Solutions for nonlinear Fokker-Planck equations with measures as initial data and McKean-Vlasov equations

Abstract

One proves the existence and uniqueness of a generalized (mild) solution for the nonlinear Fokker-Planck equation (FPE) \begin{align*} &u_t-\Delta (\beta(u))+{\mathrm{ div}}(D(x)b(u)u)=0, \quad t\geq0,\ x\in\mathbb{R}^d,\ d\ne2, \\ &u(0,\cdot)=u_0,\mbox{in }\mathbb{R}^d, \end{align*} where $u_0\in L^1(\mathbb{R}^d)$, $\beta\in C^2(\mathbb{R})$ is a nondecreasing function, $b\in C^1$, bounded, $b\ge0$, $D\in {L^\infty}(\mathbb{R}^d;\mathbb{R}^d)$, ${\rm div}\,D\in L^2(\mathbb{R}^d)+L^\infty(\mathbb{R}^d),$ with ${({\rm div}\, D)^-}\in L^\infty(\mathbb{R}^d)$, $\beta$ strictly increasing, if $b$ is not constant. Moreover, $t\to u(t,u_0)$ is a semigroup of contractions in $L^1(\mathbb{R}^d)$, which leaves invariant the set of probability density functions in $\mathbb{R}^d$. If ${\rm div}\,D\ge0$, $\beta'(r)\ge a|r|^{\alpha-1}$, and $|\beta(r)|\le C r^\alpha$, $\alpha\ge1,$ $d\ge3$, then $|u(t)|_{L^\infty}\le Ct^{-\frac d{d+(\alpha-1)d}}\ |u_0|^{\frac2{2+(m-1)d}},$ $t>0$, and, if $D\in L^2(\mathbb{R}^d;\mathbb{R}^d)$, the existence extends to initial data $u_0$ in the space $\mathcal{M}_b$ of bounded measures in $\mathbb{R}^d$. As a consequence for arbitrary initial laws, we obtain weak solutions to a class of McKean-Vlasov SDEs with coefficients which have singular dependence on the time marginal laws.