{"title":"The modified scattering of two dimensional semi-relativistic Hartree equations","authors":"Soonsik Kwon, Kiyeon Lee, Changhun Yang","doi":"10.1007/s00028-024-00982-7","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we consider the asymptotic behaviors of small solutions to the semi-relativistic Hartree equations in two dimension. The nonlinear term is the cubic one convolved with the Coulomb potential <span>\\(|x|^{-1}\\)</span>, and it produces the<i> long-range interaction</i> in the sense of scattering phenomenon. From this observation, one anticipates that small solutions converge to modified scattering states, although they decay as linear solutions. We show the global well-posedness and the modified scattering for small solutions in weighted Sobolev spaces. Our proof follows a road map of exploiting the space-time resonance by Germain et al. (Int Math Res Not 2009(3):414–432, 2008), and Pusateri (Commun Math Phys 332(3):1203–1234, 2014). Compared to the result in three dimensional case (Pusateri 2014), weaker time decay in two dimension is one of the main obstacles.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":null,"pages":null},"PeriodicalIF":1.1000,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Evolution Equations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00028-024-00982-7","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we consider the asymptotic behaviors of small solutions to the semi-relativistic Hartree equations in two dimension. The nonlinear term is the cubic one convolved with the Coulomb potential \(|x|^{-1}\), and it produces the long-range interaction in the sense of scattering phenomenon. From this observation, one anticipates that small solutions converge to modified scattering states, although they decay as linear solutions. We show the global well-posedness and the modified scattering for small solutions in weighted Sobolev spaces. Our proof follows a road map of exploiting the space-time resonance by Germain et al. (Int Math Res Not 2009(3):414–432, 2008), and Pusateri (Commun Math Phys 332(3):1203–1234, 2014). Compared to the result in three dimensional case (Pusateri 2014), weaker time decay in two dimension is one of the main obstacles.
本文考虑了二维半相对论哈特里方程小解的渐近行为。非线性项是与库仑势 \(|x|^{-1}\)相卷积的立方项，它会产生散射现象意义上的长程相互作用。根据这一观察结果，我们可以预见小解会收敛于修正的散射态，尽管它们会衰减为线性解。我们证明了小解在加权索波列夫空间中的全局好求和修正散射。我们的证明遵循了杰曼等人（Int Math Res Not 2009(3):414-432, 2008）和普萨特里（Commun Math Phys 332(3):1203-1234, 2014）利用时空共振的路线图。与三维情况下的结果（Pusateri，2014 年）相比，二维情况下的时间衰减较弱是主要障碍之一。
期刊介绍：
The Journal of Evolution Equations (JEE) publishes high-quality, peer-reviewed papers on equations dealing with time dependent systems and ranging from abstract theory to concrete applications.
Research articles should contain new and important results. Survey articles on recent developments are also considered as important contributions to the field.
Particular topics covered by the journal are:
Linear and Nonlinear Semigroups
Parabolic and Hyperbolic Partial Differential Equations
Reaction Diffusion Equations
Deterministic and Stochastic Control Systems
Transport and Population Equations
Volterra Equations
Delay Equations
Stochastic Processes and Dirichlet Forms
Maximal Regularity and Functional Calculi
Asymptotics and Qualitative Theory of Linear and Nonlinear Evolution Equations
Evolution Equations in Mathematical Physics
Elliptic Operators
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