McKean-Vlasov SDEs with Local Distributional Interactions: Well-Posedness and Entropy-Cost Estimates

We study McKean-Vlasov SDEs with interaction kernels in ${\mathtt{W}}^{-\mathtt{\text{dd}},\mathtt{k}},$ the local negative Sobolev space on ${\mathbb{R}}^d$ with indexes $\text{dd}\in [0,\mathrm{\infty })$ and $k\in [1,\mathrm{\infty }]\mathrm{.}$ We derive the local well-posedness for any singular indexes $(\text{dd},k)\in [0,\mathrm{\infty })\times [1,\mathrm{\infty }],$ and prove the global well-posedness for any initial distributions provided $\text{dd}+\text{ff}dk<1$. Moreover, the relative entropy and the $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\text{dd},k\ast }$-distance induced by ${\mathtt{W}}^{-\mathtt{\text{dd}},\mathtt{k}}$ are estimated for the time-marginal distributions of solutions by using the Wasserstein distance of initial distributions, which describe the regularity of the solution in initial distribution. In particular, the main results apply to Nemytskii-type SDEs which depend on higher order derivatives of the density functions, as well as McKean-Vlasov SDEs with interactions more singular than Riesz kernels.