Pointwise convergence problem of the Korteweg–de Vries–Benjamin–Ono equation. (October 2022)

• Record Type: Journal Article
• Authors: Zhao, Yajuan; Yan, Xiangqian; Yan, Wei; Li, Yongsheng
• Is Part Of: Nonlinear analysis. Volume 67(2022)
• Publication Date: 2022-10
• Subjects: Korteweg–de Vries–Benjamin–Ono equation -- Pointwise convergence -- Hausdorff dimension -- Stochastic continuity; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2022.103611
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: In this paper we investigate the pointwise convergence problem for the Korteweg–de Vries–Benjamin–Ono equation u t + γ H ( ∂ x 2 u ) − β ∂ x 3 u = 0, ( x, t ) ∈ R × R, u ( x, 0 ) = f ( x ) ∈ H s ( R ), where γ β > 0 . We prove that the solution u ( x, t ) = U t f ( x ) converges pointwisely to the initial data f ( x ) for a.e. x ∈ R when f ∈ H s ( R ) with s ≥ 1 4, and that the Hausdorff dimension of the divergence set of points of the solution is α U ( s ) = 1 − 2 s when 1 4 ≤ s ≤ 1 2 . We also obtain the stochastic continuity for the initial data with much less regularity, i.e. for a large class of the initial data in L 2 ( R ), via the randomization technique.