Random data Cauchy problem for the fourth order Schrödinger equation with the second order derivative nonlinearities. (January 2020)

• Record Type: Journal Article
• Authors: Chen, Mingjuan; Zhang, Shuai
• Is Part Of: Nonlinear analysis. Volume 190(2020)
• Publication Date: 2020-01
• Subjects: Random initial data -- Fourth order Schrödinger equation -- Supercritical well-posedness; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.111608
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider the Cauchy problem of the fourth order nonlinear Schrödinger equation (4NLS) ( i ∂ t + ε Δ + Δ 2 ) u = P m ( ( ∂ x α u ) | α | ≤ 2, ( ∂ x α u ¯ ) | α | ≤ 2 ), m ≥ 3 on R d ( d ≥ 2 ) with random initial data. Here, P m is a homogeneous polynomial of degree m containing the second order derivative. We show the almost sure local well-posedness and small data global existence in H s ( R d ) with some range of s < s c, where s c is the scaling critical regularity, i.e., s c ≔ d ∕ 2 − 2 ∕ ( m − 1 ) . Our results contain the cases s ∈ ( s c − 1 ∕ 2, s c ] when d ≥ 3 and m ≥ 5 . Similar supercritical well-posedness results also hold for d = 2, m ≥ 4 and d ≥ 3, 3 ≤ m < 5 .