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Discrete & Continuous Dynamical Systems - A

2014 , Volume 34 , Issue 4

Special Issue on Optimal Transport and Applications

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Alessio Figalli and  Filippo Santambrogio
2014, 34(4): i-ii doi: 10.3934/dcds.2014.34.4i +[Abstract](35) +[PDF](73.8KB)
Optimal mass transportation can be traced back to Gaspard Monge's paper in 1781. There, for engineering/military reasons, he was studying how to minimize the cost of transporting a given distribution of mass from one location to another, giving rise to a challenging mathematical problem. This problem, an optimization problem in a certain class of maps, had to wait for almost two centuries before seeing significant progress (starting with Leonid Kantorovich in 1942), even on the very fundamental question of the existence of an optimal map. Due to these connections with several other areas of pure and applied mathematics, optimal transportation has received much renewed attention in the last twenty years. Indeed, it has become an increasingly common and powerful tool at the interface between partial differential equations, fluid mechanics, geometry, probability theory, and functional analysis. At the same time, it has led to significant developments in applied mathematics, with applications ranging from economics, biology, meteorology, design, to image processing. Because of the success and impact that this subject is still receiving, we decided to create a special issue collecting selected papers from leading experts in the area.

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A global existence result for the semigeostrophic equations in three dimensional convex domains
Luigi Ambrosio , Maria Colombo , Guido De Philippis and  Alessio Figalli
2014, 34(4): 1251-1268 doi: 10.3934/dcds.2014.34.1251 +[Abstract](27) +[PDF](457.7KB)
Exploiting recent regularity estimates for the Monge-Ampère equation, under some suitable assumptions on the initial data we prove global-in-time existence of Eulerian distributional solutions to the semigeostrophic equations in 3-dimensional convex domains.
Prescription of Gauss curvature on compact hyperbolic orbifolds
Jérôme Bertrand
2014, 34(4): 1269-1284 doi: 10.3934/dcds.2014.34.1269 +[Abstract](30) +[PDF](400.9KB)
In this paper, we generalize a result by Alexandrov on the Gauss curvature prescription for Euclidean convex bodies. We prove an analogous result for hyperbolic orbifolds. In addition to the duality theory for convex sets, our main tool comes from optimal mass transport.
Approximation of a simple Navier-Stokes model by monotonic rearrangement
Yann Brenier
2014, 34(4): 1285-1300 doi: 10.3934/dcds.2014.34.1285 +[Abstract](32) +[PDF](389.1KB)
We consider the very simple Navier-Stokes model for compressible fluids in one space dimension, where there is no temperature equation and both the pressure and the viscosity are proportional to the density. We show that the resolution of this Navier-Stokes system can be reduced, through the crucial intervention of a monotonic rearrangement operator, to the time discretization of a very elementary differential equation with noise. In addition, our result can be easily extended to a related Navier-Stokes-Poisson system.
Optimal location problems with routing cost
Giuseppe Buttazzo , Serena Guarino Lo Bianco and  Fabrizio Oliviero
2014, 34(4): 1301-1317 doi: 10.3934/dcds.2014.34.1301 +[Abstract](25) +[PDF](2648.9KB)
In the paper a model problem for the location of a given number $N$ of points in a given region $\Omega$ and with a given resources density $\rho(x)$ is considered. The main difference between the usual location problems and the present one is that in addition to the location cost an extra routing cost is considered, that takes into account the fact that the resources have to travel between the locations on a point-to-point basis. The limit problem as $N\to\infty$ is characterized and some applications to airfreight systems are shown.
Uniqueness for Keller-Segel-type chemotaxis models
José Antonio Carrillo , Stefano Lisini and  Edoardo Mainini
2014, 34(4): 1319-1338 doi: 10.3934/dcds.2014.34.1319 +[Abstract](60) +[PDF](514.6KB)
We prove uniqueness in the class of integrable and bounded nonnegative solutions in the energy sense to the Keller-Segel (KS) chemotaxis system. Our proof works for the fully parabolic KS model, it includes the classical parabolic-elliptic KS equation as a particular case, and it can be generalized to nonlinear diffusions in the particle density equation as long as the diffusion satisfies the classical McCann displacement convexity condition. The strategy uses Quasi-Lipschitz estimates for the chemoattractant equation and the above-the-tangent characterizations of displacement convexity. As a consequence, the displacement convexity of the free energy functional associated to the KS system is obtained from its evolution for bounded integrable initial data.
