The invariance principle for nonlinear Fokker-Planck equations

Barbu V, Röckner M (2022)
Journal of Differential Equations 315: 200-221.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Autor*in
Barbu, Viorel; Röckner, MichaelUniBi
Abstract / Bemerkung
One studies here, via the invariance principle for nonlinear semigroups in Banach spaces [8], the properties of the omega-limit set omega(u(0)) corresponding to the orbit gamma(u(0)) ={u(t, u(0)); t >= 0}, where u = u(t, u(0)) is the solution to the nonlinear Fokker-Planck equation u(t) - Delta beta(u) + div(Db(u)u) = 0 in (0,infinity) x R-d, u(0,x) = u(0)(x), x is an element of R-d, u(0) is an element of L-1 (R-d), d >= 3. Here, ss is an element of C-1( R) and ss '(r) > 0, for all r not equal 0. Moreover, ss is a sublinear function, possibly degenerate in the origin, b is an element of C-1(R), bbounded, b >= b(0) is an element of(0, infinity), D is bounded such that D= -del Phi, where Phi is an element of C(R-d) is such that Phi >= 1, Phi(x) -> infinity as vertical bar x vertical bar -> infinity and satisfies a condition of the form Delta Phi - alpha vertical bar del Phi vertical bar(2) <= 0, a.e. on R-d. The main conclusion is that the equation has an equilibrium state and the set omega(u(0)) is a nonempty, compact subset of L-1(R-d) while, for each t >= 0, the operator u(0) -> u(t, u(0)) is an isometry on omega(u(0)). In the nondegenerate case 0 < gamma(0) <= ss ' <= gamma(1) studied in [2], it follows that lim (t ->infinity) S(t)u(0) = u(infinity) in L-1(R-d), where u(infinity) is the unique bounded stationary solution to the equation. (c) 2022 Elsevier Inc. All rights reserved.
Stichworte
Fokker-Planck equation; McKean-Vlasov equations; Generalized solution; Nonlinear semigroup
Erscheinungsjahr
2022
Zeitschriftentitel
Journal of Differential Equations
Band
315
Seite(n)
200-221
ISSN
0022-0396
eISSN
1090-2732
Page URI
https://pub.uni-bielefeld.de/record/2961636

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Barbu V, Röckner M. The invariance principle for nonlinear Fokker-Planck equations. Journal of Differential Equations . 2022;315:200-221.
Barbu, V., & Röckner, M. (2022). The invariance principle for nonlinear Fokker-Planck equations. Journal of Differential Equations , 315, 200-221. https://doi.org/10.1016/j.jde.2022.01.043
Barbu, V., and Röckner, M. (2022). The invariance principle for nonlinear Fokker-Planck equations. Journal of Differential Equations 315, 200-221.
Barbu, V., & Röckner, M., 2022. The invariance principle for nonlinear Fokker-Planck equations. Journal of Differential Equations , 315, p 200-221.
V. Barbu and M. Röckner, “The invariance principle for nonlinear Fokker-Planck equations”, Journal of Differential Equations , vol. 315, 2022, pp. 200-221.
Barbu, V., Röckner, M.: The invariance principle for nonlinear Fokker-Planck equations. Journal of Differential Equations . 315, 200-221 (2022).
Barbu, Viorel, and Röckner, Michael. “The invariance principle for nonlinear Fokker-Planck equations”. Journal of Differential Equations 315 (2022): 200-221.

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