On the well-posedness of solutions with finite energy for nonlocal equations of porous medium type
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We study well-posedness and equivalence of different notions of solutions with finite energy for nonlocal porous medium type equations of the form ∂tu − Aϕ(u) = 0. These equations are possibly degenerate nonlinear diffusion equations with a general nondecreasing continuous nonlinearity ϕ and the largest class of linear symmetric nonlocal diffusion operators A considered so far. The operators are defined from a bilinear energy form E and may be degenerate and have some x-dependence. The fractional Laplacian, symmetric finite differences, and any generator of symmetric pure jump Lévy processes are included. The main results are (i) an Ole˘ınik 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.