Nonlinear Fokker–Planck equations as smooth Hilbertian gradient flows

Abstract

Under suitable assumptions on $\beta :\mathbb{R}\to \mathbb{R}$, $D:{\mathbb{R}}^d\to {\mathbb{R}}^d$, and $b:{\mathbb{R}}^d\to \mathbb{R}$, the nonlinear Fokker–Planck equation $u_t-\Delta \beta (u)+\mathrm{div}(Db(u)u)=0$, in $(0,\infty )\times {\mathbb{R}}^d$ where $D=-\nabla \Phi$, can be identified as a smooth gradient flow $\frac{d^{+}}{dt}u(t)+\nabla E_{u(t)}=0$, $\forall t>0$. Here, $E:{\mathcal{P}}^{\ast }\cap L^{\infty }({\mathbb{R}}^d)\to \mathbb{R}$ is the energy function associated with the equation, where ${\mathcal{P}}^{\ast }$ is a certain convex subset of the space of probability densities. ${\mathcal{P}}^{\ast }$ 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 $⟨\nabla E_u,z_u{⟩}_u=\mathrm{diff}{E}_u\cdot z_u$ for all vector fields $z_u$ on ${\mathcal{P}}^{\ast }$, where $⟨\cdot ,\cdot {⟩}_u$ is a scalar product on a suitable tangent space ${\mathcal{T}}_u({\mathcal{P}}^{\ast })\subset {\mathcal{D}}^{\prime }({\mathbb{R}}^d)$.