The invariance principle for nonlinear Fokker-Planck equations

doi:10.1016/j.jde.2022.01.043

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	The invariance principle for nonlinear Fokker-Planck equations
	Barbu, Viorel 1,2; Rockner, Michael 3,4
	2022-04-05
发表期刊	JOURNAL OF DIFFERENTIAL EQUATIONS
ISSN	0022-0396
卷号	315 页码:200-221
摘要	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.
关键词	Fokker-Planck equation McKean-Vlasov equations Generalized solution Nonlinear semigroup
DOI	10.1016/j.jde.2022.01.043
收录类别	SCI
语种	英语
WOS研究方向	Mathematics
WOS类目	Mathematics
WOS记录号	WOS:000754804600007
出版者	ACADEMIC PRESS INC ELSEVIER SCIENCE
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/60021
专题	中国科学院数学与系统科学研究院
通讯作者	Barbu, Viorel
作者单位	1.Romanian Acad, Octav Mayer Inst Math, Iasi, Romania 2.AlI Cuza Univ, Iasi, Romania 3.Univ Bielefeld, Fak Math, D-33501 Bielefeld, Germany 4.Chinese Acad Sci, Acad Math & Syst Sci, Beijing, Peoples R China
推荐引用方式 GB/T 7714	Barbu, Viorel,Rockner, Michael. The invariance principle for nonlinear Fokker-Planck equations[J]. JOURNAL OF DIFFERENTIAL EQUATIONS,2022,315:200-221.
APA	Barbu, Viorel,&Rockner, Michael.(2022).The invariance principle for nonlinear Fokker-Planck equations.JOURNAL OF DIFFERENTIAL EQUATIONS,315,200-221.
MLA	Barbu, Viorel,et al."The invariance principle for nonlinear Fokker-Planck equations".JOURNAL OF DIFFERENTIAL EQUATIONS 315(2022):200-221.