Nonlinear Fokker-Planck equations with time-dependent coefficients

Abstract

An operatorial based approach is used here to prove the existence and uniqueness of a strong solution $u$ to the time-varying nonlinear Fokker--Planck equation $u_t(t,x)-\Delta(a(t,x,u(t,x))u(t,x))+{\rm div}(b(t,x,u(t,x))u(t,x))=0$ in $(0,\infty)\times \mathbb{R}$ $u(0,x)=u_0(x),\ x\in\mathbb{R}^d$ in the Sobolev space $H^{-1}(\mathbb{R}^d)$, under appropriate conditions on the $a:[0,T]\times\mathbb{R}^d\times\mathbb{R}\to\mathbb{R}$ and $b:[0,T]\times\mathbb{R}^d\times\mathbb{R}\to\mathbb{R}^d.$ It is proved also that, if $u_0$ is a density of a probability measure, so is $u(t,\cdot)$ for all $t\ge0$. Moreover, we construct a weak solution to the McKean-Vlasov SDE associated with the Fokker-Planck equation such that $u(t)$ is the density of its time marginal law. MSC: 60H15, 47H05, 47J05. Keywords: Fokker--Planck equation, Cauchy problem, stochastic differential equation, Sobolev space, periodic solution.