Extended dynamic Monge–Kantorovich convergence conjecture

Let $\Omega$ be the domain, let $f=f^+-f^-$ be the prescribed source term, and let $(\mu(t),u(t))$ solve the extended dynamic Monge–Kantorovich equations with initial datum $\mu_0$. For $0\leq\beta\leq1$, define

$$p=\frac{2-\beta}{1-\beta}.$$

Let $u_p$ solve the $p$-Poisson equation. Extended dynamic Monge–Kantorovich convergence conjecture. For every initial datum $\mu_0$, the pair $(\mu(t),u(t))$ converges to

$$\left(\lvert\nabla u_p\rvert^{p-2},u_p\right).$$

This conjecture identifies the long-time equilibrium of the extended dynamics with a $p$-Poisson solution. The surrounding results establish Lyapunov decrease and identify the minimizer in the relevant parameter range, but the full convergence assertion is not established.