes the diffusion of a particle in a one-dimensional double-well energy landscape, in the limit of small noise. The wells have different depth, and in the small-noise limit the process converges to a Markov process on a two-state system, in which jumps only happen from the higher to the lower well. This transmutation of a gradient system into a variational evolution of non-gradient type is a model for how many one-directional chemical reactions emerge as limit of reversible ones. The $Gamma$-convergence proved in this paper both identifies the `fate of the gradient system for these reactions and the variational structure of the limiting irreversible reactions.
We study functions of bounded variation (and sets of finite perimeter) on a convex open set $Omegasubseteq X$, $X$ being an infinite dimensional real Hilbert space. We relate the total variation of such functions, defined through an integration by parts formula, to the short-time behaviour of the semigroup associated with a perturbation of the Ornstein--Uhlenbeck operator.
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