On the twist condition and $c$-monotone transport plans
Thierry Champion and  Luigi De Pascale
2014, 34(4): 1339-1353 doi: 10.3934/dcds.2014.34.1339 +[Abstract](30) +[PDF](436.7KB)
A usual approach for proving the existence of an optimal transport map, be it in ${\mathbb R}^d$ or on more general manifolds, involves a regularity condition on the transport cost (the so-called Left Twist condition, i.e. the invertibility of the gradient in the first variable) as well as the fact that any optimal transport plan is supported on a cyclically-monotone set. Under the classical assumption that the initial measure does not give mass to sets with $\sigma$-finite $\mathcal{H}^{d-1}$ measure and a stronger regularity condition on the cost (the Strong Left Twist), we provide a short and self-contained proof of the fact that any feasible transport plan (optimal or not) satisfying a $c$-monotonicity assumption is induced by a transport map. We also show that the usual costs induced by Tonelli Lagrangians satisfy the Strong Left Twist condition we propose.
Gradient flow structures for discrete porous medium equations
Matthias Erbar and  Jan Maas
2014, 34(4): 1355-1374 doi: 10.3934/dcds.2014.34.1355 +[Abstract](39) +[PDF](442.7KB)
We consider discrete porous medium equations of the form $\partial_t\rho_t = \Delta \phi(\rho_t)$, where $\Delta$ is the generator of a reversible continuous time Markov chain on a finite set $\boldsymbol{\chi} $, and $\phi$ is an increasing function. We show that these equations arise as gradient flows of certain entropy functionals with respect to suitable non-local transportation metrics. This may be seen as a discrete analogue of the Wasserstein gradient flow structure for porous medium equations in $\mathbb{R}^n$ discovered by Otto. We present a one-dimensional counterexample to geodesic convexity and discuss Gromov-Hausdorff convergence to the Wasserstein metric.
On the Lagrangian structure of quantum fluid models
Philipp Fuchs , Ansgar Jüngel and  Max von Renesse
2014, 34(4): 1375-1396 doi: 10.3934/dcds.2014.34.1375 +[Abstract](34) +[PDF](443.4KB)
Some quantum fluid models are written as the Lagrangian flow of mass distributions and their geometric properties are explored. The first model includes magnetic effects and leads, via the Madelung transform, to the electromagnetic Schrödinger equation in the Madelung representation. It is shown that the Madelung transform is a symplectic map between Hamiltonian systems. The second model is obtained from the Euler-Lagrange equations with friction induced from a quadratic dissipative potential. This model corresponds to the quantum Navier-Stokes equations with density-dependent viscosity. The fact that this model possesses two different energy-dissipation identities is explained by the definition of the Noether currents.
Optimal transport and large number of particles
Wilfrid Gangbo and  Andrzej Świech
2014, 34(4): 1397-1441 doi: 10.3934/dcds.2014.34.1397 +[Abstract](95) +[PDF](675.6KB)
We present an approach for proving uniqueness of ODEs in the Wasserstein space. We give an overview of basic tools needed to deal with Hamiltonian ODE in the Wasserstein space and show various continuity results for value functions. We discuss a concept of viscosity solutions of Hamilton-Jacobi equations in metric spaces and in some cases relate it to viscosity solutions in the sense of differentials in the Wasserstein space.
Metric cycles, curves and solenoids
Vladimir Georgiev and  Eugene Stepanov
2014, 34(4): 1443-1463 doi: 10.3934/dcds.2014.34.1443 +[Abstract](36) +[PDF](402.0KB)
We prove that every one-dimensional real Ambrosio-Kirchheim current with zero boundary (i.e. a cycle) in a lot of reasonable spaces (including all finite-dimensional normed spaces) can be represented by a Lipschitz curve parameterized over the real line through a suitable limit of Cesàro means of this curve over a subsequence of symmetric bounded intervals (viewed as currents). It is further shown that in such spaces, if a cycle is indecomposable, i.e. does not contain ``nontrivial'' subcycles, then it can be represented again by a Lipschitz curve parameterized over the real line through a limit of Cesàro means of this curve over every sequence of symmetric bounded intervals, that is, in other words, such a cycle is a solenoid.
