  We develop a theory of existence, uniqueness and regularity for a porous
medium equation with fractional diffusion, $\frac{\partial u}{\partial t} +
(-\Delta)^{1/2} (|u|^{m-1}u)=0$ in $\mathbb{R}^N$, with $m>m_*=(N-1)/N$,
$N\ge1$ and $f\in L^1(\mathbb{R}^N)$. An $L^1$-contraction semigroup is
constructed and the continuous dependence on data and exponent is established.
Nonnegative solutions are proved to be continuous and strictly positive for all
$x\in\mathbb{R}^N$, $t>0$.
