二维Boussinesq方程组相关模型的研究

发布时间：2018-04-19 01:35

  本文选题：Boussinesq方程组 + 非齐次边界　； 参考：《东华大学》2017年博士论文

【摘要】：随着科学的发展,出现越来越多的流体动力学方程(组),在实际应用中,包含时间变量t的方程(组)被称为非线性发展方程(组).Boussinesq方程组有着极强的物理背景和数学意义,能够描述大气科学与海洋运动中旋转和分层两个最显著的特征,一经提出就引起广泛关注.近年来,很多学者为扩大Boussinesq方程组的适用范围,又提出了多种改进形式的方程,这些方程的广泛研究与应用,对民生、经济发展等问题的研究具有较大的理论意义和应用指导价值.在介绍了二维Boussinesq方程组的研究背景和现状,以及本文所需的一些基本理论和常用不等式之后,本文主要研究了多个Boussinesq方程组模型的整体适定性,包含半粘性非线性Boussinesq方程组、带有非齐次边界的Boussinesq方程组以及分数阶Boussinesq方程组,并得到了一些有意义的结果.主要相关结论如下:(1)研究了无热扩散系数的Boussinesq方程组的初边值问题.当粘性系数为依赖于温度的函数时,该方程组的整体适定性仍是开放性问题.本章通过构造方程组的近似解的方法,在讨论相关收敛性之后,证得方程组准强解的更高阶正则性和唯一性.(2)利用对非齐次方程的传统处理方法,研究了带有非齐次边界条件Boussinesq方程组解的适定性.通过重新计算方程组的扰动变量,将带有扰动变量的方程组转换成与其等价的可以用传统方法计算的方程组,最终对方程组解的整体适定性进行了讨论.(3)研究了亚临界条件下分数阶Boussinesq方程组的初边值问题.充分利用α,β∈(23,1)的条件,讨论了方程的持续正则性.同时,我们还求得了其衰减估计.结合现有的相关研究成果,为了完善分数阶二维Boussinesq方程组正则性的理论体系,还将其推广至Sobolev空间中更进一步地进行了讨论.(4)研究了次临界条件下分数阶各向异性Boussinesq方程组的Cauchy问题.这一部分,以第四章的模型为基础,在周期域中次临界情形下研究了只在水平方向耗散的分数阶Boussinesq方程组的Cauchy问题,证明了其强解的适定性问题,并讨论了整体吸引子的存在性.最后,对主要工作再次进行了总结并对未来的研究提出了展望.
[Abstract]:With the development of science, more and more hydrodynamic equations have emerged. In practical applications, the equations containing time variable t are called nonlinear evolution equations (group. Boussinesq equations have strong physical background and mathematical significance.It can describe the most prominent characteristics of rotation and stratification in atmospheric science and ocean motion.In recent years, in order to expand the scope of application of Boussinesq equations, many scholars have put forward a variety of improved equations, which are widely studied and applied to people's livelihood.The research of economic development and other issues has great theoretical significance and application guidance value.After introducing the research background and present situation of two-dimensional Boussinesq equations, and some basic theories and common inequalities needed in this paper, this paper mainly studies the global fitness of multiple Boussinesq equations, including semi-viscous nonlinear Boussinesq equations.The Boussinesq equations with nonhomogeneous boundaries and the fractional Boussinesq equations are obtained, and some meaningful results are obtained.The main conclusions are as follows: 1) the initial-boundary value problem of Boussinesq equations without thermal diffusion coefficient is studied.When the viscosity coefficient is a temperature-dependent function, the global fitness of the equations is still an open problem.In this chapter, the higher order regularity and uniqueness of the quasi strong solutions of the equations are obtained by constructing the approximate solutions of the equations, and after discussing the convergence of the correlation, we use the traditional methods to deal with the inhomogeneous equations.In this paper, the solution of Boussinesq equations with inhomogeneous boundary conditions is studied.By recalculating the perturbed variables of the equations, the equations with the perturbation variables are converted into the equivalent equations which can be calculated by the traditional method.Finally, the problem of initial boundary value of fractional Boussinesq equations under subcritical condition is discussed.The persistence regularity of the equation is discussed by using the conditions of 伪, 尾 鈭，

本文编号：1771032