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Register a new userGlobal attractors for gradient flows in metric spaces
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We develop the long-time analysis for gradient flow equations in metric spaces. In particular, we consider two notions of solutions for metric gradient flows, namely energy and generalized solutions. While the former concept coincides with the notion of curves of maximal slope, we introduce the latter to include limits of time-incremental approximations constructed via the Minimizing Movements approach. For both notions of solutions we prove the existence of the global attractor. Since the evolutionary problems we consider may lack uniqueness, we rely on the theory of generalized semiflows introduced by J.M. Ball. The notions of generalized and energy solutions are quite flexible and can be used to address gradient flows in a variety of contexts, ranging from Banach spaces to Wasserstein spaces of probability measures. We present applications of our abstract results by proving the existence of the global attractor for the energy solutions both of abstract doubly nonlinear evolution equations in reflexive Banach spaces, and of a class of evolution equations in Wasserstein spaces, as well as for the generalized solutions of some phase-change evolutions driven by mean curvature.
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We propose a variational form of the BDF2 method as an alternative to the commonly used minimizing movement scheme for the time-discrete approximation of gradient flows in abstract metric spaces. Assuming uniform semi-convexity --- but no smoothness --- of the augmented energy functional, we prove well-posedness of the method and convergence of the discrete approximations to a curve of steepest descent. In a smooth Hilbertian setting, classical theory would predict a convergence order of two in time, we prove convergence order of one-half in the general metric setting and under our weak hypotheses. Further, we illustrate these results with numerical experiments for gradient flows on a compact Riemannian manifold, in a Hilbert space, and in the $L^2$-Wasserstein metric.
We study two notions of Dirichlet problem associated with BV energy minimizers (also called functions of least gradient) in bounded domains in metric measure spaces whose measure is doubling and supports a $(1,1)$-Poincare inequality. Since one of the two notions is not amenable to the direct method of the calculus of variations, we construct, based on an approach of [23, 29], solutions by considering the Dirichlet problem for $p$-harmonic functions, $p>1$, and letting $pto 1$. Tools developed and used in this paper include the inner perimeter measure of a domain.
The Penrose-Fife system for phase transitions is addressed. Dirichlet boundary conditions for the temperature are assumed. Existence of global and exponential attractors is proved. Differently from preceding contributions, here the energy balance equation is both singular at 0 and degenerate at infinity. For this reason, the dissipativity of the associated dynamical process is not trivial and has to be proved rather carefully.
In this note we consider problems related to parabolic partial differential equations in geodesic metric measure spaces, that are equipped with a doubling measure and a Poincare inequality. We prove a location and scale invariant Harnack inequality for a minimizer of a variational problem related to a doubly non-linear parabolic equation involving the p-Laplacian. Moreover, we prove the sufficiency of the Grigoryan--Saloff-Coste theorem for general p > 1 in geodesic metric spaces. The approach used is strictly variational, and hence we are able to carry out the argument in the metric setting.
We provide new characterizations of Sobolev ad BV spaces in doubling and Poincare metric spaces in the spirit of the Bourgain-Brezis-Mironescu and Nguyen limit formulas holding in domains of R^N.
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