Notation.- Notation.- Gradient Flow in Metric Spaces.- Curves and Gradients in Metric Spaces.- Existence of Curves of Maximal Slope and their Variational Approximation.- Proofs of the Convergence Theorems.- Uniqueness, Generation of Contraction Semigroups, Error Estimates.- Gradient Flow in the Space of Probability Measures.- Preliminary Results on Measure Theory.- The Optimal Transportation Problem.- The Wasserstein Distance and its Behaviour along Geodesics.- Absolutely Continuous Curves in p(X) and the Continuity Equation.- Convex Functionals in p(X).- Metric Slope and Subdifferential Calculus in (X).- Gradient Flows and Curves of Maximal Slope in p(X).

Gradient Flows: In Metric Spaces and in the Space of Probability Measures

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 that 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.

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The geometry of dissipative evolution equations: the porous medium equation

The Origins of Local Regression.- Local Regression Methods.- Fitting with LOCFIT.- Local Likelihood Estimation.- Density Estimation.- Flexible Local Regression.- Survival and Failure Time Analysis.- Discrimination and Classification.- Variance Estimation and Goodness of Fit.- Bandwidth Selection.- Adaptive Parameter Choice.- Computational Methods.- Optimizing Local Regression.

http://cyber.sibsutis.ru:82/Monarev/docs/nauka/PROBABILITY/MVsa_Statistics%20and%20applications/Loader%20C.%20Local%20Regression%20and%20Likelihood%20(Springer,%201999)(305s).pdf

Local Regression and Likelihood

This article summarizes various aspects and results for some general formulations of the classical chemotaxis models also known as Keller-Segel models. It is intended as a survey of results for the most common formulation of this classical model for positive chemotactical movement and offers possible generalizations of these results to more universal models. Furthermore it collects open questions and outlines mathematical progress in the study of the Keller-Segel model since the first presentation of the equations in 1970.

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From 1970 until present: the Keller-Segel model in chemotaxis and its consequences

The structure conditions are the ellipticity of a and the (weak) monotonicity of b, and b has to be a subgradient in case m > 1. First we treat the case that b is continuous, and later (Sect. 4) we include Stefan problems, that is, we allow b to have jumps. The special cases of an elliptic equation with time as parameter, that is, b(z)= 0, and the standard parabolic equation, that is, b(z)=z are included. Some special single equations of mixed elliptic and parabolic type are given in the following. The gas flow through a porous medium is described by the equation

Quasilinear elliptic-parabolic differential equations

A system of partial differential equations modelling chemotactic aggregation is analysed (Keller-Segel model). Conditions on the system of paramaters are given implying global existence of smooth solutions. In two space dimensions and radially symmetric situations, explosion of the bacteria concentration in finite time is shown for a class of initial values

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On explosions of solutions to a system of partial differential equations modelling chemotaxis

In this paper we apply the method of implicit time discretization to the mean curvature flow equation including outer forces. In the framework ofBV-functions we construct discrete solutions iteratively by minimizing a suitable energy-functional in each time step. Employing geometric and variational arguments we show an energy estimate which assures compactness of the discrete solutions. An additional convergence condition excludes a loss of area in the limit. Thus existence of solutions to the continuous problem can be derived. We append a brief discussion of the related Mullins-Sekerka equation.

Implicit time discretization for the mean curvature flow equation

On donne un resultat de regularite partielle interieur pour des minimiseurs de certaines fonctionnelles entre des varietes de Riemann

Partial hölder continuity for minima of certain energies among maps into a Riemannian manifold

This paper discusses the asymptotic behavior as ɛ → 0+ of the chemical potentials λɛ associated with solutions of variational problems within the Van der Waals-Cahn-Hilliard theory of phase transitions in a fluid with free energy, per unit volume, given by ɛ2¦▽ϱ¦2+ W(ϱ), where ϱ is the density. The main result is that λɛ is asymptotically equal to ɛEλ/d+o(ɛ), with E the interfacial energy, per unit surface area, of the interface between phases, λ the (constant) sum of principal curvatures of the interface, and d the density jump across the interface. This result is in agreement with a formula conjectured by M. Gurtin and corresponds to the Gibbs-Thompson relation for surface tension, proved by G. Caginalp within the context of the phase field model of free boundaries arising from phase transitions.

Stephan Luckhaus

Papers

Quasilinear elliptic-parabolic differential equations

On explosions of solutions to a system of partial differential equations modelling chemotaxis

Implicit time discretization for the mean curvature flow equation

Partial hölder continuity for minima of certain energies among maps into a Riemannian manifold

The Gibbs-Thompson relation within the gradient theory of phase transitions