I'm new to the topics of optimal transport and gradient flows, so probably the question is obvious, but I can't figure it out. Currently I'm reading as an Introduction: { Euclidean, Metric, and Wasserstein } Gradient Flows: an overview
Regarding the Porous Media Equation the paper states:
If one considers the case of particles which are advected by a potential and subject to this kind of diffusion, the PDE reads $$\partial_t\varrho-\Delta\varrho^m-\nabla\cdot(\varrho\nabla V)= 0$$ for an exponent $m$ > 1. One can formally check that this is the equation of the gradient flow of the energy $$F(\varrho)=\frac{1}{m-1}\int\varrho^m(x)dx+\int V(x)\varrho(x)dx$$(set to $+\infty$ if $\varrho\notin L^m$). [...] Note that, in the discrete step $\min_\varrho F(\varrho)+\frac{W^2_2(\varrho,\varrho_0)}{2\tau}$, the solution $\varrho$ satisfies $$\begin{cases} \frac{m}{m-1}\varrho^{m-1}+ V +\frac{\varphi}{\tau}=C & \varrho-\text{a.e.} \\ \frac{m}{m-1}\varrho^{m-1}+ V +\frac{\varphi}{\tau}\geq C & \text{on} \{\varrho =0\}. \end{cases}$$ This allows to express $\varrho^{m-1}=\frac{m}{m-1}(C-V-\frac{\varphi}{\tau})_+$ and implies that $\varrho$ is compactly supported if $\varrho_0$ is compactly supported, as soon as $V$ has some growth conditions.
Here $W_2$ is the 2-Wasserstein distance, $\varphi$ the Kantorovich potential and $C$ some constant.
I don't undertstand why $\varrho$ is compactly supported in this case.
I suppose that the fact that $\varrho_0$ has compact support has influence through $\varphi$. Sure, if $\varphi$ grows faster than $\vert V\vert$ for large values of $\vert x\vert$, then this would be true. I know that $x\mapsto \frac{x^2}{2}-\varphi (x)$ is convex and that the transport map is given by $T(x)=x-\nabla \varphi (x)$.