一类模式演化方程的全局吸引子及其维数估计

研究了一类源自模式演化问题的非线性发展方程所产生的动力系统,并考虑了其全局吸引子的存在性及维数估计问题.这类模式演化方程与化学反应和火焰燃烧有密切关系,因此具有重要的物理背景,而且因为它含有关于空间变量的四阶微分算子,还具有重要的理论价值.借助插值不等式以及sobolev嵌入定理,可以进行一系列精细的估计,最终根据一个经典的结果,证明了在维数不超过三维的空间中的有界集合上,系统的全局吸引子存在.进一步应用Sobolev-Lieb-Thirring不等式进行估计,可以得到全局吸引子的分形维数的界.

Global Attractors for a Class of Pattern Formation Equations

A class of pattern formation equations is treated, the existence of its global attractor is considered and some estirnates of its dimension are given. This sort of pattern formation equations relate to turbulence phenomenon in chemistry and combustion, so it is of great physical importance. Besides, its important theoretical value lies in that the equation contains differential operator of the fourth order in space. The global attractor exists after a series sophisticated estimates by the means of interpolation inequity and embedding theorems in sobolev spaces. Furthermore, some estimates on its fractal dimension are given.