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

Abstract

In this paper we study the initial boundary value problem for the system\\ $-\mbox{div}\left[(I+\mathbf{m} \mathbf{m}^T)\nabla p\right]=s(x),\ \ \mathbf{m}_t-α^2Δ\mathbf{m}+|\mathbf{m}|^{2(γ-1)}\mathbf{m}=β^2(\mathbf{m}\cdot\nabla p)\nabla p$ 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 $|\nabla p|$ and $|\nabla \mathbf{m}|$ 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 $v\equiv (I+\mathbf{m}{\mathbf{m}}^T)\nabla p\cdot \nabla p$, and then suitably applying the De Giorge iteration method to the equation.

References in corpus (1)

• Global existence of strong solutions to a groundwater flow problem