There is a very nice theory of gradient flows in metric spaces by [Ambrosio, Gigli and Savaré](http://www.springer.com/birkhauser/mathematics/book/978-3-7643-8721-1). 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](http://en.wikipedia.org/wiki/Wasserstein_metric). This theory has been succesfully applied in PDE's (see e.g Otto's [geometry of dissipative evolution equations](http://www.mis.mpg.de/preprints/1999/preprint1999_8.pdf)) 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](http://www.springer.com/birkhauser/mathematics/book/978-3-7643-8721-1)] for EDI formulations of gradient flows in metric spaces)



Reference request: Wasserstein metric spaces for non linear weights/mobility?
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.


