非线性粘弹性方程组解的动力学性质研究

发布时间：2018-04-24 03:37

  本文选题：粘性波方程 + 非线性　； 参考：《曲阜师范大学》2017年硕士论文

【摘要】：粘弹性力学是研究粘弹性材料在荷载作用下应力和应变所满足的规律.粘弹性力学是物理学和数学的交叉学科.早期关于粘弹性体的研究并未引起科学界与工程界的广泛注意,发展比较缓慢.但近四十余年来,粘弹性力学及其相应的数学理论得到了快速的发展.在材料科学中的数学理论这一颇受国际应用数学界重视的前沿领域中,现已成为十分活跃的研究课题.粘弹性力学中研究的方程大部分都是偏微分方程.特别地,粘弹性波方程的能量衰减研究引起了学者们的广泛关注.本文主要考察非齐次粘性波动方程组解的衰减估计,文章分为两章:第一章我们考虑下面带有边界控制的非线性粘弹性波动方程组的定解问题(?)其中μ,λ是拉梅常数,u= (u1,…,un)是一个向量函数,divu=ux11+ux22+…+uxnn是u的梯度,△=(?)且(?)这里Ω是Rn(n≥ 1)的一个具有光滑边界αΩ的有界区域,r 0且g是定义在R+上的正的递减函数,Γ :=αΩ,Γ = Γ=Γ0 ∪Γ1, m(Γ0∩Γ1) =0,Γ0,Γ1测度大于零,n是αΩ的单位外法向量.第二章我们考虑下面的具有Dirichlet齐次边界的非线性粘弹性波动方程组的定解问题(?)这里Ω是Rn(n≥ 1)中具有光滑边界αΩ的一个有界区域,r 0且g是定义在R+上的正的递减函数.我们的目标是用迭代法得到解的一般(General)能量衰减率.
[Abstract]:Viscoelastic mechanics is the law of stress and strain of viscoelastic materials under load. Viscoelastic mechanics is an interdisciplinary discipline in physics and mathematics. The early studies on viscoelastic materials have not attracted much attention from scientific and engineering circles, and their development is slow. However, viscoelastic mechanics and its corresponding mathematical theory have developed rapidly in the past forty years. The mathematical theory in material science, which has been paid much attention by the international applied mathematics, has become a very active research topic. Most of the equations studied in viscoelastic mechanics are partial differential equations. In particular, the energy attenuation of viscoelastic wave equations has attracted much attention. In this paper, we mainly study the decay estimation of solutions of nonhomogeneous viscous wave equations. This paper is divided into two chapters: in chapter 1, we consider the problem of determining solutions of nonlinear viscoelastic wave equations with boundary control. Where 渭, 位 is the Ramie constant u = u 1, 鈥，

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