Longtime behavior and weak-strong uniqueness for a nonlocal porous media\n equation

2018/12/18 by Esther S. Daus, Daus, Esther S., Maria Pia Gualdani +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1812.07326

openalex publication_date 2018/12/18 · openalex created_date 2022/08/01 · openalex updated_date 2026/07/28

Abstract

In this manuscript we consider a non-local porous medium equation with\nnon-local diffusion effects given by a fractional heat operator\n n
partial_t u =
mboxdiv(u
nabla p),
qquad\n
partial_t p = -(-
Delta)^s p + u², in three space\ndimensions for 3/4\≤ s < 1 and analyze the long time asymptotics. The proof\nis based on energy methods and leads to algebraic decay towards the stationary\nsolution u=0 and \∇ p=0 in the L²(\ℝ³)-norm. The decay rate\ndepends on the exponent s. We also show weak-strong uniqueness of solutions\nand continuous dependence from the initial data. As a side product of our\nanalysis we also show that existence of weak solutions, previously shown in\n[Caffarelli, Gualdani, Zamponi 2018] for 3/4\≤ s \≤ 1, holds for 1/2 <\ns\≤ 1 if we consider our problem in the torus.\n

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