Remarks on multi-marginal symmetric Monge-Kantorovich problems
Nassif Ghoussoub and  Bernard Maurey
2014, 34(4): 1465-1480 doi: 10.3934/dcds.2014.34.1465 +[Abstract](27) +[PDF](405.1KB)
Symmetric Monge-Kantorovich transport problems involving a cost function given by a family of vector fields were used by Ghoussoub-Moameni to establish polar decompositions of such vector fields into $m$-cyclically monotone maps composed with measure preserving $m$-involutions ($m\geq 2$). In this note, we relate these symmetric transport problems to the Brenier solutions of the Monge and Monge-Kantorovich problem, as well as to the Gangbo-Święch solutions of their multi-marginal counterparts, both of which involving quadratic cost functions.
The Abresch-Gromoll inequality in a non-smooth setting
Nicola Gigli and  Sunra Mosconi
2014, 34(4): 1481-1509 doi: 10.3934/dcds.2014.34.1481 +[Abstract](23) +[PDF](547.5KB)
We prove that the Abresch-Gromoll inequality holds on infinitesimally Hilbertian $CD(K,N)$ spaces in the same form as the one available on smooth Riemannian manifolds.
Hessian metrics, $CD(K,N)$-spaces, and optimal transportation of log-concave measures
Alexander V. Kolesnikov
2014, 34(4): 1511-1532 doi: 10.3934/dcds.2014.34.1511 +[Abstract](32) +[PDF](464.0KB)
We study the optimal transportation mapping $\nabla \Phi : \mathbb{R}^d \mapsto \mathbb{R}^d$ pushing forward a probability measure $\mu = e^{-V} \ dx$ onto another probability measure $\nu = e^{-W} \ dx$. Following a classical approach of E. Calabi we introduce the Riemannian metric $g = D^2 \Phi$ on $\mathbb{R}^d$ and study spectral properties of the metric-measure space $M=(\mathbb{R}^d, g, \mu)$. We prove, in particular, that $M$ admits a non-negative Bakry--Émery tensor provided both $V$ and $W$ are convex. If the target measure $\nu$ is the Lebesgue measure on a convex set $\Omega$ and $\mu$ is log-concave we prove that $M$ is a $CD(K,N)$ space. Applications of these results include some global dimension-free a priori estimates of $\| D^2 \Phi\|$. With the help of comparison techniques on Riemannian manifolds and probabilistic concentration arguments we proof some diameter estimates for $M$.
A survey of the Schrödinger problem and some of its connections with optimal transport
Christian Léonard
2014, 34(4): 1533-1574 doi: 10.3934/dcds.2014.34.1533 +[Abstract](81) +[PDF](723.1KB)
This article is aimed at presenting the Schrödinger problem and some of its connections with optimal transport. We hope that it can be used as a basic user's guide to Schrödinger problem. We also give a survey of the related literature. In addition, some new results are proved.
Congestion-driven dendritic growth
Bertrand Maury , Aude Roudneff-Chupin and  Filippo Santambrogio
2014, 34(4): 1575-1604 doi: 10.3934/dcds.2014.34.1575 +[Abstract](31) +[PDF](602.2KB)
In order to observe growth phenomena in biology where dendritic shapes appear, we propose a simple model where a given population evolves feeded by a diffusing nutriment, but is subject to a density constraint. The particles (e.g., cells) of the population spontaneously stay passive at rest, and only move in order to satisfy the constraint $\rho\leq 1$, by choosing the minimal correction velocity so as to prevent overcongestion. We treat this constraint by means of projections in the space of densities endowed with the Wasserstein distance $W_2$, defined through optimal transport. This allows to provide an existence result and suggests some numerical computations, in the same spirit of what the authors did for crowd motion (but with extra difficulties, essentially due to the fact that the total mass may increase). The numerical simulations show, according to the values of the parameter and in particular of the diffusion coefficient of the nutriment, the formation of dendritic patterns in the space occupied by cells.
