Generalized solutions to nonlinear Fokker–Planck equations

Abstract

The nonlinear Fokker–Planck equationut+divx(D(x,u)u)−Δβ(u)=0 in (0,∞)×Rd;u(0,x)=u0(x), where β:R→R is monotonically increasing and continuous, is treated as a nonlinear Cauchy problem of accretive type in the space L1(Rd). Two distinct cases: D(x,u)≡b(u) and that of linear drift term D(x,u)≡D(x) are treated separately. In the first case, one proves the existence of an entropy generalized solution u=u(t,x) in sense of S. Kružkov [11] and, in the second, one proves the existence of a generalized solution under additional hypothesis β,β−1∈Lip(R). In both cases, the generalized solution u=u(t,u0) defines a continuous semiflow u0→u(t,u0) of contractions in L1(Rd).