Numerical a-posteriori regularity for solutions of a surface growth model

表面生长模型解的数值后验正则性

基本信息

批准号：
282524798
负责人：
Professor Dr. Dirk Blömker, Ph.D.
金额：
--
依托单位：
Lehrstuhl für Nichtlineare Analysis
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2015
资助国家：
德国
起止时间：
2014-12-31 至 2018-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/282524798?language=en
关键词：
Numerical posteriori regularity solutions surface

项目摘要

The project investigates the practicability of a-posteriori-regularity introduced by Chernyshenko, Constantin, Robinson, and Titi for the Navier-Stokes equation in dimension three. The key idea is to use numerical data or other approximations in analytic a-priori estimates, in order to prove rigorous bounds on solutions for fixed initial conditions. This rules out the possibility of a blow up in finite time for the unique smooth local solution, and thus establishes its global existence. This solves the problem of global uniqueness of smooth solutions at least for the given initial condition and a small neighbourhood around it. The calculation of the numerical simulation does not need to be rigorous, only the evaluation of the derived analytic bounds by using the numerical data.Instead of the final goal of the full 3D Navier-Stokes equation, we first test and optimize the method on a model from surface growth, which is both numerically and analytically much easier to access. For the start we will focus even on the one-dimensional model, which already exhibits similar problems than 3D-Navier Stokes. We intend to incorporate numerical data for the spectrum of the linearisation into the analytic estimates. Especially, because this has the potential of taking care of linear instabilities in the equation. For this aim, rigorous numerical calculation for maximal eigenvalues will be applied.In the second half of the project we will treat the two-dimensional surface growth equation, where the theory of global existence is not fully settled yet. Moreover, the method should be applied and tested with other models, even if the existence and uniqueness of global solutions is already settled, in order to verify the quality of the method. Of interest are here equations with similar structure as, for example, the Kuramoto-Sivashinsky equation, where in dimension two, the global existence of solutions is not fully settled,at least for squares.

该项目调查了Chernyshenko，Constantin，Robinson和Titi引入的A-Posteriori-rendibolity的实用性，以在第三维中使用Navier-Stokes方程。关键思想是在分析A-Priori估计中使用数值数据或其他近似值，以证明针对固定初始条件的解决方案的严格界限。这排除了在有限的时间内为独特的平滑本地解决方案爆炸的可能性，从而确立了其全球存在。这至少在给定的初始条件和周围的小社区中解决了平滑解决方案的全球唯一性问题。数值模拟的计算不需要严格，只能通过使用数值数据评估衍生的分析界限。该目标是完整的3D Navier-Stokes方程的最终目标，我们首先从表面增长中测试并优化了模型上的方法，这在数值和分析上既可以易于访问。首先，我们将重点放在一维模型上，该模型已经表现出比3D-Navier Stokes相似的问题。我们打算将线性化频谱的数值数据纳入分析估计。尤其是因为这有可能照顾方程式中的线性不稳定性。为此，将应用对最大特征值的严格数值计算。此外，即使已经解决了全局解决方案的存在和独特性，也应将该方法应用于其他模型，以验证该方法的质量。感兴趣的是具有类似结构的方程式，例如，库拉莫托 - 西瓦辛斯基方程，在第二个维度，至少对于正方形，在第二个方程中，解决方案的整体存在并未完全解决。