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

Abstract

One proves the existence and uniqueness of a generalized (mild) solution for the nonlinear Fokker–Planck equation (FPE)ut−Δ(β(u))+div(D(x)b(u)u)=0,t≥0,x∈Rd,d≠2,u(0,⋅)=u0,in Rd, where u0∈L1(Rd), β∈C2(R) is a nondecreasing function, b∈C1, bounded, b≥0, D∈L∞(Rd;Rd) with divD∈(L2+L∞)(Rd), and (divD)−∈L∞(Rd), β strictly increasing, if b is not constant. Moreover, t→u(t,u0) is a semigroup of contractions in L1(Rd), which leaves invariant the set of probability density functions in Rd. If divD≥0, β′(r)≥a|r|α−1, and |β(r)|≤Crα, α≥1, d≥3, then |u(t)|L∞≤Ct−dd+(α−1)d|u0|22+(m−1)d, t>0, and if D∈L2(Rd;Rd) the existence extends to initial data u0 in the space Mb of bounded measures in Rd. 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.