A note on the applications of Wick products and Feynman diagrams in the study of singular partial differential equations

被引:3
作者
Yamazaki, Kazuo [1 ]
机构
[1] Texas Tech Univ, Dept Math & Stat, Lubbock, TX 79409 USA
关键词
Feynman diagrams; Gaussian hypercontractivity; Magnetohydrodynamics system; Quantum field theory; Space-time white noise; Wick products; NAVIER-STOKES EQUATIONS; DRIVEN;
D O I
10.1016/j.cam.2020.113338
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
The study of singular partial differential equations has seen rapid significant developments very recently. In particular, the works of Hairer (2013, 2014), and Gubinelli et al. (2015) paved the way for others to follow, providing blueprints for further study. Yet, many manuscripts in this field consist of extensive applications of techniques from physics, specifically quantum field theory, such as Wick products which are best explained in terms of Feynman diagrams. The purpose of this short note is to describe how the necessity of Wick products comes about, their applications using Feynman diagrams, as well as the utility of Gaussian hypercontractivity theorem. We also conclude with a description of an open problem that seems to be very mathematically challenging and physically meaningful. The author's intention is to make this note as accessible as possible to a wide audience by providing sufficient details, as well as relatively self-contained by including all relevant results which are necessary for our discussions. (C) 2020 Elsevier B.V. All rights reserved.
引用
收藏
页数:15
相关论文
共 42 条
[1]   KINETIC FOR MULATION AND GLOBAL EXISTENCE FOR THE HALL-MAGNETO-HYDRODYNAMICS SYSTEM [J].
Acheritogaray, Marion ;
Degond, Pierre ;
Frouvelle, Amic ;
Liu, Jian-Guo .
KINETIC AND RELATED MODELS, 2011, 4 (04) :901-918
[2]   THE AMPLITUDE EQUATION NEAR THE CONVECTIVE THRESHOLD - APPLICATION TO TIME-DEPENDENT HEATING EXPERIMENTS [J].
AHLERS, G ;
CROSS, MC ;
HOHENBERG, PC ;
SAFRAN, S .
JOURNAL OF FLUID MECHANICS, 1981, 110 (SEP) :297-334
[3]  
[Anonymous], 2003, Introduction to PDEs and Waves for the Atmosphere and Ocean
[4]  
Bahouri H, 2011, GRUNDLEHR MATH WISS, V343, P1, DOI 10.1007/978-3-642-16830-7_1
[5]   PARACONTROLLED DISTRIBUTIONS AND THE 3-DIMENSIONAL STOCHASTIC QUANTIZATION EQUATION [J].
Catellier, Remi ;
Chouk, Khalil .
ANNALS OF PROBABILITY, 2018, 46 (05) :2621-2679
[6]   Well-posedness for Hall-magnetohydrodynamics [J].
Chae, Dongho ;
Degond, Pierre ;
Liu, Jian-Guo .
ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE, 2014, 31 (03) :555-565
[7]  
Da Prato G., 1996, PUBBLICAZIONE DIPART, V505
[8]  
Da Prato G., 2007, WICK POWERS STOCHAST, P1
[9]   Reconnection events in two-dimensional Hall magnetohydrodynamic turbulence [J].
Donato, S. ;
Servidio, S. ;
Dmitruk, P. ;
Carbone, V. ;
Shay, M. A. ;
Cassak, P. A. ;
Matthaeus, W. H. .
PHYSICS OF PLASMAS, 2012, 19 (09)
[10]  
Evans L. C., 1998, PARTIAL DIFFERENTIAL