以测度为初值的非线性Fokker-Planck方程和McKean-Vlasov方程的解
Solutions for nonlinear Fokker-Planck equations with measures as initial data and McKean-Vlasov equations
摘要：
证明了非线性Fokker-Planck方程（FPE）\begin{align*} &u_t-Δ(β(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*}广义（温和）解的存在性和唯一性，其中$u_0\in L^1(\mathbb{R}^d)$，$β\in C^2(\mathbb{R})$是非减函数，$b\in C^1$，有界，$b\ge0$，$D\in {L^\infty}(\mathbb{R}^d;\mathbb{R}^d)$，${\rm div}\,D\in L^2(\mathbb{R}^d)+L^\infty(\mathbb{R}^d),$且${({\rm div}\, D)^-}\in L^\infty(\mathbb{R}^d)$，$β$严格递增，如果$b$不是常数.此外，$t\to u(t,u_0)$是$L^1(\mathbb{R}^d)$中的压缩半群，它使$\mathbb{R}^d$中的概率密度函数集保持不变。如果${\rm div}\,D\ge0$，$β'(r)\ge a|r|^{α-1}$，和$|β(r)|\le C r^α$，$α\ge1,$ $d\ge3$，那么$|u(t)|_{L^\infty}\le Ct^{-\frac d{d+(α-1)d}}\ |u_0|^{\frac2{2+(m-1)d}},$ $t>0$，和，如果$D\in L^2(\mathbb{R}^d;\mathbb{R}^d)$，存在性扩展到初始数据$u_0$在空间$\mathcal{M}_b$中的有界测量在$\mathbb{R}^d$。作为一个结果，任意的初始法律，我们得到了弱解的一类McKean-Vlasov方程的系数有奇异依赖的时间边际法律。
英文摘要：
One proves the existence and uniqueness of a generalized (mild) solution for the nonlinear Fokker-Planck equation (FPE) \begin{align*} &u_t-Δ(β(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 $u_0\in L^1(\mathbb{R}^d)$, $β\in C^2(\mathbb{R})$ 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 ${({\rm div}\, D)^-}\in L^\infty(\mathbb{R}^d)$, $β$ 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 ${\rm div}\,D\ge0$, $β'(r)\ge a|r|^{α-1}$, and $|β(r)|\le C r^α$, $α\ge1,$ $d\ge3$, then $|u(t)|_{L^\infty}\le Ct^{-\frac d{d+(α-1)d}}\ |u_0|^{\frac2{2+(m-1)d}},$ $t>0$, 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.
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