Partial regularity of weak solutions to a PDE system with cubic nonlinearity

TL;DR

This paper studies the regularity of weak solutions to a biological network PDE system with cubic nonlinearities, providing partial regularity results and estimates on the size of potential singular sets.

Contribution

It establishes a partial regularity theorem for weak solutions of a complex PDE system with cubic nonlinearities, advancing understanding of singularity formation.

Findings

Partial regularity theorem for weak solutions

Estimate of the Hausdorff dimension of singular sets

Insights into regularity properties of biological network models

Abstract

In this paper we investigate regularity properties of weak solutions to a PDE system that arises in the study of biological transport networks. The system consists of a possibly singular elliptic equation for the scalar pressure of the underlying biological network coupled to a diffusion equation for the conductance vector of the network. There are several different types of nonlinearities in the system. Of particular mathematical interest is a term that is a polynomial function of solutions and their partial derivatives and this polynomial function has degree three. That is, the system contains a cubic nonlinearity. Only weak solutions to the system have been shown to exist. The regularity theory for the system remains fundamentally incomplete. In particular, it is not known whether or not weak solutions develop singularities. In this paper we obtain a partial regularity theorem, which gives an estimate for the parabolic Hausdorff dimension of the set of possible singular points.