Gradient Flows In Metric Spaces Research Articles

One-dimensional viscoelastic von Kármán theories derived from nonlinear thin-walled beams

We derive an effective one-dimensional limit from a three-dimensional Kelvin–Voigt model for viscoelastic thin-walled beams, in which the elastic and the viscous stress tensor comply with a frame-indifference principle. The limiting system of equations comprises stretching, bending, and twisting both in the elastic and the viscous stress. It coincides with the model already identified via Friedrich and Kružík (Arch Ration Mech Anal 238:489–540, 2020) and Friedrich and Machill (Nonlinear Differ Equ Appl NoDEA 29, Article number: 11, 2022) by a successive dimension reduction, first from 3D to a 2D theory for von Kármán plates and then from 2D to a 1D theory for ribbons. In the present paper, we complement the previous analysis by showing that the limit can also be obtained by sending the height and width of the beam to zero simultaneously. Our arguments rely on the static Gamma -convergence in Freddi et al. (Math Models Methods Appl Sci 23:743–775, 2013), on the abstract theory of metric gradient flows (Ambrosio et al. in Gradient flows in metric spaces and in the space of probability measures. Lectures mathematics, ETH Zürich, Birkhäuser, Basel, 2005), and on evolutionary Gamma -convergence (Sandier and Serfaty in Commun Pure Appl Math 57:1627–1672, 2004).

Minimal solutions to generalized Lambda-semiflows and gradient flows in metric spaces

Generalized Lambda-semiflows are an abstraction of semiflows with nonperiodic solutions, for which there may be more than one solution corresponding to given initial data. A select class of solutions to generalized Lambda-semiflows is introduced. It is proved that such minimal solutions are unique corresponding to given ranges and generate all other solutions by time reparametrization. Special qualities of minimal solutions are shown. The concept of minimal solutions is applied to gradient flows in metric spaces and generalized semiflows. Generalized semiflows have been introduced by Ball.

Derivation of a one-dimensional von K\xe1rm\xe1n theory for viscoelastic ribbons

We consider a two-dimensional model of viscoelastic von Kármán plates in the Kelvin’s-Voigt’s rheology derived from a three-dimensional model at a finite-strain setting in Friedrich and Kružík (Arch Ration Mech Anal 238: 489–540, 2020). As the width of the plate goes to zero, we perform a dimension-reduction from 2D to 1D and identify an effective one-dimensional model for a viscoelastic ribbon comprising stretching, bending, and twisting both in the elastic and the viscous stress. Our arguments rely on the abstract theory of gradient flows in metric spaces by Sandier and Serfaty (Commun Pure Appl Math 57:1627–1672, 2004) and complement the Gamma -convergence analysis of elastic von Kármán ribbons in Freddi et al. (Meccanica 53:659–670, 2018). Besides convergence of the gradient flows, we also show convergence of associated time-discrete approximations, and we provide a corresponding commutativity result.

Derivation of von Kármán Plate Theory in the Framework of Three-Dimensional Viscoelasticity

We apply a quasistatic nonlinear model for nonsimple viscoelastic materials at a finite-strain setting in the Kelvin's-Voigt's rheology to derive a viscoelastic plate model of von K\'arm\'an type. We start from time-discrete solutions to a model of three-dimensional viscoelasticity where the viscosity stress tensor complies with the principle of time-continuous frame-indifference. Combining the derivation of nonlinear plate theory by Friesecke, James and M\"{u}ller, and the abstract theory of gradient flows in metric spaces by Sandier and Serfaty we perform a dimension-reduction from 3D to 2D and identify weak solutions of viscoelastic form of von K\'arm\'an plates.

A variational formulation of the BDF2 method for metric gradient flows

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 L2-Wasserstein metric.

Γ-convergence and relaxations for gradient flows in metric spaces: a minimizing movement approach

We present new abstract results on the interrelation between the minimizing movement scheme for gradient flows along a sequence of Γ-converging functionals and the gradient flow motion for the corresponding limit functional, in a general metric space. We are able to allow a relaxed form of minimization in each step of the scheme, and so we present new relaxation results too.

Weighted Energy-Dissipation principle for gradient flows in metric spaces

This paper develops the so-called Weighted Energy-Dissipation (WED) variational approach for the analysis of gradient flows in metric spaces. This focuses on the minimization of the parameter-dependent global-in-time functional of trajectoriesIε[u]=∫0∞e−t/ε(12|u′|2(t)+1εϕ(u(t)))dt, featuring the weighted sum of energetic and dissipative terms. As the parameter ε is sent to 0, the minimizers uε of such functionals converge, up to subsequences, to curves of maximal slope driven by the functional ϕ. This delivers a new and general variational approximation procedure, hence a new existence proof, for metric gradient flows. In addition, it provides a novel perspective towards relaxation.

Second-order in time schemes for gradient flows in Wasserstein and geodesic metric spaces

The time discretization of gradient flows in metric spaces uses variants of the celebrated implicit Euler-type scheme of Jordan, Kinderlehrer, and Otto [9]. We propose in this Note a different approach, which allows us to construct two second-order in time numerical schemes. In a metric space framework, we show that the schemes are well defined and prove the convergence for one of them under some regularity assumptions. For the particular case of a Fokker–Planck gradient flow in the Wasserstein space, we obtain (theoretically and numerically) the second-order convergence.

