Almost Sure Global Well-posedness for Fractional Cubic Schr "odinger\n equation on torus

2014/04/21 by Seckin Demirbas, Demirbas, Seckin
Engineering · Mathematics · #35Q55 #Advanced Mathematical Physics Problems #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods for differential equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1404.5270

openalex publication_date 2014/04/21 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

Abstract

In [12], we proved that 1-d periodic fractional Schr "odinger equation with\ncubic nonlinearity is locally well-posed in H^s for s>\(1-\α)/(2)\nand globally well-posed for s>\(5\α-1)/(6). In this paper we define an\ninvariant probability measure \μ on H^s for s<\α-\(1)/(2), so\nthat for any \ε>0 there is a set \Ω\⊂ H^s such that\n\μ(\Ω^c)<\ε and the equation is globally well-posed for initial\ndata in \Ω. We see that this fills the gap between the local\nwell-posedness and the global well-posedness range in almost sure sense for\n\(1-\α)/(2)<\α-\(1)/(2), i.e. \α>\(2)/(3) in almost\nsure sense.\n

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