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460
A macroscopic crowd motion model of gradient flow type
 MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES
, 2010
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On the inviscid limit of a model for crack propagation
 MATH. MODELS METH. APPL. SCI.
, 2007
"... We study the evolution of a single crack in an elastic body and assume that the crack path is known in advance. The motion of the crack tip is modeled as a rateindependent process on the basis of Griffith’s local energy release rate criterion. According to this criterion, the system may stay in a l ..."
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Cited by 40 (9 self)
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We study the evolution of a single crack in an elastic body and assume that the crack path is known in advance. The motion of the crack tip is modeled as a rateindependent process on the basis of Griffith’s local energy release rate criterion. According to this criterion, the system may stay in a local minimum before it performs a jump. The goal of this paper is to prove existence of such an evolution and to shed light on the discrepancy between the local energy release rate criterion and models which are based on a global stability criterion (as for example the Francfort/Marigo model). We construct solutions to the local model via the vanishing viscosity method and compare different notions of weak, local and global solutions.
Computability of probability measures and MartinLöf randomness over metric spaces
, 709
"... In this paper we investigate algorithmic randomness on more general spaces than the Cantor space, namely computable metric spaces. To do this, we first develop a unified framework allowing computations with probability measures. We show that any computable metric space with a computable probability ..."
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Cited by 38 (11 self)
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In this paper we investigate algorithmic randomness on more general spaces than the Cantor space, namely computable metric spaces. To do this, we first develop a unified framework allowing computations with probability measures. We show that any computable metric space with a computable probability measure is isomorphic to the Cantor space in a computable and measuretheoretic sense. We show that any computable metric space admits a universal uniform randomness test (without further assumption). Key words: Computability, computable metric spaces, computable measures, Kolmogorov complexity, algorithmic randomness, randomness tests. 1
ON THE EQUIVALENCE OF THE ENTROPIC CURVATUREDIMENSION CONDITION AND BOCHNER’S INEQUALITY ON METRIC MEASURE SPACES
, 2013
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RICCI FLOW, ENTROPY AND OPTIMAL TRANSPORTATION
"... Abstract. Let a smooth family of Riemannian metrics g(τ) satisfy the backwards Ricci flow equation on a compact oriented ndimensional manifold M. Suppose two families of normalized nforms ω(τ) ≥ 0 and ˜ω(τ) ≥ 0 satisfy the forwards (in τ) heat equation on M generated by the connection Laplacian ..."
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Cited by 37 (1 self)
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Abstract. Let a smooth family of Riemannian metrics g(τ) satisfy the backwards Ricci flow equation on a compact oriented ndimensional manifold M. Suppose two families of normalized nforms ω(τ) ≥ 0 and ˜ω(τ) ≥ 0 satisfy the forwards (in τ) heat equation on M generated by the connection Laplacian ∆g(τ). If these nforms represent two evolving distributions of particles over M, the minimum rootmeansquare distance W2(ω(τ), ˜ω(τ), τ) to transport the particles of ω(τ) onto those of ˜ω(τ) is shown to be nonincreasing as a function of τ, without sign conditions on the curvature of (M, g(τ)). Moreover, this contractivity property is shown to characterize supersolutions to the Ricci flow.
Critical mass for a PatlakKellerSegel model with degenerate diffusion in higher dimensions
, 2008
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Local and Global WellPosedness for Aggregation Equations and PatlakKellerSegel Models with Degenerate Diffusion
, 2010
"... Recently, there has been a wide interest in the study of aggregation equations and PatlakKellerSegel (PKS) models for chemotaxis with degenerate diffusion. The focus of this paper is the unification and generalization of the wellposedness theory of these models. We prove local wellposedness on b ..."
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Cited by 34 (9 self)
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Recently, there has been a wide interest in the study of aggregation equations and PatlakKellerSegel (PKS) models for chemotaxis with degenerate diffusion. The focus of this paper is the unification and generalization of the wellposedness theory of these models. We prove local wellposedness on bounded domains for dimensions d ≥ 2 and in all of space for d ≥ 3, the uniqueness being a result previously not known for PKS with degenerate diffusion. We generalize the notion of criticality for PKS and show that subcritical problems are globally wellposed. For a fairly general class of problems, we prove the existence of a critical mass which sharply divides the possibility of finite time blow up and global existence. Moreover, we compute the critical mass for fully general problems and show that solutions with smaller mass exists globally. For a class of supercritical problems we prove finite time blow up is possible for initial data of arbitrary mass. 1
A family of nonlinear fourth order equations of gradient flow type
, 2009
"... Global existence and longtime behavior of solutions to a family of nonlinear fourth order evolution equations on Rd are studied. These equations constitute gradient flows for the perturbed information functionals Fα,λ(u) = 1 ..."
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Cited by 33 (9 self)
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Global existence and longtime behavior of solutions to a family of nonlinear fourth order evolution equations on Rd are studied. These equations constitute gradient flows for the perturbed information functionals Fα,λ(u) = 1