High-frequency limit of non-autonomous gradient flows of functionals with time-periodic forcing

We study the high-frequency limit of non-autonomous gradient flows in metric spaces of energy functionals comprising an explicitly time-dependent perturbation term which might oscillate in a rapid way. On grounds of the existence results by Ferreira and Guevara (2015) on non-autonomous gradient flows (which we also extend to a broader range of energy functionals), we prove that the associated solution curves converge to a solution of the time-averaged evolution equation in the limit of infinite frequency. Under additional assumptions on the energy, we obtain an explicit rate of convergence. Furthermore, we specifically investigate nonlinear drift-diffusion equations with time-dependent drift which formally are gradient flows with respect to the L2-Wasserstein distance. We prove that a family of weak solutions obtained as a limit of the Minimizing Movements scheme exhibits the above-mentioned behavior in the high-frequency limit.

Measure solutions for non-local interaction PDEs with two species

This paper presents a systematic existence and uniqueness theory of weak measure solutions for systems of non-local interaction PDEs with two species, which are the PDE counterpart of systems of deterministic interacting particles with two species. The main motivations behind those models arise in cell biology, pedestrian movements, and opinion formation. In case of symmetrizable systems (i.e. with cross-interaction potentials one multiple of the other), we provide a complete existence and uniqueness theory within (a suitable generalization of) the Wasserstein gradient flow theory in Ambrosio et al (2008 Gradient Flows in Metric Spaces and in the Space of Probability Measures (Lectures in Mathematics ETH Zürich) 2nd edn (Basel: Birkhäuser)) and Carrillo et al (2011 Duke Math. J. 156 229–71), which allows the consideration of interaction potentials with a discontinuous gradient at the origin. In the general case of non-symmetrizable systems, we provide an existence result for measure solutions which uses a semi-implicit version of the Jordan–Kinderlehrer–Otto (JKO) scheme (Jordan et al 1998 SIAM J. Math. Anal. 29 1–17), which holds in a reasonable non-smooth setting for the interaction potentials. Uniqueness in the non-symmetrizable case is proven for C2 potentials using a variant of the method of characteristics.

Unilateral gradient flow of the Ambrosio–Tortorelli functional by minimizing movements

Motivated by models of fracture mechanics, this paper is devoted to the analysis of a unilateral gradient flow of the Ambrosio–Tortorelli functional, where unilaterality comes from an irreversibility constraint on the fracture density. Solutions of such evolution are constructed by means of an implicit Euler scheme. An asymptotic analysis in the Mumford–Shah regime is then carried out. It shows the convergence towards a generalized heat equation outside a time increasing crack set. In the spirit of gradient flows in metric spaces, a notion of curve of maximal unilateral slope is also investigated, and analogies with the unilateral slope of the Mumford–Shah functional are also discussed.

Erratum to: A Trotter product formula for gradient flows in metric spaces

d dt d2(u(t), y) ≤ φ(y)− φ(u(t)). The rescaled function v : [0, T∗/2] → X defined by v(t) = u(2t) satisfies the equations which have been stated for u. If the factor 1 2 is omitted, the theorem is correct. The proof is correct as stated and shows convergence to the correct equation, as can be seen, for example, in the last displayed formula of the proof on page 418. Further errors that need to be amended are given below: • On line 8 of page 413, [t j−1, t j ) should be replaced by [t j−1 h , t j h ). • The second line of the proof to Lemma 3.1 should be: “For the convenience of the reader, we provide a simple direct proof of (1) in the case that μ L d .”

A variational principle for gradient flows in metric spaces

We present a novel variational approach to gradient-flow evolution in metric spaces. In particular, we advance a functional defined on entire trajectories, whose minimizers converge to curves of maximal slope for geodesically convex energies. The crucial step of the argument is the reformulation of the variational approach in terms of a dynamic programming principle, and the use of the corresponding Hamilton–Jacobi equation. The result is applicable to a large class of nonlinear evolution PDEs including nonlinear drift-diffusion, Fokker–Planck, and heat flows on metric-measure spaces.

A Trotter product formula for gradient flows in metric spaces

We prove a Trotter product formula for gradient flows in metric spaces. This result is applied to establish convergence in the L 2-Wasserstein metric of the splitting method for some Fokker-Planck equations and porous medium type equations perturbed by a potential.

Global attractors for gradient flows in metric spaces

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 of Ambrosio et al. (2005) [5], we introduce the latter to include limits of time-incremental approximations constructed via the Minimizing Movements approach (De Giorgi, 1993; Ambrosio, 1995 [3,15]). 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 in Ball (1997) [7]. 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.

A metric approach to a class of doubly nonlinear evolution equations and applications

E Abstract. This paper deals with the analysis of a class of doubly nonlinear evo- lution equations in the framework of a general metric space. We propose for such equations a suitable metric formulation (which in fact extends the notion of Curve of Maximal Slope for gradient flows in metric spaces, see (5)), and prove the exis- tence of solutions for the related Cauchy problem by means of an approximation scheme by time discretization. Then, we apply our results to obtain the existence of solutions to abstract doubly nonlinear equations in reflexive Banach spaces. The metric approach is also exploited to analyze a class of evolution equations in L 1 spaces.

Inverse Total Variation Flow

In this paper we analyze iterative regularization with the Bregman distance of the total variation seminorm. Moreover, we prove existence of a solution of the corresponding flow equation as introduced in [M. Burger, G. Gilboa, S. Osher, and J. Xu, Commun. Math. Sci., 4 (2006), pp. 179–212] in a functional analytical setting using methods from convex analysis. The results are generalized to variational denoising methods with ${\rm L}^p$-norm fit-to-data terms and Bregman distance regularization terms. For the associated flow equations well-posedness is derived using recent results on metric gradient flows from [L. Ambrosio, N. Gigli, and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005]. In contrast to previous work the results of this paper apply for the analysis of variational denoising methods with the Bregman distance under adequate noise assumptions. Aside from the theoretical results we introduce a level set technique based on Bregman distance regularization for denoising of surfaces and demonstrate the efficiency of this method. (A corrected version of this paper has been appended to the PDF file.)