Refined probabilistic global well-posedness for the weakly dispersive\n NLS

2020/10/25 by Chenmin Sun, Nikolay Tzvetkov, Sun, Chenmin +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #FOS: Mathematics #Navier-Stokes equation solutions #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2010.13065

openalex publication_date 2020/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We continue our study of the cubic fractional NLS with very weak dispersion\n\α>1 and data distributed according to the Gibbs measure. We construct\nthe natural strong solutions for\n\α>\α₀=\(31-\√(233))/(14)\≈ 1.124 which is strictly\nsmaller than \(8)/(7), the threshold beyond which the first nontrivial\nPicard iteration has no longer the Sobolev regularity needed for the\ndeterministic well-posedness theory. This also improves our previous result in\nSun-Tzvetkov citeSun-Tz2. We rely on recent ideas of Bringmann\n citeBringmann and Deng-Nahmod-Yue citeDeng2. In particular we adapt to\nour situation the new resolution ansatz in citeDeng2 which captures the most\nsingular frequency interaction parts in the X^s,b type space. To overcome\nthe difficulties caused by the weakly dispersive effect, our specific strategy\nis to benefit from the "almost" transport effect of these singular parts and to\nexploit their L^\∞ as well as the Fourier-Lebesgue property in order to\ninherit the random feature from the linear evolution of high frequency\nportions.\n

Citations

• Nonlinear fractional Schrödinger equations in one dimension
• On the Continuum Limit for Discrete NLS with Long-Range Lattice Interactions
• Invariant Gibbs measures and global strong solutions for nonlinear Schrödinger equations in dimension two

Cited by

• Probabilistic well-posedness of dispersive PDEs beyond variance blowup I: Benjamin-Bona-Mahony equation

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