[1]周光发.具有Radon测度数据的单向流问题[J].南京师范大学学报(自然科学版),2020,43(03):12-15.[doi:10.3969/j.issn.1001-4616.2020.03.003]
　Zhou Guangfa.A Unidirectional Flow Problem with Radon Measure Data[J].Journal of Nanjing Normal University(Natural Science Edition),2020,43(03):12-15.[doi:10.3969/j.issn.1001-4616.2020.03.003]
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具有Radon测度数据的单向流问题()
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《南京师范大学学报》（自然科学版）[ISSN:1001-4616/CN:32-1239/N]

卷:
第43卷
期数:
2020年03期
页码:
12-15
栏目:
·数学·
出版日期:
2020-09-30

文章信息/Info

Title:
A Unidirectional Flow Problem with Radon Measure Data
文章编号:
1001-4616(2020)03-0012-04
作者:
周光发
江苏警官学院基础课教研部,江苏 南京 210031
Author(s):
Zhou Guangfa
Department of General Courses Jiangsu Police Institute,Nanjing 210031,China
关键词:
Radon测度; 弱解; 单相流
Keywords:
Radon measure; weak solutions; unidirectional flow
分类号:
35K57 35B40
DOI:
10.3969/j.issn.1001-4616.2020.03.003
文献标志码:
A
摘要:
单相流问题可以从不可压流体力学中的一个简化的Boussinesq方程推导出来. Boussinesq方程由不可压Navier-Stokes方程和一个非线性热传导方程耦合而成. 它在大气科学和海洋科学中有重要的应用. 本文的目的是要证明具有Radon测度数据的单相流问题至少存在一个整体的弱解. 我们利用正则化方法完成定理的证明. 首先,构造一系列逼近的正则化解; 然后,利用方程的非线性微妙结构和一个推广的 Gronwall 不等式,建立良好的一致估计; 最后,利用标准的Aubin-Lions-Simon紧致化原理、Lebesgue控制收敛定理及非线性泛函分析中的一些结论,完成定理的证明. 本文的创新之处在于充分利用方程的细微非线性结构,获得精细的一致先验估计.
Abstract:
A unidirectional flow model is derived from a simplified Boussinesq system,which consists of a nonlinear heat equation coupled with the incompressible Navier-Stokes system. It has many important applications in atmosphere and ocean sciences. The aim of this paper is to prove the global existence of weak solutions to the unidirectional flow problem with Radon measure data. To achieve this,the regularized method is used. First,we construct the approximation strong solutions. Then,we apply a generalized Gronwall lemma to establish the uniform a priori estimates. Finally,we apply the standard compactness principle due to Aubin-Lions-Simon and thus the proof is finished. Here it should be note that the Gronwall inequality and the Lebesgue dominated convergence theorem are also used. The novelty of this paper may be lying in using the nonlinear subtle structure to obtain some fine uniform a priori estimates.

参考文献/References:

[1] XU X. A unidirectional flow with temperature dependent viscosity[J]. Nonlinear analysis-theory methods and application,1994,23(3):369-386.[2]CHESKIDOV A,SHVYDKOY R. A unified approach to regularity problems for the 3D Navier-Stokes and Euler equations:the use of Kolmogorov’s dissipation range[J]. Journal of mathematical fluid mechanics,2014,16(2):263-273.[3]HAN P. Decay results of the Nonstationary Navier-Stokes flows in Half-spaces[J]. Archive for rational mechanics and analysis,2018,230:977-1015. [4]BENAMEUR J. Long time decay to Lei-Lin solution of 3D Navier-Stokes equations[J]. Journal of mathematical analysis and application,2015,422:424-434.[5]TAO T,ZHANG L. On the continuous periodic weak solution of Boussinesq equations[J]. Siam journal on mathematical analysis,2018,50:1120-1162.[6]TAO T,ZHANG L. Hölder continuous solution of Boussinesq equations with compact support[J]. Journal of functional analysis,2017,272:4334-4402.[7]BOCCARDO L,GALLOUE T. Nonlinear elliptic and parabolic equations involving measure data[J]. Journal of functional analysis,1989,87:149-169.[8]SIMON J. Compact sets in . Annali di mathematica pura ed applicata,1987,196:65-96.

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备注/Memo

备注/Memo:
收稿日期:2019-07-19.
基金项目:“十三五”江苏省重点建设学科建设工程资助项目(苏教研[2016]9号).
通讯作者:周光发,副教授,研究方向:应用数学. E-mail:zhouguangfa@jspi.edu.cn
更新日期/Last Update: 2020-09-15