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

TL;DR

This paper establishes well-posedness, including existence, uniqueness, and equivalence of solution notions, for nonlocal porous medium equations with finite energy, covering a broad class of operators like fractional Laplacians and jump processes.

Contribution

It introduces new well-posedness results and equivalence between solution concepts for nonlocal nonlinear diffusion equations with general operators.

Findings

Proves Olenik type uniqueness for energy solutions

Establishes existence and uniqueness of distributional solutions with finite energy

Shows equivalence between different solution notions and provides quantitative estimates

Abstract

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.