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

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 $E=\int |\nabla h|\ln |\nabla h|dx$ 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.

15 pages

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