Results 1  10
of
283
The Geometry of Dissipative Evolution Equations: The Porous Medium Equation
"... We show that the porous medium equation has a gradient flow structure which is both physically and mathematically natural. In order to convince the reader that it is mathematically natural, we show the time asymptotic behavior can be easily understood in this framework. We use the intuition and the ..."
Abstract

Cited by 405 (11 self)
 Add to MetaCart
We show that the porous medium equation has a gradient flow structure which is both physically and mathematically natural. In order to convince the reader that it is mathematically natural, we show the time asymptotic behavior can be easily understood in this framework. We use the intuition and the calculus of Riemannian geometry to quantify this asymptotic behavior.
On the geometry of metric measure spaces
 II, ACTA MATH
, 2004
"... We introduce and analyze lower (’Ricci’) curvature bounds Curv(M, d,m) ≥ K for metric measure spaces (M, d,m). Our definition is based on convexity properties of the relative entropy Ent(.m) regarded as a function on the L2Wasserstein space of probability measures on the metric space (M, d). Amo ..."
Abstract

Cited by 247 (9 self)
 Add to MetaCart
(Show Context)
We introduce and analyze lower (’Ricci’) curvature bounds Curv(M, d,m) ≥ K for metric measure spaces (M, d,m). Our definition is based on convexity properties of the relative entropy Ent(.m) regarded as a function on the L2Wasserstein space of probability measures on the metric space (M, d). Among others, we show that Curv(M, d,m) ≥ K implies estimates for the volume growth of concentric balls. For Riemannian manifolds, Curv(M, d,m) ≥ K if and only if RicM (ξ, ξ) ≥ K · ξ2 for all ξ ∈ TM. The crucial point is that our lower curvature bounds are stable under an appropriate notion of Dconvergence of metric measure spaces. We define a complete and separable metric D on the family of all isomorphism classes of normalized metric measure spaces. The metric D has a natural interpretation, based on the concept of optimal mass transportation. We also prove that the family of normalized metric measure spaces with doubling constant ≤ C is closed under Dconvergence. Moreover, the family of normalized metric measure spaces with doubling constant ≤ C and radius ≤ R is compact under Dconvergence.
Generalization Of An Inequality By Talagrand, And Links With The Logarithmic Sobolev Inequality
 J. Funct. Anal
, 2000
"... . We show that transport inequalities, similar to the one derived by Talagrand [30] for the Gaussian measure, are implied by logarithmic Sobolev inequalities. Conversely, Talagrand's inequality implies a logarithmic Sobolev inequality if the density of the measure is approximately logconcave, ..."
Abstract

Cited by 244 (12 self)
 Add to MetaCart
. We show that transport inequalities, similar to the one derived by Talagrand [30] for the Gaussian measure, are implied by logarithmic Sobolev inequalities. Conversely, Talagrand's inequality implies a logarithmic Sobolev inequality if the density of the measure is approximately logconcave, in a precise sense. All constants are independent of the dimension, and optimal in certain cases. The proofs are based on partial dierential equations, and an interpolation inequality involving the Wasserstein distance, the entropy functional and the Fisher information. Contents 1. Introduction 1 2. Main results 5 3. Heuristics 11 4. Proof of Theorem 1 18 5. Proof of Theorem 3 24 6. An application of Theorem 1 30 7. Linearizations 31 Appendix A. A nonlinear approximation argument 35 References 36 1. Introduction Let M be a smooth complete Riemannian manifold of dimension n, with the geodesic distance d(x; y) = inf 8 < : s Z 1 0 j _ w(t)j 2 dt; w 2 C 1 ((0; 1); M); w(0) = x; w(1) = y 9 ...
Ricci curvature for metricmeasure spaces via optimal transport
 ANN. OF MATH
, 2005
"... We define a notion of a measured length space X having nonnegative NRicci curvature, for N ∈ [1, ∞), or having ∞Ricci curvature bounded below by K, for K ∈ R. The definitions are in terms of the displacement convexity of certain functions on the associated Wasserstein metric space P2(X) of proba ..."
Abstract

Cited by 231 (10 self)
 Add to MetaCart
We define a notion of a measured length space X having nonnegative NRicci curvature, for N ∈ [1, ∞), or having ∞Ricci curvature bounded below by K, for K ∈ R. The definitions are in terms of the displacement convexity of certain functions on the associated Wasserstein metric space P2(X) of probability measures. We show that these properties are preserved under measured GromovHausdorff limits. We give geometric and analytic consequences.
Globalintime weak measure solutions, finitetime aggregation and confinement for nolocal interaction equations
, 2009
"... In this paper, we provide a wellposedness theory for weak measure solutions of the Cauchy problem for a family of nonlocal interaction equations. These equations are continuum models for interacting particle systems with attractive/repulsive pairwise interaction potentials. The main phenomenon of ..."
Abstract

Cited by 68 (19 self)
 Add to MetaCart
(Show Context)
In this paper, we provide a wellposedness theory for weak measure solutions of the Cauchy problem for a family of nonlocal interaction equations. These equations are continuum models for interacting particle systems with attractive/repulsive pairwise interaction potentials. The main phenomenon of interest is that, even with smooth initial data, the solutions can concentrate mass in finite time. We develop an existence theory that enables one to go beyond the blowup time in classical norms and allows for solutions to form atomic parts of the measure in finite time. The weak measure solutions are shown to be unique and exist globally in time. Moreover, in the case of sufficiently attractive potentials, we show the finite time total collapse of the solution onto a single point, for compactly supported initial measures. Finally, we give conditions on compensation between the attraction at large distances and local repulsion of the potentials to have globalintime confined systems for compactly supported initial data. Our approach is based on the theory of gradient flows in the space of probability measures endowed with the Wasserstein metric. In addition to classical tools, we exploit the stability of the flow with respect to the transportation distance to greatly simplify many problems by reducing them to questions about particle approximations.
Convergence of the masstransport steepest descent scheme for the subcritical PatlakKellerSegel model
 SIAM J. Numer. Anal
"... Abstract. Variational steepest descent approximation schemes for the modified PatlakKellerSegel equation with a logarithmic interaction kernel in any dimension are considered. We prove the convergence of the suitably interpolated in time implicit Euler scheme, defined in terms of the Euclidean W ..."
Abstract

