# Global Strong Solution With BV Derivatives to Singular Solid-on-Solid model With Exponential Nonlinearity

## TL;DR

This paper proves the existence of global strong solutions with bounded variation derivatives for a singular fourth-order solid-on-solid model with exponential nonlinearity, using a gradient flow approach and energy corrections.

## Contribution

It introduces a novel approach with a logarithmic energy correction to establish global solutions with BV derivatives for a highly singular model.

## Key findings

- Proved the evolution variational inequality solution maintains positive, bounded gradient.
- Established the existence of global strong solutions allowing asymmetric singularities.
- Demonstrated the effectiveness of gradient flow structure with energy correction in singular PDEs.

## Abstract

In this work, we consider the one dimensional very singular fourth-order equation for solid-on-solid model in attachment-detachment-limit regime with exponential nonlinearity $$h_t = \nabla \cdot (\frac{1}{|\nabla h|} \nabla e^{\frac{\delta E}{\delta h}}) =\nabla \cdot (\frac{1}{|\nabla h|}\nabla e^{- \nabla \cdot (\frac{\nabla h}{|\nabla h|})})$$ where total energy $E=\int |\nabla h|$ is the total variation of $h$. Using a logarithmic correction $E=\int |\nabla h|\ln|\nabla h| d x$ and gradient flow structure with a suitable defined functional, we prove the evolution variational inequality solution preserves a positive gradient $h_x$ which has upper and lower bounds but in BV space. We also obtain the global strong solution to the solid-on-solid model which allows an asymmetric singularity $h_{xx}^+$ happens.