Generalized solutions to nonlinear Fokker–Planck equations

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...

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Bibliographic Details
Published in:Journal of Differential Equations 2016-08, Vol.261 (4), p.2446-2471
Main Author: Barbu, Viorel
Format: Article
Language:English
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Summary: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).
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2016.05.001