Global generalized solutions to a chemotaxis model of capillary-sprout growth during tumor angiogenesis

This paper considers a chemotaxis-convection model of capillary-sprout growth during tumor angiogenesis

$ \begin{equation*} \begin{split} \left\{ {\begin{array}{*{20}{l}} {{u_t} = \Delta u - \nabla \cdot (u\nabla v) + \nabla \cdot (u\nabla w),}&{x \in \Omega ,t > 0,} \\ {{v_t} = \Delta v + \nabla \cdot (v\nabla w) - v + u,}&{x \in \Omega ,t > 0,} \\ {0 = \Delta w - w + u,}&{x \in \Omega ,t > 0,} \end{array}} \right. \end{split} \end{equation*} $

under Neumann initial-boundary conditions in a smooth bounded domain. In the two-dimensional setting, introducing a generalized solution concept according to (Winkler, 2015 [35]) and constructing an appropriate regularized system, we prove the global existence of at least one such solution with suitably regular initial data by an approximation procedure. To overcome the difficulty in taking the limit to its regularized system, we establish some technical estimates related to several energy integrals with special structures like $ \int_0^T\int_{\Omega}{\frac{{{u_\varepsilon }v_\varepsilon^p}}{{1 +\varepsilon {u_\varepsilon }}}} $, $ p>1 $ and $ \int_0^T \int_\Omega {\frac{{{u_\varepsilon }}}{{1+\varepsilon {u_\varepsilon }}}\ln^{k}({u_\varepsilon } + 1)}dxdt $, $ k\in(1,2) $.