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

  • Félix del Teso

    The Norwegian University of Science and Technology (NTNU), Trondheim, Norway
  • Jørgen Endal

    The Norwegian University of Science and Technology (NTNU), Trondheim, Norway
  • Espen R. Jakobsen

    The Norwegian University of Science and Technology (NTNU), Trondheim, Norway
On the well-posedness of solutions with finite energy for nonlocal equations of porous medium type cover
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Abstract

We study well-posedness and equivalence of different notions of solutions with finite energy for nonlocal porous medium type equations of the form

tuAφ(u)=0.\partial_t u – A \varphi (u) = 0.

These equations are possibly degenerate nonlinear diffusion equations with a general nondecreasing continuous nonlinearity φ\varphi, and the largest class of linear symmetric nonlocal diffusion operators AA considered so far. The operators are defined from a bilinear energy form E\mathcal E and may be degenerate and have some xx-dependence. The fractional Laplacian, symmetric finite differences, and any generator of symmetric pure jump Lévy processes are included. The main results are (i) an Oleĭnik 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 wellposedness results for both notions of solutions. We also obtain quantitative energy and related LpL^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.