Nonlinear Fokker-Planck equations as smooth Hilbertian gradient flows
Viorel Barbu, Michael Röckner
TL;DR
The paper casts the nonlinear Fokker–Planck equation with nonlinear diffusion and drift as a smooth gradient flow on a refined probability space ${\\mathcal P}^*$ endowed with a Hilbertian tangent structure. It proves $H^{-1}$-regularity of the semigroup and derives an explicit gradient of the energy $E(u) = \\int (\\eta(u) + \\Phi u) dx$, yielding the evolution $\\frac{d}{dt}u(t) = -\\nabla E_{u(t)}$ together with an energy-dissipation law. The framework provides a rigorous geometric description of NFPE dynamics, with well-posed, dissipative evolution and a connection to Wasserstein gradient flow in the special case $b(r) \\equiv 1$. This offers a robust variational view of nonlinear diffusion processes and McKean–Vlasov-type dynamics, with implications for long-time behavior and equilibrium formation.
Abstract
Under suitable assumptions on $β:\mathbb{R}\!\to\!\mathbb{R}, \,D:\mathbb{R}^d\!\to\!\mathbb{R}^d$ and $b:\mathbb{R}^d\!\to\!\mathbb{R}$, the nonlinear Fokker-Planck equation $u_t-Δβ(u)+{\rm div}(Db(u)u)=0$, in $(0,\infty)\times\mathbb{R}^d$ where $D=-\nablaΦ$, can be identified as a smooth gradient flow $\frac{d^+}{dt}\,u(t)+\nabla E_{u(t)}=0$, $\forall t>0$. Here, $E:\mathcal{P}^*\cap L^\infty(\mathbb{R}^d)\to\mathbb{R}$ is the energy function associated to the equation, where $\mathcal{P}^*$ is a certain convex subset of the space of probability densities. $\mathcal{P}^*$ is invariant under the flow and $\nabla E_u$ is the gradient of $E$, that is, the tangent vector field to $\mathcal{P}$ at $u$ defined by $\left<\nabla E_u,z_u\right>_u={\rm diff}\,E_u\cdot z_u$ for all vector fields $z_u$ on $\mathcal{P}^*$, where $\left<\cdot,\cdot\right>_u$ is a scalar product on a suitable tangent space $\mathcal{T}_u(\mathcal{P}^*)\subset\mathcal{D}'(\mathbb{R}^d)$.