The invariance principle for nonlinear Fokker–Planck equations

Abstract

One studies here, via the invariance principle for nonlinear semigroups in Banach spaces [8], the properties of the ω-limit set ω(u0) corresponding to the orbit γ(u0)={u(t,u0);t≥0}, where u=u(t,u0) is the solution to the nonlinear Fokker–Planck equationut−Δβ(u)+div(Db(u)u)=0in (0,∞)×Rd,u(0,x)=u0(x),x∈Rd,u0∈L1(Rd),d≥3. Here, β∈C1(R) and β′(r)>0, ∀r≠0. Moreover, β is a sublinear function, possibly degenerate in the origin, b∈C1(R), b bounded, b≥b0∈(0,∞), D is bounded such that D=−∇Φ, where Φ∈C(Rd) is such that Φ≥1, Φ(x)→∞ as |x|→∞ and satisfies a condition of the form ΔΦ−α|∇Φ|2≤0, a.e. on Rd. The main conclusion is that the equation has an equilibrium state and the set ω(u0) is a nonempty, compact subset of L1(Rd) while, for each t≥0, the operator u0→u(t,u0) is an isometry on ω(u0). In the nondegenerate case 0<γ0≤β′≤γ1 studied in [2], it follows that limt→∞⁡S(t)u0=u∞ in L1(Rd), where u∞ is the unique bounded stationary solution to the equation.