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

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.