  We consider degenerate porous medium equations with a divergence type of
drift terms. We establish the existence of $L^{q}$-weak solutions (satisfying
energy estimates or even further with moment and speed estimates in Wasserstein
spaces), in case the drift term belongs to a sub-scaling (including scaling
invariant) class depending on $q$ and $m$ caused by the nonlinear structure of
diffusion, which is a major difference compared to that of a linear case. It is
noticeable that the classes of drift terms become wider if the drift term is
divergence-free. Similar conditions of gradients of drift terms are also
provided to ensure the existence of such weak solutions. Uniqueness results
follow under an additional condition on the gradients of the drift terms with
the aid of methods developed in Wasserstein spaces. One of our main tools is so
called the splitting method to construct a sequence of approximated solutions,
which implies, bypassing to the limit, the existence of weak solutions
satisfying not only an energy inequality but also moment and speed estimates.
One of the crucial points in the construction is uniform H\"{o}lder continuity
up to initial time for homogeneous porous medium equations, which seems to be
of independent interest. As an application, we improve a regularity result for
solutions of a repulsive Keller-Segel system of porous medium type.

Analysis of PDEs