On the well-posedness of solutions with finite energy for nonlocal equations of porous medium type

Félix del Teso, Jørgen Endal, Espen R. Jakobsen

Published 2016-10-07Version 1

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. Finally, we obtain quantitative energy and related $L^p$-estimates for distributional solutions.