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

该项目研究了Chernyshenko、Constantin、Robinson和Titi为三维Navier-Stokes方程引入的后正则性的实用性。关键思想是在分析先验估计中使用数值数据或其他近似，以证明固定初始条件下解的严格界。这就排除了唯一光滑局部解在有限时间内爆炸的可能性，从而确定了它的全局存在性。这解决了光滑解的全局唯一性问题，至少对于给定的初始条件及其周围的小邻域是如此。数值模拟的计算不需要严格，只需要利用数值数据求出推导出的解析界即可。我们的最终目标不是完整的3D Navier-Stokes方程，而是首先在表面生长的模型上测试和优化该方法，这在数值和分析上都更容易获得。首先，我们将关注一维模型，它已经表现出与3D-Navier Stokes相似的问题。我们打算将线性化谱的数值数据纳入分析估计中。特别是，因为这有可能处理方程中的线性不稳定性。为此目的，将应用严格的最大特征值数值计算。在项目的后半部分，我们将处理二维表面生长方程，其中全局存在理论尚未完全解决。此外，即使已经确定了全局解的存在性和唯一性，该方法也应该与其他模型一起应用和测试，以验证该方法的质量。我们感兴趣的是结构类似的方程，比如Kuramoto-Sivashinsky方程，在二维中，解的整体存在性没有完全确定，至少对于平方来说是这样。

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.