Couplings and changes of variables.- Three examples of coupling techniques.- The founding fathers of optimal transport.- Qualitative description of optimal transport.- Basic properties.- Cyclical monotonicity and Kantorovich duality.- The Wasserstein distances.- Displacement interpolation.- The Monge-Mather shortening principle.- Solution of the Monge problem I: global approach.- Solution of the Monge problem II: Local approach.- The Jacobian equation.- Smoothness.- Qualitative picture.- Optimal transport and Riemannian geometry.- Ricci curvature.- Otto calculus.- Displacement convexity I.- Displacement convexity II.- Volume control.- Density control and local regularity.- Infinitesimal displacement convexity.- Isoperimetric-type inequalities.- Concentration inequalities.- Gradient flows I.- Gradient flows II: Qualitative properties.- Gradient flows III: Functional inequalities.- Synthetic treatment of Ricci curvature.- Analytic and synthetic points of view.- Convergence of metric-measure spaces.- Stability of optimal transport.- Weak Ricci curvature bounds I: Definition and Stability.- Weak Ricci curvature bounds II: Geometric and analytic properties.

Optimal Transport: Old and New

1980 Preface * 1999 Preface * 1999 Acknowledgements * Introduction * 1 Circular Logic * 2 Phase Singularities (Screwy Results of Circular Logic) * 3 The Rules of the Ring * 4 Ring Populations * 5 Getting Off the Ring * 6 Attracting Cycles and Isochrons * 7 Measuring the Trajectories of a Circadian Clock * 8 Populations of Attractor Cycle Oscillators * 9 Excitable Kinetics and Excitable Media * 10 The Varieties of Phaseless Experience: In Which the Geometrical Orderliness of Rhythmic Organization Breaks Down in Diverse Ways * 11 The Firefly Machine 12 Energy Metabolism in Cells * 13 The Malonic Acid Reagent ('Sodium Geometrate') * 14 Electrical Rhythmicity and Excitability in Cell Membranes * 15 The Aggregation of Slime Mold Amoebae * 16 Numerical Organizing Centers * 17 Electrical Singular Filaments in the Heart Wall * 18 Pattern Formation in the Fungi * 19 Circadian Rhythms in General * 20 The Circadian Clocks of Insect Eclosion * 21 The Flower of Kalanchoe * 22 The Cell Mitotic Cycle * 23 The Female Cycle * References * Index of Names * Index of Subjects

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/596B270197FE27DE5FABB598B6210622/S0025557200150892a.pdf/div-class-title-span-class-italic-the-geometry-of-biological-time-span-by-a-t-winfree-pp-544-dm68-corrected-second-printing-1990-isbn-3-540-52528-9-springer-div.pdf

The geometry of biological time , by A. T. Winfree. Pp 544. DM68. Corrected Second Printing 1990. ISBN 3-540-52528-9 (Springer)

关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

Sphere Packings, Lattices and Groups

A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

We give a variational proof of global stability for bistable travelling waves of scalar reaction-diffusion equations on the real line. In particular, we recover some of the classical results by P. Fife and J.B. McLeod (1977) without any use of the maximum principle. The method that is illustrated here in the simplest possible setting has been successfully applied to more general parabolic or hyperbolic gradient-like systems.

/pdf/a-variational-proof-of-global-stability-for-bistable-na7d1uvlog.pdf

A variational proof of global stability for bistable travelling waves

When the steady states at infinity become unstable through a
pattern forming bifurcation, a travelling wave may bifurcate
into a modulated front which is time-periodic in a moving
frame. This scenario has been studied by B. Sandstede and A. Scheel
for a class of reaction-diffusion systems on the real line.
Under general assumptions, they showed that the modulated
fronts exist and are spectrally stable near the bifurcation
point. Here we consider a model problem for which
we can prove the nonlinear stability of these solutions with
respect to small localized perturbations.
This result does not follow from the spectral stability, because
the linearized operator around the modulated front has essential
spectrum up to the imaginary axis. The analysis is illustrated by
numerical simulations.

/pdf/stable-transport-of-information-near-essentially-unstable-2lo4jubagl.pdf

Stable transport of information near essentially unstable localized structures

Burgers vortices are stationary solutions of the three-dimensional Navier–Stokes equations in the presence of a background straining flow. These solutions are given by explicit formulas only when the strain is axisymmetric. In this paper we consider a weakly asymmetric strain and prove in that case that non-axisymmetric vortices exist for all values of the Reynolds number. In the limit of large Reynolds numbers, we recover the asymptotic results of Moffatt, Kida & Ohkitani [11]. We also show that the asymmetric vortices are stable with respect to localized two-dimensional perturbations.

/pdf/existence-and-stability-of-asymmetric-burgers-vortices-35wkjfcode.pdf

Existence and Stability of Asymmetric Burgers Vortices

We give a variational proof of global stability for bistable travelling waves of scalar reaction-diffusion equations on the real line. In particular, we recover some of the classical results by P. Fife and J.B. McLeod without any use of the maximum principle. The method that is illustrated here in the simplest possible setting has been successfully applied to more general parabolic or hyperbolic gradient-like systems.

The Ginzburg–Landau equation \(\) on the real line has spatially periodic steady states of the form \(\), with \(\) and \(\). For \(\), \(\), we construct solutions which converge for all t>0 to the limiting pattern \(\) as \(\). These solutions are stable with respect to sufficiently small \(\) perturbations, and behave asymptotically in time like \(\), where \(\) is uniquely determined by the boundary conditions \(\). This extends a previous result of [BrK92] by removing the assumption that \(\) should be close to zero. The existence of the limiting profile \(\) is obtained as an application of the theory of monotone operators, and the long-time behavior of our solutions is controlled by rewriting the system in scaling variables and using energy estimates involving an exponentially growing damping term.

Thierry Gallay

Papers

A variational proof of global stability for bistable travelling waves

Stable transport of information near essentially unstable localized structures

Existence and Stability of Asymmetric Burgers Vortices

A variational proof of global stability for bistable travelling waves

Diffusive Mixing of Stable States in the Ginzburg-Landau Equation