A new family of transportation costs with applications to reaction–diffusion and parabolic equations with boundary conditions

Abstract

Abstract. This paper introduces a family of transportation costs between non-negative measures. This family is used to obtain parabolic and reaction-diffusion equations with drift, subject to Dirichlet boundary condition, as the gradient flow of the entropy functional Ω ρ log ρ + V ρ + 1 dx. In [5], Figalli and Gigli study a transportation cost that can be used to obtain parabolic equations with drift subject to Dirichlet boundary condition. However, the drift and the boundary condition are coupled in that work. The costs in this paper allow the drift and the boundary condition to be detached.

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“…To the best of our knowledge, [20] contains the only other proof of this metric characterization 2 in a (Wasserstein-like) space of measures for a PDE with Dirichlet boundary conditions. In the frameworks of [8,13,15,17], as well as ours, the main obstacles to proving that the limit of the scheme is, in fact, a curve of maximal slope are the need for lower semicontinuity of the descending slope of the entropy and the lack of geodesic convexity. Indeed, as observed in the premise to [15,Theorem 1.8], «it is difficult to find an explicit form of [the slope] and to show its lower semicontinuity [...].…”

“…For the proof we need a lemma, to which we will also often refer later. This lemma, inspired by [17,Lemma 5.8] allows to control (µ − ν) ∂Ω in terms of T (µ, ν) and of the restrictions µ Ω and ν Ω . This fact is convenient for two reasons:…”

“…The proof of the first Step in Proposition 5.13, i.e., the case q = 1, and of the preliminary lemma Lemma 5.16 follow the lines of [13,Proposition 3.7 (24)] and [17,Proposition 5.3]. In all these proofs, the key is to leverage the optimality of µ by constructing small variations.…”

“…Like in the proofs of [13,Theorem 3.5] and [17,Theorem 4.1], the key to Proposition 5.19 is to first determine (see Lemma 5.22) that the measures constructed with (1.7) already solve approximately the Fokker-Planck equation. In order to prove that the limit curve has the desired properties and that convergence holds in L 1 loc (0, ∞); L q (Ω) (Lemma 5.24), two other preliminary lemmas turn out to be particularly useful.…”

A non-local problem for the Fokker-Planck equation related to the Becker-Döring model

“…Let us point out that gradient flows with boundary condition are quite delicate and are first studied in [26] for the heat equation with Dirichlet boundary conditions. McKean-Vlasov equations with non-local interaction and some boundary are just recently studied in [33]. However, the particular boundary condition (1.5) together with the non-local constraint (1.4) has been to our knowledge not studied in the literature so far.…”