  We study well-posedness and equivalence of different notions of solutions
with finite energy for nonlocal porous medium type equations of the form
$$\partial_tu-A\varphi(u)=0.$$ These equations are possibly degenerate
nonlinear diffusion equations with a general nondecreasing continuous
nonlinearity $\varphi$ and the largest class of linear symmetric nonlocal
diffusion operators $A$ considered so far. The operators are defined from a
bilinear energy form $\mathcal{E}$ and may be degenerate and have some
$x$-dependence. The fractional Laplacian, symmetric finite differences, and any
generator of symmetric pure jump L\'evy processes are included. The main
results are (i) an Ole\u{\i}nik type uniqueness result for energy solutions;
(ii) an existence (and uniqueness) result for distributional solutions with
finite energy; and (iii) equivalence between the two notions of solution, and
as a consequence, new well-posedness results for both notions of solutions. We
also obtain quantitative energy and related $L^p$-estimates for distributional
solutions. Our uniqueness results are given for a class of functions defined
from test functions by completion in a certain topology. We study rigorously
several cases where this space coincides with standard function spaces. In
particular, for operators comparable to fractional Laplacians, we show that
this space is a parabolic homogeneous fractional Sobolev space.
