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

• Applicant: Professor Dr. Dirk Blömker, Ph.D.
• Subject Area: Mathematics
• Term: from 2015 to 2019
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 282524798

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.

• DFG Programme: Research Grants