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

Publication ,  Journal Article
Liu, JG; Xu, X
Published in: Journal of Differential Equations
April 15, 2018

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.

## Published In

Journal of Differential Equations

1090-2732

0022-0396

April 15, 2018

264

8

## Start / End Page

5489 / 5526

• General Mathematics
• 4904 Pure mathematics
• 4901 Applied mathematics
• 0102 Applied Mathematics
• 0101 Pure Mathematics

### Citation

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Liu, J. G., & Xu, X. (2018). Partial regularity of weak solutions to a PDE system with cubic nonlinearity. Journal of Differential Equations, 264(8), 5489–5526. https://doi.org/10.1016/j.jde.2018.01.001
Liu, J. G., and X. Xu. “Partial regularity of weak solutions to a PDE system with cubic nonlinearity.” Journal of Differential Equations 264, no. 8 (April 15, 2018): 5489–5526. https://doi.org/10.1016/j.jde.2018.01.001.
Liu JG, Xu X. Partial regularity of weak solutions to a PDE system with cubic nonlinearity. Journal of Differential Equations. 2018 Apr 15;264(8):5489–526.
Liu, J. G., and X. Xu. “Partial regularity of weak solutions to a PDE system with cubic nonlinearity.” Journal of Differential Equations, vol. 264, no. 8, Apr. 2018, pp. 5489–526. Scopus, doi:10.1016/j.jde.2018.01.001.
Liu JG, Xu X. Partial regularity of weak solutions to a PDE system with cubic nonlinearity. Journal of Differential Equations. 2018 Apr 15;264(8):5489–5526.

## Published In

Journal of Differential Equations

1090-2732

0022-0396

April 15, 2018

264

8

5489 / 5526