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

Research paper by Viorel Barbu, Michael Röckner

Indexed on: 06 May '20Published on: 05 May '20Published in: arXiv - Mathematics - Analysis of PDEs



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, $b\geq 0$, $D\in(L^2\cap L^\infty)(\mathbb{R}^d;\mathbb{R}^d)$ with ${\rm div}\, D\in L^\infty(\mathbb{R}^d)$, and ${\rm div}\,D\geq0$, $\beta$ 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 ${\rm div}\,D\geq0$, $\beta'(r)\geq a|r|^{\alpha-1}$, and $|\beta(r)|\leq C r^\alpha$, $\alpha\geq1,$ $\alpha>\frac{d-2}d$, $d\geq3$, then $|u(t)|_{L^\infty}\le Ct^{-\frac d{d+(\alpha-1)d}}\ |u_0|^{\frac2{2+(m-1)d}},$ $t>0$, and the existence extends to initial data $u_0$ in the space $\mathcal{M}_b$ of bounded measures in $\mathbb{R}^d$. The solution map $\mu\mapsto S(t)\mu$, $t\geq0$, is a Lipschitz contractions on $\mathcal{M}_b$ and weakly continuous in $t\in[0,\infty)$. 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.