Nonlinear mobility continuity equations and generalized displacement convexity

By José Antonio Carrillo, S. Lisini, Giuseppe Savar'e, Dejan Slepvcev. 2009

112 citations recorded by the source corpus.

We consider the geometry of the space of Borel measures endowed with a distance that is defined by generalizing the dynamical formulation of the Wasserstein distance to concave, nonlinear mobilities. We investigate the energy landscape of internal, potential, and interaction energies. For the internal energy, we give an explicit sufficient condition for geodesic convexity which generalizes the condition of McCann. We take an eulerian approach that does not require global information on the geodesics. As by-product, we obtain existence, stability, and contraction results for the semigroup obtained by solving the homogeneous Neumann boundary value problem for a nonlinear diffusion equation in a convex bounded domain. For the potential energy and the interaction energy, we present a non-rigorous argument indicating that they are not displacement semiconvex.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/881c507423e3d234ebcd6651383e6e6ddab83e5d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Gradient flows of the entropy for finite Markov chains (2011, background): “In a continuous setting, metrics with these properties have been studied in the recent papers [6, 9].”
Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces (2015, background): “If h is concave and the Bakry-Émery condition BE(0,∞) holds, it has been proved in [22] that we still have d dt H(%, φ) ≥ 0.”
Weak Solutions for First Order Mean Field Games with Local Coupling (2013, background): “Second, [10] strongly benefited from the L∞−estimate on m obtained for optimal transport problems by Carrillo, Lisini, Savaré and D. Slepcev [16].”
A User’s Guide to Optimal Transport (2013, unknown): “Savaré [33] and [26] of J.”
Cahn–Hilliard and thin film equations with nonlinear mobility as gradient flows in weighted-Wasserstein metrics (2012, methodology): “ regular for every u∈ Xr(Ω), t≥ 0, and for every regular curve γ. (31) 2.3. Survey: weighted transport distances. We shall now review the weighted transport distances W m introduced in [12] (see also [7] and [23]) without going into details about their formal deﬁnition. When M = ∞ they could in fact”
Geodesics for a class of distances in the space of probability measures (2012, unknown): “This estimate will follow from a generalized displacement convexity argument of Carrillo, Lisini, Savar´e and Slepcev [6].”