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There is a very nice theory of gradient flows in metric spaces by Ambrosio, Gigli and Savaré. One particularly important application is the quadratic Wasserstein setting, where the metric space in question is $(\mathcal{P},\mathcal{W}_2)$, $\mathcal{P}=\mathcal{P}(X)$ being the space of Borel probability measures on a polish space $X$ and $\mathcal{W}_2$ the associated quadratic Wasserstein distance. This theory has been succesfully applied in PDE's (see e.g Otto's geometry of dissipative evolution equations) and has deep connections with optimal mass transport.

In this particular context one uses continuity equations $\partial_t\mu_t+\operatorname{div}(\mu_t v_t)=0$ for measures $\mu\in \mathcal{P}(X)$ in order to formally view $\mathcal{P}$ as a Riemanian Manifold $\mathcal{M}$. The norm of a tangent vector $\dot{\mu_t}\in T_{\mu_t}\mathcal{M}$ can be defined as weigthed $L^2$ $$ |\dot{\mu}_t|_{T_{\mu_t}\mathcal{M}}^2=\int |v_t|^2d\mu_t. $$ I apologize for not giving more details but this is a whole theory of its own.

Here is my question:

Has anyone seen a similar theory with non-linear weights? More precisely: assume that one is only interested in measures $\mu=\rho(x)dx$ that are absolutely continuous w.r.t Lebesgue's measure (say on a base space $X=\Omega\subset \mathbb{R}^d$ for simplicity), and that we are given a fixed "mobility function" $\eta:[0,\infty]\to [0,\infty]$ such that $\eta(0)=0$. Can one use nonlinear continuity equations $\partial_t\rho_t+\operatorname{div}(\eta(\rho_t)v_t)=0$ and weighted norms $$ |\dot{\mu}_t|_{\mu_t}^2\approx\int |v_t|^2\eta(\rho_t)dx $$ to build a similar "theory"? (or at least apply some ideas, I do not want to redo the whole theory)

This question arised in my research from a specific physical system with conservation of mass that enjoys a gradient-flow structure through an Energy-Dissipation-Inequality, but with a prescribed nonlinear mobility function as above. (see [1] for EDI formulations of gradient flows in metric spaces)

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Yes, this issue has been considered. You can start having a look at `A new class of transport distances between measures' by Dolbeault, Nazaret and Savaré (http://link.springer.com/article/10.1007%2Fs00526-008-0182-5).

The basic assumption needed for the theory to work is that the mobility $\eta$ is increasing and concave, as discussed in the paper.

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  • $\begingroup$ A quick look indicates that this is exactly what I needed, thank you Nicola. $\endgroup$ – leo monsaingeon Nov 6 '14 at 23:17

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