A fractional porous medium equation

We develop a theory of existence, uniqueness and regularity for the following porous medium equation with fractional diffusion, {partial derivative u/partial derivative t + (-Delta)1/2(vertical bar u vertical bar(m-1)u) = 0, x is an element of R(N), t >0, u(x, 0) = f(x), x is an element of R(N), with m > m(*) = (N - 1)/N, N >= 1 and f is an element of L(1) (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 E RN, > 0. (C) 2010 Elsevier Inc. All rights reserved.