【作者】 杨韬；

【导师】 罗宏；

【作者基本信息】 四川师范大学 ， 数学， 2020， 硕士

【摘要】 本文研究了六阶Cahn-Hilliard方程解的存在唯一性和全局吸引子的正则性.主要结果包括下面两个部分.第一部分主要考虑弱解的存在唯一性.先利用Galerkin方法和能量估计研究了方程在H-1(Ω)空间的弱解,再应用空间序列方法和T弱连续算子理论研究了L2(Ω)空间的弱解,最后根据方程的变分结构,得到了方程在H2(Ω)空间解的存在唯一性.第二部分主要研究了该方程全局吸引子的正则性.使用能量方法和紧算子理论得到了方程在H-1(Ω)空间和L2(Ω)空间存在吸引子,再根据梯度型方程的能量随时间递减的特点得到方程在H2(Ω)空间存在吸引子;最后,利用迭代方法、线性半群的正则性估计和全局吸引子的存在性定理,证明了对任意的k≥ 0该方程在Hk(Ω)空间存在一个全局吸引子.更多还原

【Abstract】 In this thesis,we study the existence and uniqueness of weak solutions of a sixth-order Cahn-Hilliard equation,as well as the regularity of its global attractor.The main results are divided into the following two parts.The first part of this thesis is devoted to the existence and uniqueness of weak solutions.First of all,we study weak solutions of the equation in H-1(Ω)space by using Galerkin method and energy estimates.Then,we study weak solutions in L2(Ω)space by applying the method of spacial sequences and the theory of T weakly continuous operator.Last but not least,from the variational structure of the equation,we obtain the existence and uniqueness of its solution in H2(Ω)space.The second part of this thesis is devoted to the regularity of the global attractor of this equation.First of all,by using energy method and compact operator theory,we obtain the result that the equation possesses a global attractor in both H-1(Ω)space and L2(Ω)space.Then,from a characteristic of gradient type equations that energy decreases with time,we obtain the existence of the global attractor of the equation in H2(Ω)space.Finally,by using an iteration procedure,regularity estimates for semigroups of linear operators and a theorem for the existence of global attractors,we prove that this equation possesses a global attractor in Hk(Ω)space for any k≥ 0.更多还原