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

Motivated by applications to vortex rings, we study the Cauchy problem for the three-dimensional axisymmetric Navier-Stokes equations without swirl, using scale invariant function spaces. If the axisymmetric vorticity ω θ is integrable with respect to the two-dimensional measure dr dz, where (r, θ, z) denote the cylindrical coordinates in R 3 , we show the existence of a unique global solution, which converges to zero in L 1 norm as t → ∞. The proof of local well-posedness follows exactly the same lines as in the two-dimensional case, and our approach emphasizes the similarity between both situations. The solutions we construct have infinite energy in general, so that energy dissipation cannot be invoked to control the long-time behavior. We also treat the more general case where the initial vorticity is a finite measure whose atomic part is small enough compared to viscosity. Such data include point masses, which correspond to vortex filaments in the three-dimensional picture.

/pdf/remarks-on-the-cauchy-problem-for-the-axisymmetric-navier-2ni1zqstez.pdf

Remarks on the Cauchy problem for the axisymmetric Navier-Stokes equations

We propose two different proofs of the fact that Oseen's vortex is the unique solution of the two-dimensional Navier–Stokes equation with a Dirac mass as initial vorticity. The first argument, due to C. E. Wayne and the second named author, is based on an entropy estimate for the vorticity equation in self-similar variables. The second proof is new and relies on symmetrization techniques for parabolic equations. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

On the uniqueness of the solution of the two‐dimensional Navier–Stokes equation with a Dirac mass as initial vorticity

We consider the damped hyperbolic equation $$ \epsilon u_{tt} + u_t \,=\, u_{xx} + \FF(u) , \quad x \in \mathbf{R} , \quad t \ge 0 , \leqno(1) $$ where $\epsilon$ is a positive, not necessarily small parameter. We assume that $\FF(0) = \FF(1) = 0$ and that $\FF$ is concave on the interval [0,1]. Under these hypotheses, (1) has a family of monotone traveling wave solutions (or propagating fronts) connecting the equilibria u=0 and u=1. This family is indexed by a parameter $c \ge c_*$ related to the speed of the front. In the critical case $c=c_*$, we prove that the traveling wave is asymptotically stable with respect to perturbations in a weighted Sobolev space. In addition, we show that the perturbations decay to zero like $t^{-3/2}$ as $t \to +\infty$ and approach a universal self-similar profile, which is independent of $\epsilon$, F, and the initial data. In particular, our solutions behave for large times like those of the parabolic equation obtained by setting $\epsilon = 0$ in (1). The proof of our ...

Scaling Variables and Stability of Hyperbolic Fronts

Existence and stability of propagating fronts for an autocatalytic reaction-diffusion system

https://dmpeli.math.mcmaster.ca/PaperBank/StabilityWaveNLSper.pdf

Thierry Gallay

Papers

Remarks on the Cauchy problem for the axisymmetric Navier-Stokes equations

On the uniqueness of the solution of the two‐dimensional Navier–Stokes equation with a Dirac mass as initial vorticity

Scaling Variables and Stability of Hyperbolic Fronts

Existence and stability of propagating fronts for an autocatalytic reaction-diffusion system

Orbital stability in the cubic defocusing NLS equation: I. Cnoidal periodic waves