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We study the general nonlinear diffusion equation $u_t=\nabla\cdot(u^{m-1}\nabla (-\Delta)^{-s}u)$ that describes a flow through a porous mediumwhich is driven by a nonlocal pressure. We consider constant parameters $m>1$and $0<s<1$, we assume that the solutions are non-negative and the problem isposed in the whole space. In this paper we prove existence of weak solutionsfor all integrable initial data $u_0 \ge 0$ and for all exponents $m>1$ bydeveloping a new approximation method that allows to treat the range $m\ge 3$that could not be covered by previous works. We also extend the class ofinitial data to include any non-negative measure $\mu$ with finite mass. Inpassing from bounded initial data to measure data we make strong use of an$L^1$-$L^\infty$ smoothing effect and other functional estimates. Finite speedof propagation is established for all $m\ge 2$, and this property implies theexistence of free boundaries. The authors had already proved that finitepropagation does not hold for $m<2$.

Existence of weak solutions for a general porous medium equation with nonlocal pressure