On the well-posedness of solutions with finite energy for nonlocal equations of porous medium type
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 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 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- October 2016
- DOI:
- 10.48550/arXiv.1610.02221
- arXiv:
- arXiv:1610.02221
- Bibcode:
- 2016arXiv161002221D
- Keywords:
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- Mathematics - Analysis of PDEs;
- 35A02;
- 35B30;
- 35D30;
- 35K55;
- 35K65;
- 35R09;
- 35R11
- E-Print:
- 38 pages. We now include properties of operators comparable to fractional Laplacians. In this case the uniqueness and existence classes coincide. Especially, discussion and rigorous results on homogeneous fractional Sobolev spaces are needed in our setting. To appear in "EMS Series of Congress Reports: A Volume in Honor of Helge Holden's 60th Birthday"