大初值扰动下 Boussinesq-Burgers 方程的动力学性态研究

Boussinesq-Burgers 方程是描述波浪运动现象的一类重要数学模型，也是数学物理特别是孤立子理论研究中的重要内容之一。本项目拟研究大初值扰动下带非线性和粘性的 Boussinesq-Burgers 方程的 Cauchy 问题的解的动力学性态。基于非线性强弱和粘性程度，将影响类型分为四种，分别为弱非线性和全粘性影响、弱非线性和部分粘性影响、强非线性和全粘性影响以及强非线性和部分粘性影响。从方程的结构特点考虑，对于大初值扰动的经典方程情形，本项目通过定性理论和谱分析，拟建立行波解的稳定性。进一步通过 L^p 估计理论，拟获得部分粘性影响的大初值解的整体存在性及收敛性。 对于大初值扰动的广义方程情形，由于方程中含有高次幂指数非线性项，本项目通过构造 Lyapunov 泛函与创建含幂函数的不等式相结合，拟获得大初值解的整体存在性、渐近稳定性及演化特征。

Boussinesq-Burgers equations are a kind of important mathematical models for describling wave motion and are one of the forefront topics in the study of mathematical physics, especially in soliton theory. The project will consider dynamic behaviors of solutions to the Cauchy problem of Boussinesq-Burgers equations perturbed by the large initial data with nonlinearities and viscosities. Based on the effect strength of nonlinearities and viscosities, the combinations of effects consist of four types including weak nonlinearities and full viscosities, weak nonlinearities and partial viscosities, strong nonlinearities and full viscosities, strong nonlinearities and parital viscosities. Consider the structural characteristics of equations, in case of classical equations perturbed by the large initial data, the project plans to establish the stability of traveling wave solutions by combining the qualitative theory with spectral analysis. Furthermore, the project plans to obtain the global existence and convergence rate of large data solutions of the partial viscous case by the L^p estimate theory. In the case of generalized equations perturbed by the large initial data, since equations contain a high power nonlinear term, the project plans to obtain the global existence, asymptotic stability and evolutive features of large data solutions by combining the Lypunov functional with constructed power functional inequalities.