论文标题
McKean-Vlasov SDE的弱解决方案，具有超临界漂移
Weak solutions of McKean-Vlasov SDEs with supercritical drifts
论文作者
论文摘要
请考虑以下McKean-Vlasov sde：$$ d x_t = \ sqrt {2} d w_t+\ int _ {{\ Mathbb r}^d} $ x_t $和$ k（t，x）：{\ mathbb r} _+\ times {\ mathbb r}^d \ to {\ mathbb r}^d $是时间依赖时间差异的vivergence free vector fiel。在l^q_t（\ widetilde l_x^p）的假设下，带有$ \ frac dp+\ frac2q <2 $，其中$ \ widetilde l^p_x $代表局部化的$ l^p $ -Space，我们显示了上述SDE的弱解决方案。作为应用程序，我们为以量度为初始涡度的2D-Navier-Stokes方程的弱解决方案提供了新的证明。
Consider the following McKean-Vlasov SDE: $$ d X_t=\sqrt{2}d W_t+\int_{{\mathbb R}^d}K(t,X_t-y)μ_{X_t}(dy)d t,\ \ X_0=x, $$ where $μ_{X_t}$ stands for the distribution of $X_t$ and $K(t,x): {\mathbb R}_+\times{\mathbb R}^d\to{\mathbb R}^d$ is a time-dependent divergence free vector field. Under the assumption $K\in L^q_t(\widetilde L_x^p)$ with $\frac dp+\frac2q<2$, where $\widetilde L^p_x$ stands for the localized $L^p$-space, we show the existence of weak solutions to the above SDE. As an application, we provide a new proof for the existence of weak solutions to 2D-Navier-Stokes equations with measure as initial vorticity.
