分数阶长短波方程的长时间行为
发布时间：2019-04-21 11:59
【摘要】：本文主要研究了分数阶长短波方程,(2+1)维分数阶长短波方程的长时间行为,得到了广义非自治分数阶长短波方程周期初边值问题整体光滑解的存在唯一性,广义非自治分数阶长短波方程一致吸引子的存在性,并且证明了(2+1)维分数阶长短波方程周期初边值问题存在唯一整体光滑解以及(2+1)维分数阶长短波方程整体吸引子的存在性.本文共分为五个部分.第一部分,主要阐述了分数阶微积分,无穷维动力系统以及长短波方程的物理背景及相关理论知识,回顾已有的部分研究成果,最后简单介绍了一下本文的主要研究工作.第二部分,考虑了广义分数阶长短波方程的周期初边值问题.首先运用Gagliardo-Nirenberg不等式,Young不等式和Gronwall不等式进行一致先验估计,其次利用Gal?rkin方法,研究了广义非自治分数阶长短波方程的周期初边值问题光滑解的整体存在性和唯一性.第三部分,证明了广义分数阶非自治长短波方程一致吸引子的存在性.首先通过一致先验估计以及Gal?rkin方法即可证明该周期初边值问题解的存在唯一性,其次利用非自治系统一致吸引子的有关理论得到了该方程强紧一致吸引子的存在性.第四部分,考虑了(2+1)维分数阶长短波方程的周期初边值问题.利用一致先验估计和Gal?rkin方法证明了该方程周期初边值问题整体光滑解的存在性.第五部分,证明了(2+1)维分数阶长短波方程整体吸引子的存在性.结合解半群性质,运用弱收敛方法证明了(2+1)维分数阶长短波方程的周期初边值问题整体强吸引子存在.
[Abstract]:In this paper, we mainly study the long-time behavior of fractional-order long-wave equation and (21)-dimensional fractional-order long-wave equation, and obtain the existence and uniqueness of global smooth solution for the periodic initial-boundary value problem of generalized non-autonomous fractional-order long-wave equation. Existence of uniform Attractors for Generalized non-autonomous Fractional order long short Wave equations, It is also proved that there exists a unique global smooth solution for the periodic initial boundary value problem of the (21) dimensional fractional order long short wave equation and the existence of the global attractor for the (21) dimensional fractional order long short wave equation. This paper is divided into five parts. In the first part, the physical background and related theoretical knowledge of fractional calculus, infinite-dimensional dynamical system and long-short wave equation are described, and the existing research results are reviewed. Finally, the main research work of this paper is briefly introduced. In the second part, we consider the periodic initial-boundary value problem of the generalized fractional order short-wave equation. In this paper, we first use Gagliardo-Nirenberg inequality, Young inequality and Gronwall inequality to obtain uniform prior estimates. Secondly, we use Gal?rkin method to study the global existence and uniqueness of smooth solutions to the periodic initial boundary value problems of generalized fractional order short wave equations. In the third part, the existence of uniform attractors for generalized fractional order non-autonomous short-wave equations is proved. First, the existence and uniqueness of the solution of the periodic initial-boundary value problem can be proved by using the uniform prior estimate and the Gal?rkin method. Secondly, the existence of the strongly compact uniform attractor for the equation is obtained by using the theory of the uniform attractor of the non-autonomous system. In the fourth part, the periodic initial boundary value problem of the (21) dimensional fractional order long short wave equation is considered. The existence of global smooth solutions to the periodic initial boundary value problem of the equation is proved by means of uniform prior estimates and Gal?rkin 's method. In the fifth part, the existence of the global attractor for the (21) dimensional fractional order short wave equation is proved. In this paper, the existence of global strong attractors for the periodic initial boundary value problem of (21) dimensional fractional order short wave equation is proved by using the weak convergence method combined with the properties of the solution semigroup.
【学位授予单位】：鲁东大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O175
本文编号：2462186
[Abstract]:In this paper, we mainly study the long-time behavior of fractional-order long-wave equation and (21)-dimensional fractional-order long-wave equation, and obtain the existence and uniqueness of global smooth solution for the periodic initial-boundary value problem of generalized non-autonomous fractional-order long-wave equation. Existence of uniform Attractors for Generalized non-autonomous Fractional order long short Wave equations, It is also proved that there exists a unique global smooth solution for the periodic initial boundary value problem of the (21) dimensional fractional order long short wave equation and the existence of the global attractor for the (21) dimensional fractional order long short wave equation. This paper is divided into five parts. In the first part, the physical background and related theoretical knowledge of fractional calculus, infinite-dimensional dynamical system and long-short wave equation are described, and the existing research results are reviewed. Finally, the main research work of this paper is briefly introduced. In the second part, we consider the periodic initial-boundary value problem of the generalized fractional order short-wave equation. In this paper, we first use Gagliardo-Nirenberg inequality, Young inequality and Gronwall inequality to obtain uniform prior estimates. Secondly, we use Gal?rkin method to study the global existence and uniqueness of smooth solutions to the periodic initial boundary value problems of generalized fractional order short wave equations. In the third part, the existence of uniform attractors for generalized fractional order non-autonomous short-wave equations is proved. First, the existence and uniqueness of the solution of the periodic initial-boundary value problem can be proved by using the uniform prior estimate and the Gal?rkin method. Secondly, the existence of the strongly compact uniform attractor for the equation is obtained by using the theory of the uniform attractor of the non-autonomous system. In the fourth part, the periodic initial boundary value problem of the (21) dimensional fractional order long short wave equation is considered. The existence of global smooth solutions to the periodic initial boundary value problem of the equation is proved by means of uniform prior estimates and Gal?rkin 's method. In the fifth part, the existence of the global attractor for the (21) dimensional fractional order short wave equation is proved. In this paper, the existence of global strong attractors for the periodic initial boundary value problem of (21) dimensional fractional order short wave equation is proved by using the weak convergence method combined with the properties of the solution semigroup.
【学位授予单位】：鲁东大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O175
【参考文献】
相关期刊论文 前1条
1 周毓麟,郭柏灵;高阶广义 Korteweg-de Vries 型方程组的周期边界问题与初值问题[J];数学学报;1984年02期
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