Distance of attractors of evolutionary equations.
Distance of attractors of evolutionary equations.
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发表时间：
2013
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通讯作者：
E. S. Martín
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作者：
E. S. Martín
这篇论文致力于当我们知道吸引子的行为是连续的时，得到吸引子距离的良好估计。我们的目的是研究如何提高文献中已知的吸引子的收敛速度.我们得到：如果我们有一个由Morse Smer梯度映射生成的有限动力系统，它有一个吸引子，我们以这样的方式扰动这个系统，使得这个扰动系统也有一个吸引子，那么这些吸引子的收敛速度是相应的时间一映射的距离的数量级.此外，我们将这一结果推广为如下结果：如果我们有一个由具有吸引子的Morse Smer梯度映射生成的有限动力系统，并且我们考虑了它的非自治摄动，使得这个非自治摄动动力系统有一个拉回吸引子，则吸引子和拉回吸引子的收敛速度的量级是相应的时间一映射的收敛速度。这些结果是在有限维框架下得到的，证明方法是使用Morse Smer映射所具有的跟踪性。由于我们的框架是有限维的，并且我们最终想要将这一技术应用于无限维系统，我们需要一个工具来将系统简化为有限维的单独的惯性流形。这些光滑的有限维流形，在流动下是正不变的，并且指数吸引，如果系统有吸引子，则包含吸引子。为了研究系统在扰动下的行为并得到吸引子的良好收敛速度，我们研究了这些惯性流形在系统扰动下的行为，给出了它在C0拓扑和C1拓扑下的良好收敛速度.最后，我们应用上述技巧解决了一个特定的反应扩散方程对应的吸引子的距离问题和它在薄域中的扰动问题.针对文献中存在的这一特殊问题，本文改进了吸引子的收敛速度。
This thesis is devoted to obtain good estimates of the distance of attractors once we know they behavecontinuously. Our aim is to study what we have to ask to the system to improve the known rate ofconvergence of attractors existing in the literature.We have obtained that, if we have a finite dynamical system generated by a Morse Smale gradient mapwhich has an attractor and we perturb this system in such way so that the perturbed system has also anattractor, then the rate of convergence of these attractors is of order the distance of the corresponding time one maps. Moreover, we generalize this result with the following one: If we have a finite dynamical systemgenerated by a Morse Smale gradient map which has an attractor, and we consider a nonautonomousperturbation of it, such that this non autonomous perturbed dynamical system has a pullback attractor, thenthe rate of convergence of the attractor and pullback attractor is of order the rate of convergence of thecorresponding time one maps. These results are obtained in a finite dimensional framework and the method of proof is using the Shadowing properties that Morse Smale maps have.Since our framework for these results is finite dimensional, and we eventually want to apply thistechnique to infinite dimensional systems, we need a tool that reduces the system to a finite dimensionalone, Inertial Manifolds. These smooth finite dimensional manifolds, positive invariant under the flow andexponentially attractive, contain the attractor if the system has one. Since we want to study the behavior of the systems under perturbations and obtain good rates for the convergence of the attractors, we study thebehavior of these inertial manifolds under perturbations of the system, providing good rates of itsconvergence in the C 0 topology and also in the C 1 topology.Finally, we apply the techniques mentioned above to address the problem of the distance ofattractors corresponding to a particular reaction diffusion equation and a perturbation of itdescribed in a thin domain. This thesis improves the rate of convergence of attractors for thisparticular problem existing in the literature.