由Marcus型Lévy噪声驱动的随机Hartree方程的全局适定性
Global Well-Posedness of Stochastic Hartree Equation Driven by Lévy Noise in Marcus Form
摘要: 研究随机Hartree方程在Marcus型Lévy噪声扰动下的全局适定性。选择特定条件的噪声，利用噪声规范化，得到积分方程；通过Strichartz估计和压缩映射定理研究解的局部适定性；利用伊藤公式得到解的质量守恒；应用解的一致估计研究解的全局适定性。研究结论表明：初值在Lx2空间中的随机Hartree方程在Marcus型Lévy噪声扰动下具有全局适定性和质量守恒。
Abstract: In this paper, Global well-posedness of Hartree equation driven by Lévy noise in Marcus form was established. Choosing special noise, taking noise in canonical form, the integral equation was give. Through Strichartz estimate and contraction mapping, local well-posedness of solution was inves-tigating, and conservation law was given by Ito formula. Finally global well-posedness was given by consistent estimate. Result of our research indicated that the Stochastic Hartree equation driven by Lévy noise has unique global solution and conservative in time when the initial value was in Lx2.
文章引用：张朔霖. 由Marcus型Lévy噪声驱动的随机Hartree方程的全局适定性[J]. 理论数学, 2023, 13(10): 2744-2754. https://doi.org/10.12677/PM.2023.1310282

参考文献

[1] Villarroel, J. and Montero, M. (2010) On the Effect of Random Inhomogeneities in Kerr Media Modelled by a Nonlinear Schrödinger Equation. Journal of Physics B: Atomic, Molecular and Optical Physics, 43, Article ID: 135404. [Google Scholar] [CrossRef] 
[2] Ginibre, J. and Velo, G. (1980) On a Class of Non Linear Schrödinger Equations with Non Local Interaction. Mathematische Zeitschrift, 170, 109-136. [Google Scholar] [CrossRef] 
[3] Arora, A.K. and Roudenko, S. (2022) Global Behavior of Solutions to the Focusing Generalized Hartree Equation. Michigan Mathematical Journal, 71, 619-672.
[4] Miao, C., Xu, G. and Zhao, L. (2007) Global Well-Posedness and Scattering for the Energy-Critical, Defocusing Hartree Equation for Radial Data. Journal of Functional Analysis, 253, 605-627. [Google Scholar] [CrossRef] 
[5] Miao, C., Xu, G. and Zhao, L. (2008) The Cauchy Problem of the Hartree Equation. Journal of Partial Differential Equations, 21, 22.
[6] Miao, C., Xu, G. and Zhao, L. (2010) Global Well-Posedness and Scattering for the Energy-Critical, Defo-cusing Hartree Equation in R1+n. Communications in Partial Differential Equations, 36, 729-776. [Google Scholar] [CrossRef] 
[7] 罗虎啸. 带有非线性阻尼项的广义Hartree方程的整体适定性[J]. 数学学报(中文版), 2023, 66(4): 675-686.
[8] Chen, Y. and Gao, H. (2015) The Cauchy Problem for the Hartree Equations under Random Influences. Journal of Differential Equations, 259, 5192-5219. [Google Scholar] [CrossRef] 
[9] de Bouard, A. and Debussche, A. (2003) The Stochastic Nonlinear Schrödinger Equation in H1. Stochastic Analysis and Applications, 21, 97-126. [Google Scholar] [CrossRef] 
[10] Brzeźniak, Z. and Millet, A. (2014) On the Stochastic Strichartz Es-timates and the Stochastic Nonlinear Schrödinger Equation on a Compact Riemannian Manifold. Potential Analysis, 41, 269-315. [Google Scholar] [CrossRef] 
[11] Brzeźniak, Z. and Manna, U. (2019) Stochastic Lan-dau-Lifshitz-Gilbert Equation with Anisotropy Energy Driven by Pure Jump Noise. Computers & Mathematics with Applications, 77, 1503-1512. [Google Scholar] [CrossRef] 
[12] Marcus, S. (1978) Modeling and Analysis of Stochastic Dif-ferential Equations Driven by Point Processes. IEEE Transactions on Information Theory, 24, 164-172. [Google Scholar] [CrossRef] 
[13] Brzeźniak, Z. and Manna, U. (2019) Weak Solutions of a Sto-chastic Landau-Lifshitz-Gilbert Equation Driven by Pure Jump Noise. Communications in Mathematical Physics, 371, 1071-1129. [Google Scholar] [CrossRef] 
[14] Brzeźniak, Z., Liu, W. and Zhu, J. (2021) The Sto-chastic Strichartz Estimates and Stochastic Nonlinear Schrödinger Equations Driven by Levy Noise. Journal of Func-tional Analysis, 281, Article ID: 109021. [Google Scholar] [CrossRef] 
[15] Zhu, J., Brzezniak, Z. and Liu, W. (2019) Maximal Inequalities and Exponential Estimates for Stochastic Convolutions Driven by Lévy-Type Processes in Banach Spaces with Application to Stochastic Quasi-Geostrophic Equations. SIAM Journal on Mathematical Analysis, 51, 2121-2167. [Google Scholar] [CrossRef] 

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