In this work, one proves the well-posedness in the Sobolev space \(H^{-1}({\mathbb {R}^d})\) of the Cauchy problem \(\begin{array}{r} \displaystyle \frac{{\partial }u}{{\partial }t}-\Delta \beta (u)+\textrm{div }((D(x)b(u)+K*u)u)=0 \\ \text{ in } (0,{\infty })\times {\mathbb {R}^d}; u(0,x)=u_0(x),\end{array}\) where \(d\ge 2\) , \(\beta \) is a continuous monotonically increasing function, \(D:{\mathbb {R}^d}\rightarrow {\mathbb {R}^d}\) , \(b:{\mathbb {R}}\rightarrow {\mathbb {R}}^+\) are appropriate functions and \(K\!\in \! C^1({\mathbb {R}^d}\!\setminus \!0)\) is a singular kernel. One proves also the uniqueness of distributional solutions \(u\in L^1\cap L^{\infty }\) which are weakly (narrowly) continuous in t from \([0,{\infty })\) to \(L^1({\mathbb {R}^d})\) and the uniqueness for the corresponding linearized equations, As a consequence, it follows the existence and uniqueness of strong solutions to the McKean–Vlasov stochastic differential equation (SDE) \(\begin{array}{l} dX_t=(D(X)b(u(t,X_t)+(K*u(t,\cdot ))(X_t))dt +\left( \displaystyle \frac{2\beta (u(t,X_t))}{u(t,X_t)}\right) ^{\!\!\frac{1}{2}}dW_t\\ {\mathcal {L}}_{X_t}(x)=u(t,x),\ X(0)=X_0,\ u_0=\mathbb {P}\circ X^{-1}_0, \end{array}\) where \({\mathcal {L}}(X_t)\) is the density of the probability law of the process \(X_t\) with respect to the Lebesgue measure on \({\mathbb {R}^d}\) . Moreover, the laws of solutions to this equation have the Markov property.
