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

One proves the existence and uniqueness of a generalized (mild) solution for the nonlinear Fokker-Planck equation (FPE) u(t) - Delta(beta(u)) + div(D(x)b(u)u) = 0, t >= 0, x is an element of R-d, d not equal 2, u(0, .) = u(0), in R-d, where u(0) is an element of L-1 (R-d ), beta is an element of C-2 (R) is a nondecreasing function, b is an element of C-1, bounded, b >= 0, D is an element of L-infinity (R-d; R-d) with div D is an element of (L-2+L infinity)(R-d), and (div D)(-) is an element of L-infinity (R-d), beta strictly increasing, if b is not constant. Moreover, t -> u(t, u(0)) is a semigroup of contractions in L-1 (R-d), which leaves invariant the set of probability density functions in R-d. If div D >= 0, beta'(r) >= a vertical bar r vertical bar(alpha-)(1) and vertical bar beta(r)vertical bar <= Cr-alpha, alpha >= 1, d >= 3, then vertical bar u(t)vertical bar L-infinity <= Ct(-d/d+(alpha-1)d) vertical bar u(0)vertical bar(2/2+(m-1d)), t > 0, and if D is an element of L-2 (R-d;R-d) the existence extends to initial data u(0) in the space M-b of bounded measures in 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. (C) 2021 Elsevier Inc. All rights reserved.