Global Regularity and Uniqueness of Solutions in a Surface Growth Model Using Rigorous A-Posteriori Methods

The use of rigorous numerical methods to approach problems which can not be solved using standard methods (yet) has increased signifiantly in recent years.
In this book, riogorous a-posteriori methods are used to study the time evolution of a surface growth model, given by a fourth order semi-linear parabolic partial differential equation, where standard methods fail to verify global uniqueness and smoothness of solutions.

Based on an arbitrary numerical approximation, a-posteriori error-analysis is applied in order to prevent a blow up analytically. This is a method that in a similar way also applies to the three dimensional Navier-Stokes equations. The main idea consists of energy-estimates for the error between solution and approximation that yields a scalar differential equation controlling the norm of the error with coefficients depending solely on the numerical data. This allows the solution of the differential equation to be bounded using only numerical data.

A key technical tool is a rigorous eigenvalue bound for the nonlinear operator linearized around the numerical approximation. The presented method succeeds to show global uniqueness for relatively large initial conditions, which is demonstrated in many numerical examples.