Regularity theorems for a biological network formulation model in two space dimensions

TL;DR

This paper proves regularity results for a biological network model in two dimensions, showing that stationary weak solutions are actually classical, using weakly monotone functions and Hardy space techniques.

Contribution

It provides new regularity theorems for a biological network model, demonstrating that weak solutions are smooth in two dimensions, which was not previously established.

Findings

Stationary weak solutions are classical in two dimensions.

Regularity results are achieved using weakly monotone function theory.

Hardy space methods are employed in the analysis.

Abstract

We present several regularity results for a biological network formulation model originally introduced by D. Cai and D. Hu in {\it Phys. Rev. Lett.}, {\bf 111}(2013), 138701. A consequence of these result is that a stationary weak solution must be a classical one in two space dimensions. Our mathematical analysis is based upon the weakly monotone function theory and Hardy space methods.