黏弹性流体力学中分数阶微分方程解的适定性研究

项目名称： 黏弹性流体力学中分数阶微分方程解的适定性研究

项目编号： No.11526038

项目类型： 专项基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 王芳

作者单位： 长沙理工大学

项目金额： 3万元

中文摘要： 整数阶微分本构模型不能精确的描述许多真实流体的流动特性，因此与黏弹性流体力学密切相关的分数阶问题成为了研究热点之一，它为更精确的描述黏弹性流体的流动特性提供了理论依据和数值分析. 本项目利用分数阶微积分算子构建几类广义Maxwell黏弹性流体在两个同轴圆柱体之间扭转流动的模型，求出模型的精确解.通过数值模拟分析模型中分数阶参数对流场建立的影响, 讨论分数阶流体模型解的渐近行为, 研究解的稳定性. 利用算子半群理论对分数阶抽象偏微分方程的解进行定性理论研究，这一研究方法在已有的文献中很少用到. 本项目的研究将解决黏弹性流体力学中的部分分数阶问题，促进分数阶本构模型解的定性理论研究, 而一些经典的流（如Newton流体和标准的Maxwell流体）的结果都可以作为本项目的特例而简化得到.

中文关键词： 粘弹性力学方程；适定性；渐近性态；柯西问题；初边值问题

英文摘要： The traditional differential constitutive model can not describe the viscoelastic fluid accurately. So the investigation of fractional order problems related to viscoelastic fluid mechanics has drawn attentions from researchers and academics during the last decade. Research results provide better theoretical basis and numerical analysis for viscoelastic fluid mechanics. The torsional oscillatory model of some generalized Maxwell fluid between two infinite coaxial circular cylinders with fractional operators is constructed and its exact solution is obtained in this project. Then by using the numerical simulation, the project will analysis the influence of fractional parameters, discuss the asymptotic behavior and study the stability of the solution. Finally, the qualitative theory of fractional abstract partial differential equations by the theory of semigroup operators is developed in this project. This method is rarely used in the existing literature. The purpose of this project is to resolve some fractional problems and develop the qualitative theory of fractional constitutive in visco elastic fluid mechanics. But the results of some classical flows (such as Maxwell fluids and standard Newton fluid) can be regarded as a special case of this project.

英文关键词： Viscoelastic mechanics equation；well posedness；asymptotic behavior；Cauchy problem；The initial boundary value problem