Existence of weak solutions for Porous medium equation with a divergence type of drift term in a bounded domain

We study porous medium equations with a divergence form of drift terms in a bounded domain with no-flux lateral boundary conditions. We establish $L^q$-weak solutions for $1\le q<\mathrm{\infty }$ in Wasserstein space under appropriate conditions on the drift, which is an extension of authors’ previous works done in the whole space into the case of bounded domains. Applying existence results to a certain Keller-Segel equation of consumption type, construction of $L^q$-weak solutions is also made, in case that the equation of a biological organism is of porous medium type.