On a class of modified Wasserstein distances induced by concave mobility functions defined on bounded intervals

Abstract

We study a new class of distances between Radon measures similar to those studied in [13]. These distances (more correctly pseudo-distances because can assume the value +∞) are defined generalizing the dynamical formulation of the Wasserstein distance by means of a concave mobility function. We are mainly interested in the physical interesting case (not considered in [13]) of a concave mobility function defined in a bounded interval. We state the basic properties of the space of measures endowed with this pseudo-distance. Finally, we study in detail two cases: the set of measures defined in R d with finite moments and the set of measures defined in a bounded convex set. In the two cases we give sufficient conditions for the convergence of sequences with respect to the distance and we prove a property of boundedness.

Nonlinear mobility continuity equations and generalized displacement convexity

“…In view of applications to Cahn-Hiliard models, another interesting example, still leading to the heat equation, is represented by the functional 1]. Notice that in this case the positivity domain of the mobility m is the finite interval [0, 1], a case that has not been explicitly considered in [19], but that can be still covered by a careful analysis (see [31]). …”

Existence of solutions for a class of fourth order cross-diffusion systems of gradient flow type

“…i.e., W M is the canonical product of the distances W m k for each of the scalar components which have been defined in [10,22] as in (3.3) for the scalar case n = 1.…”

“…A brief review of some essential properties of W M is provided in Section 3.2. Gradient flows in W M for simpler functionals than (1.4) have been studied in [35], following up on the results [10,22,7] for scalar equations with nonlinear mobility functions. The initial motivation for studying (1.1) is its similarity to multi-component Cahn-Hilliard systems (see e.g.…”

Well-posedness of evolution equations with time-dependent nonlinear mobility: A modified minimizing movement scheme

“…We do not seek for full generality of energy functionals E here, since our main interest is focussed on the time-dependent mobility function m : R ≥0 ×R ≥0 → R ≥0 which turns (1.1) into a non-autonomous nonlinear evolution equation. When m does not explicitly depend on t, it is known that (1.1) possesses a variational structure [8,16,17], if m is nonnegative and concave on the interior of an interval [0, S], the so-called value space [29], with S ∈ R ≥0 ∪ {+∞}. In this work, we require that at each fixed time t ≥ 0, m(t, ·) is an admissible mobility in the sense of [8,16] (see condition (M1) below), where the corresponding value spaces [0, S(t)] are assumed to be expanding over time, i.e., S is nondecreasing.…”

“…We use the (pointwise in t) formal gradient structure of (1.1) from [8,16] with respect to the metric W m(t,·) induced by m(t, ·):…”