Cited by 54 (19 self)
 Add to MetaCart
(Show Context)
Abstract. Variational steepest descent approximation schemes for the modified PatlakKellerSegel equation with a logarithmic interaction kernel in any dimension are considered. We prove the convergence of the suitably interpolated in time implicit Euler scheme, defined in terms of the Euclidean Wasserstein distance, associated to this equation for subcritical masses. As a consequence, we recover the recent result about the global in time existence of weaksolutions to the modified PatlakKellerSegel equation for the logarithmic interaction kernel in any dimension in the subcritical case. Moreover, we show how this method performs numerically in one dimension. In this particular case, this numerical scheme corresponds to a standard implicit Euler method for the pseudoinverse of the cumulative distribution function. We demonstrate its capabilities to reproduce easily without the need of meshrefinement the blowup of solutions for supercritical masses. 1.
Functional inequalities, thick tails and asymptotics for the critical mass PatlakKellerSegel model
, 2011
"... We investigate the long time behavior of the critical mass PatlakKellerSegel equation. This equation has a one parameter family of steadystate solutions λ, λ> 0, with thick tails whose second moment is not bounded. We show that these steady state solutions are stable, and find basins of attrac ..."
Abstract

Cited by 51 (12 self)
 Add to MetaCart
(Show Context)
We investigate the long time behavior of the critical mass PatlakKellerSegel equation. This equation has a one parameter family of steadystate solutions λ, λ> 0, with thick tails whose second moment is not bounded. We show that these steady state solutions are stable, and find basins of attraction for them using an entropy functional Hλ coming from the critical fast diffusion equation in R 2. We construct solutions of PatlakKellerSegel equation satisfying an entropyentropy dissipation inequality for Hλ. While the entropy dissipation for Hλ is strictly positive, it turns out to be a difference of two terms, neither of which need to be small when the dissipation is small. We introduce a strategy of controlled concentration to deal with this issue, and then use the regularity obtained from the entropyentropy dissipation inequality to prove the existence of basins of attraction for each stationary state composed by certain initial data converging towards λ. In the present paper, we do not provide any estimate of the rate of convergence, but we discuss how this would result from a stability result for a certain sharp GagliardoNirenbergSobolev inequality.
Heat flow on Finsler manifolds
, 2009
"... This paper studies the heat flow on Finsler manifolds. A Finsler manifold is a smooth manifold M equipped with a Minkowski norm F(x, ·) : TxM → R+ on each tangent space. Mostly, we will require that this norm is strongly convex and smooth and that it depends smoothly on the base point x. The particu ..."
Abstract

Cited by 47 (18 self)
 Add to MetaCart
(Show Context)
This paper studies the heat flow on Finsler manifolds. A Finsler manifold is a smooth manifold M equipped with a Minkowski norm F(x, ·) : TxM → R+ on each tangent space. Mostly, we will require that this norm is strongly convex and smooth and that it depends smoothly on the base point x. The particular case of a Hilbert norm on each tangent space leads to the important subclasses of Riemannian manifolds where the heat flow is widely studied and well understood. We present two approaches to the heat flow on a Finsler manifold: • either as gradient flow on L2 (M,m) for the energy E(u) = 1
Dynamics of Labyrinthine Pattern Formation in Magnetic Fluids: A MeanField Theory
 ARCH. RATIONAL MECH. ANAL
"...
..."
Lubrication Approximation With Prescribed Nonzero Contact Angle
 Comm. Partial Differential Equations
"... We prove longtime existence for a weak solution s(t; x) 0 of the lubrication approximation @ t s + @ x (s @ 3 x s) = 0 in fs ? 0g with prescribed contact angle of, say, 4 (@ x s) 2 = 1 on @fs ? 0g : AMS code: 35K65, 35R35, 76D08, 76S05 Introduction The equation we study models an experim ..."
Abstract

Cited by 43 (5 self)
 Add to MetaCart
(Show Context)
We prove longtime existence for a weak solution s(t; x) 0 of the lubrication approximation @ t s + @ x (s @ 3 x s) = 0 in fs ? 0g with prescribed contact angle of, say, 4 (@ x s) 2 = 1 on @fs ? 0g : AMS code: 35K65, 35R35, 76D08, 76S05 Introduction The equation we study models an experiment we will introduce first. Our intention hereby is not to give a full description of the experiment and its various models, but to provide the reader with some intuition of the equation under consideration. This intuition will motivate some of the ideas used in the its analysis. The experiment is the flow of a viscous fluid in a HeleShaw cell. The HeleShaw cell is made of two parallel sheets of glass with a small spacing b. Let G ae IR 2 denote the twodimensional cross section (think of a rectangle) of the HeleShaw cell, so that G \Theta (0; b) ae IR 3 is the threedimensional gap. We assume that @G \Theta (0; b), the lateral border of this gap, is also made of glass. A vis...