A glimpse into the differential topology and geometry of optimal transport
Robert J. McCann
2014, 34(4): 1605-1621 doi: 10.3934/dcds.2014.34.1605 +[Abstract](129) +[PDF](473.3KB)
This note exposes the differential topology and geometry underlying some of the basic phenomena of optimal transportation. It surveys basic questions concerning Monge maps and Kantorovich measures: existence and regularity of the former, uniqueness of the latter, and estimates for the dimension of its support, as well as the associated linear programming duality. It shows the answers to these questions concern the differential geometry and topology of the chosen transportation cost. It also establishes new connections --- some heuristic and others rigorous --- based on the properties of the cross-difference of this cost, and its Taylor expansion at the diagonal.
Multi-marginal optimal transport and multi-agent matching problems: Uniqueness and structure of solutions
Brendan Pass
2014, 34(4): 1623-1639 doi: 10.3934/dcds.2014.34.1623 +[Abstract](33) +[PDF](413.9KB)
We prove uniqueness and Monge solution results for multi-marginal optimal transportation problems with a certain class of surplus functions; this class arises naturally in multi-agent matching problems in economics. This result generalizes a seminal result of Gangbo and Święch [17]. Of particular interest, we show that this also yields a partial generalization of the Gangbo-Święch result to manifolds; alternatively, we can think of this as a partial extension of McCann's theorem for quadratic costs on manifolds to the multi-marginal setting [23].
    We also show that the class of surplus functions considered here neither contains, nor is contained in, the class of surpluses studied in [27], another generalization of Gangbo and Święch's result.
Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in $RCD (K, \infty)$ metric measure spaces
Giuseppe Savaré
2014, 34(4): 1641-1661 doi: 10.3934/dcds.2014.34.1641 +[Abstract](36) +[PDF](565.6KB)
We prove that the linear ``heat'' flow in a $RCD (K, \infty)$ metric measure space $(X, d, m)$ satisfies a contraction property with respect to every $L^p$-Kantorovich-Rubinstein-Wasserstein distance, $p\in [1,\infty]$. In particular, we obtain a precise estimate for the optimal $W_\infty$-coupling between two fundamental solutions in terms of the distance of the initial points.
    The result is a consequence of the equivalence between the $RCD (K, \infty)$ lower Ricci bound and the corresponding Bakry-Émery condition for the canonical Cheeger-Dirichlet form in $(X, d, m)$. The crucial tool is the extension to the non-smooth metric measure setting of the Bakry's argument, that allows to improve the commutation estimates between the Markov semigroup and the Carré du Champ $\Gamma$ associated to the Dirichlet form.
    This extension is based on a new a priori estimate and a capacitary argument for regular and tight Dirichlet forms that are of independent interest.
On the local theory of prescribed Jacobian equations
Neil S. Trudinger
2014, 34(4): 1663-1681 doi: 10.3934/dcds.2014.34.1663 +[Abstract](47) +[PDF](443.4KB)
We develop the fundamentals of a local regularity theory for prescribed Jacobian equations which extend the corresponding results for optimal transportation equations. In this theory the cost function is extended to a generating function through dependence on an additional scalar variable. In particular we recover in this generality the local regularity theory for potentials of Ma, Trudinger and Wang, along with the subsequent development of the underlying convexity theory.
On landscape functions associated with transport paths
Qinglan Xia
2014, 34(4): 1683-1700 doi: 10.3934/dcds.2014.34.1683 +[Abstract](29) +[PDF](428.0KB)
In this paper, we introduce a multiple-sources version of the landscape function which was originally introduced by Santambrogio in [10]. More precisely, we study landscape functions associated with a transport path between two atomic measures of equal mass. We also study p-harmonic functions on a directed graph for nonpositive $p$. We show an equivalence relation between landscape functions associated with an $\alpha $-transport path and $ p$-harmonic functions on the underlying graph of the transport path for $ p=\alpha /(\alpha -1)$, which is the conjugate of $\alpha $. Furthermore, we prove the Lipschitz continuity of a landscape function associated with an optimal transport path on each of its connected components.

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