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

We develop a full theory for the new class of Optimal Entropy-Transport problems between nonnegative and finite Radon measures in general topological spaces. These problems arise quite naturally by relaxing the marginal constraints typical of Optimal Transport problems: given a pair of finite measures (with possibly different total mass), one looks for minimizers of the sum of a linear transport functional and two convex entropy functionals, which quantify in some way the deviation of the marginals of the transport plan from the assigned measures. As a powerful application of this theory, we study the particular case of Logarithmic Entropy-Transport problems and introduce the new Hellinger–Kantorovich distance between measures in metric spaces. The striking connection between these two seemingly far topics allows for a deep analysis of the geometric properties of the new geodesic distance, which lies somehow between the well-known Hellinger–Kakutani and Kantorovich–Wasserstein distances.

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Optimal Entropy-Transport problems and a new Hellinger–Kantorovich distance between positive measures

A second order in time, uniquely solvable, unconditionally stable numerical scheme for Cahn-Hilliard-Navier-Stokes equation

We describe the mathematical theory of diffusion and heat transport with a view to including some of the main directions of recent research. The linear heat equation is the basic mathematical model that has been thoroughly studied in the last two centuries. It was followed by the theory of parabolic equations of different types. In a parallel development, the theory of stochastic partial differential equations gives a foundation to the probabilistic study of diffusion.

The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion

We analyze the decay and instant regularization properties of the evolution semigroups generated by two-dimensional drift-diffusion equations in which the scalar is advected by a shear flow and dissipated by full or partial diffusion. We consider both the space-periodic $${\mathbb{T}^2}$$
 setting and the case of a bounded channel $${\mathbb{T} \times [0,1]}$$
 with no-flux boundary conditions. In the infinite Peclet number limit (diffusivity $${\nu\to 0}$$
 ), our work quantifies the enhanced dissipation effect due to the shear. We also obtain hypoelliptic regularization, showing that solutions are instantly Gevrey regular even with only partial diffusion. The proofs rely on localized spectral gap inequalities and ideas from hypocoercivity with an augmented energy functional with weights replaced by pseudo-differential operators (of a rather simple form). As an application, we study small noise inviscid limits of invariant measures of stochastic perturbations of passive scalars, and show that the classical Freidlin scaling between noise and diffusion can be modified. In particular, although statistically stationary solutions blow up in $${H^1}$$
 in the limit $${\nu \to 0}$$
 , we show that viscous invariant measures still converge to a unique inviscid measure.

Enhanced Dissipation, Hypoellipticity, and Anomalous Small Noise Inviscid Limits in Shear Flows

We consider Rayleigh–Benard convection as modelled by the Boussinesq equations in the infinite-Prandtl-number limit. We are interested in the scaling of the average upward heat transport, the Nusselt number Nu, in terms of the non-dimensionalized temperature forcing, the Rayleigh number Ra. Experiments, asymptotics and heuristics suggest that Nu ∼ Ra1/3. This work is mostly inspired by two earlier rigorous work on upper bounds of Nu in terms of Ra. (1) The work of Constantin and Doering establishing Nu ≲ Ra1/3ln 2/3Ra with help of a (logarithmically failing) maximal regularity estimate in L∞ on the level of the Stokes equation. (2) The work of Doering, Reznikoff and the first author establishing Nu ≲ Ra1/3ln 1/3Ra with help of the background field method. The paper contains two results. (1) The background field method can be slightly modified to yield Nu ≲ Ra1/3ln 1/15Ra. (2) The estimates behind the background field method can be combined with the maximal regularity in L∞ to yield Nu ≲ Ra1/3ln 1/3ln Ra —...

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Rayleigh–Bénard convection: Improved bounds on the Nusselt number

We consider a model for mixing binary viscous fluids under an incompressible flow. We prove the impossibility of perfect mixing in finite time for flows with finite viscous dissipation. As measures of mixedness we consider a Monge–Kantorovich–Rubinstein transportation distance and, more classically, the H−1 norm. We derive rigorous a priori lower bounds on these mixing norms which show that mixing cannot proceed faster than exponentially in time. The rate of the exponential decay is uniform in the initial data.

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Maximal mixing by incompressible fluid flows

We consider a model for mixing binary viscous fluids under an incompressible flow. We proof the impossibility of perfect mixing in finite time for flows with finite viscous dissipation. As measures of mixedness we consider a Monge--Kantorovich--Rubinstein transportation distance and, more classically, the $H^{-1}$ norm. We derive rigorous a priori lower bounds on these mixing norms which show that mixing cannot proceed faster than exponentially in time. The rate of the exponential decay is uniform in the initial data.

We consider the coarsening process of a binary viscous liquid after a temperature quench. In a first, diffusion-dominated coarsening regime (“evaporation-recondensation process”), the typical length scale $\ell$ increases according to the power law $\ell\sim t^{1/3}$, where t is the time. Siggia [Phys. Rev. A, 20 (1979), pp. 595–605] argued that in a second regime, coarsening should be mediated by viscous flow of the mixture. This leads to a crossover in the coarsening rates to the power law $\ell\sim t$. We consider a simple sharp-interface model which just allows for flow-mediated coarsening. For this model, we prove rigorously that coarsening cannot proceed faster than $\ell\sim t$. The analysis follows closely a method proposed in [R. V. Kohn and F. Otto, Comm. Math. Phys., 229 (2002), pp. 375–395], which is based on the gradient flow structure of the evolution. The analysis makes use of a Monge–Kantorowicz–Rubinstein transportation distance with logarithmic cost function as a proxy for the intrinsic ...

http://www.iam.uni-bonn.de/fileadmin/user_upload/seis/Papers/460-komplett.pdf

Upper Bounds on Coarsening Rates in Demixing Binary Viscous Liquids

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Christian Seis

Papers

Rayleigh–Bénard convection: Improved bounds on the Nusselt number

Maximal mixing by incompressible fluid flows

Maximal mixing by incompressible fluid flows

Upper Bounds on Coarsening Rates in Demixing Binary Viscous Liquids

A quantitative theory for the continuity equation