Partial regularity of suitable weak solutions of the model arising in amorphous molecular beam epitaxy

TL;DR

This paper investigates the size of singular sets in solutions to a nonlinear parabolic PDE related to amorphous molecular beam epitaxy, establishing bounds on their Hausdorff dimension depending on a parameter.

Contribution

It generalizes previous results by relating the Hausdorff dimension of singularities to the parameter  in a nonlinear PDE, extending to a 3D Navier-Stokes system.

Findings

Hausdorff measure of singular set is zero for specific  range

Generalizes prior work for =2 and zero forcing term

Applicable to a 3D modified Navier-Stokes system

Abstract

In this paper, we are concerned with the precise relationship between the Hausdorff dimension of possible singular point set $\mathcal{S}$ of suitable weak solutions and the parameter $\alpha$ in the nonlinear term in the following parabolic equation $$h_t+h_{xxxx}+\partial_{xx}|h_x|^\alpha=f.$$ It is shown that when $5/3\leq\alpha<7/3$, the $\frac{3\alpha-5}{\alpha-1}$-dimensional parabolic Hausdorff measure of $\mathcal{S}$ is zero, which generalizes the recent corresponding work of Oz\'anski and Robinson in [31,SIAM J. Math. Anal. 51: 228--255, 2019] for $\alpha=2$ and $f=0$. The same result is valid for a 3D modified Navier-Stokes system.