Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion

We show the existence of locally bounded global solutions to the chemotaxis system $$u_t = \nabla\cdot(D(u)\nabla u) - \nabla\cdot(\frac{u}{v} \nabla v)$$ $$v_t = \Delta v - uv$$ with homogeneous Neumann boundary conditions and suitably regular positive initial data in smooth bounded domains $\Omega \subset \mathbb{R}^N$, $N\geq2$, for $D(u)\geq \delta u^{m-1}$ with some $\delta>0$, provided that $m>1+\frac N4$.