项目名称: 一类非线性发展方程的定性理论
项目编号: No.11471127
项目类型: 面上项目
立项/批准年度: 2015
项目学科: 数理科学和化学
项目作者: 金春花
作者单位: 华南师范大学
项目金额: 65万元
中文摘要: 本项目旨在研究来源于物理学、生物、化学、连续介质力学等领域的一类具有鲜明的物理背景的非线性发展方程,。研究内容主要涉及到一类非线性扩散方程的一维及多维行波解问题、奇异初值解的发展趋势、可压Navier-Stokes方程的周期解问题,粘弹流方程解的长时间渐近行为,以及分数阶扩散方程解的相关理论研究. 这些都是目前人们所关注的热点、难点问题。刻画这类方程不仅需要经典的偏微分方程理论知识,并且需要根据不同的方程选择合适的研究框架和理论工具,甚至需要研究工具和方法的不断拓展和创新。本项目的研究不仅能对于解释某些实际现象提供一定的参考价值,而且研究方法与结果也将在一定程度上丰富和完善偏微分方程的理论。
中文关键词: 行波解;周期解;奇异性;整体适定性;分数阶扩散方程
英文摘要: This project is concerned with a class of nonlinear evolutionary equations coming from physics, biology, chemistry, continuum mechanics and so on. The research content involve the aspect such as the traveling wave solutions for a class of nonlinear diffusion equations, the time evolution of an initial singularity and the large time behavior,the time periodic solutions for the compressible Navier-Stokes equations, global well-posedness for compressible viscoelastic fluids , the related theory of fractional diffusion equation, these problems are always the focus and difficult issues.To characterize this class of equations, it requires not only the classical theory of partial differential equations, but also need to select appropriate research framework and theoretical tools,and even have to constantly expand and improve the prior tools and methods. The research of this project will provide valuable preferences for explaining some actual phenomena, and will enrich and perfect the theory of partial differential equations.
英文关键词: Traveling wave solution;Periodic solution;Singularity;global well-posedness;fractional diffusion equations