Global existence of strong solutions to a biological network formulation model in $2+1$ dimensions

In this paper we study the initial boundary value problem for the system\ $-\mbox{{div}}\left[(I+$\mathbf{m}$ $\mathbf{m}$^T)\nabla p\right]=s(x),\
$\mathbf{m}$_t-\alpha^2\Delta$\mathbf{m}$+|$\mathbf{m}$|^{2(\gamma-1)}$\mathbf{m}$=\beta^2($\mathbf{m}$\cdot\nabla p)\nabla p$ in two space dimensions. This problem has been proposed as a continuum model for biological transportation networks. The mathematical challenge is due to the presence of cubic nonlinearities, also known as trilinear forms, in the system. We obtain a weak solution $(\mathbf{m},p)$ with both $\mathrm{\mid }\mathrm{\nabla }p\mathrm{\mid }$ and $\mathrm{\mid }\mathrm{\nabla }\mathbf{m}\mathrm{\mid }$ being bounded. The result immediately triggers a bootstrap argument which can yield higher regularity for the weak solution. This is achieved by deriving an equation for $v\equiv (I+\mathbf{m}{\mathbf{m}}^T)\mathrm{\nabla }p\cdot \mathrm{\nabla }p$, and then suitably applying the De Giorge iteration method to the equation.