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


Journal Article

© 2018 Elsevier Inc. 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.

Full Text

Duke Authors

Cited Authors

  • Liu, JG; Xu, X

Published Date

  • April 15, 2018

Published In

Volume / Issue

  • 264 / 8

Start / End Page

  • 5489 - 5526

Electronic International Standard Serial Number (EISSN)

  • 1090-2732

International Standard Serial Number (ISSN)

  • 0022-0396

Digital Object Identifier (DOI)

  • 10.1016/j.jde.2018.01.001

Citation Source

  • Scopus