Partial regularity of weak solutions and life-span of smooth solutions
to a biological network formulation model
Abstract
In this paper we study partial regularity of weak solutions to the
initial boundary value problem for the system
$-\textup{div}\left[(I+{\bf
m}\otimes {\bf m})\nabla
p\right]=S(x),\ \
\partial_t{\bf
m}-D^2\Delta {\bf
m}-E^2({\bf
m}\cdot\nabla p)\nabla
p+|{\bf
m}|^{2(\gamma-1)}{\bf
m}=0$, where $S(x)$ is a given function and $D, E,
\gamma$ are given numbers. This problem has been
proposed as a PDE model for biological transportation networks. The
mathematical difficulty is due to the fact that the system in the model
features both a quadratic nonlinearity and a cubic nonlinearity. The
regularity issue seems to have a connection to a conjecture by De Giorgi
\cite{DE}. We also investigate the life-span of
classical solutions. Our results show that local existence of a
classical solution can always be obtained and the life-span of such a
solution can be extended as far away as one wishes as long as the term
$\|{\bf
m}(x,0)\|_{\infty,
\Omega}+\|S(x)\|_{\frac{2N}{3},
\Omega}$ is made suitably small, where $N$ is the
space dimension and
$\|\cdot\|_{q,\Omega}$
denotes the norm in $L^q(\Omega)$.