Gradient flows of time-dependent functionals in metric spaces and applications to PDEs

Abstract

We develop a gradient-flow theory for time-dependent functionals defined in abstract metric spaces. Global well-posedness and asymptotic behavior of solutions are provided. Conditions on functionals and metric spaces allow to consider the Wasserstein space P 2 (R d ) and apply the results for a large class of PDEs with time-dependent coefficients like confinement and interaction potentials and diffusion. Our results can be seen as an extension of those in Ambrosio- Gigli-Savaré (2005)[2] to the case of timedependent functionals. For that matter, we need to consider some residual terms, timeversions of concepts like λ-convexity, time-differentiability of minimizers for MoreauYosida approximations, and a priori estimates with explicit time-dependence for De Giorgi interpolation. Here, functionals can be unbounded from below and satisfy a type of λ-convexity that changes as the time evolves. AMS MSC2010: 35R20, 34Gxx, 58Exx, 49Q20, 49J40, 35Qxx, 35K15, 60J60, 28A33.

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

“…In [23], where general non-convex problems have been studied, a certain Lipschitz condition for the free energy functional comes into play which resembles our additional conditions on h from (M4). In [9,23], re-proving the classical properties of the minimizing movement scheme is more involved than for our scheme from Definition 1.2-in contrast, deriving higher regularity estimates brings additional difficulties in our case, since now the heat entropy is a time-dependent functional. Non-autonomous equations of Wasserstein gradient flow form with linear mobility have also been investigated in [22] using a time-averaged form of the classical minimizing movement scheme.…”

“…This auxiliary flow is-by construction-the heat flow. Nonautonomous evolution equations of gradient flow type have already been studied in [9,23] from the opposite point of view: there, time-dependent energy functionals on time-independent metric spaces were considered and a different modification of the minimizing movement scheme was investigated. In [23], where general non-convex problems have been studied, a certain Lipschitz condition for the free energy functional comes into play which resembles our additional conditions on h from (M4).…”

Gradient flow for the Boltzmann entropy and Cheeger’s energy on time-dependent metric measure spaces

“…To circumvent this problem we suppose that the set of differentiability points can be chosen independent of x, cf. [10,21]. To illustrate this we give the following example, which has also been discussed in [21].…”

“…Unlike in the static case we additionally have to impose a condition on the difference quotients of the functionals, cf. [10,Theorem 5.4].…”

Gradient flow formulation and longtime behaviour of a constrained Fokker–Planck equation

“…This was the starting point of interpretations of more general nonlinear, nonlocal Fokker-Planck equations as Wasserstein gradient flows with respect to time-dependent energies, on constrained manifolds and even of none dissipative equations. A selection of works introducing certain constraints into gradient flows are [11,6,21].…”