Uniqueness and properties of distributional solutions of nonlocal equations of porous medium type
Provides foundational theoretical results for a broad class of nonlocal nonlinear evolution equations, benefiting researchers in PDEs and numerical analysis.
The authors prove uniqueness, existence, and properties (including L1-contraction and a priori estimates) for bounded distributional solutions of a general class of nonlocal porous medium type equations, covering fractional Laplacians and numerical discretizations. They also establish continuous dependence and convergence of numerical approximations.
We study the uniqueness, existence, and properties of bounded distributional solutions of the initial value problem problem for the anomalous diffusion equation $\partial_tu-\mathcal{L}^μ[φ(u)]=0$. Here $\mathcal{L}^μ$ can be any nonlocal symmetric degenerate elliptic operator including the fractional Laplacian and numerical discretizations of this operator. The function $φ:\mathbb{R} \to \mathbb{R}$ is only assumed to be continuous and nondecreasing. The class of equations include nonlocal (generalized) porous medium equations, fast diffusion equations, and Stefan problems. In addition to very general uniqueness and existence results, we obtain $L^1$-contraction and a priori estimates. We also study local limits, continuous dependence, and properties and convergence of a numerical approximation of our equations.