scholarly journals Long-time asymptotics for a 1D nonlocal porous medium equation with absorption or convection

2019 ◽  
Vol 22 (03) ◽  
pp. 1950015
Author(s):  
Filomena Feo ◽  
Yanghong Huang ◽  
Bruno Volzone
Keyword(s):  
Porous Medium ◽  
Exponential Convergence ◽  
Porous Medium Equation ◽  
Entropy Method ◽  
Nonlocal Diffusion ◽  
One Dimensional ◽  
Medium Equation ◽  
Long Time ◽  
Porous Medium Equations ◽  
Long Time Asymptotics

In this paper, the long-time asymptotic behaviors of one-dimensional porous medium equations with a fractional pressure and absorption or convection are studied. In the parameter regimes when the nonlocal diffusion is dominant, the entropy method is adapted to derive the exponential convergence of relative entropy of solutions in similarity variables.

scholarly journals A simple proof of the generalized Leibniz rule on bounded Euclidean domains

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Quoc-Hung Nguyen ◽  
Yannick Sire ◽  
Juan-Luis Vázquez
Keyword(s):  
Porous Medium ◽  
Partial Differential Equations ◽  
Simple Proof ◽  
Porous Medium Equation ◽  
Forthcoming Paper ◽  
Leibniz Rule ◽  
Bounded Domains ◽  
Medium Equation ◽  
Porous Medium Equations ◽  
Laplacian Equations

Abstract This paper is devoted to a simple proof of the generalized Leibniz rule in bounded domains. The operators under consideration are the so-called spectral Laplacian and the restricted Laplacian. Equations involving such operators have lately been considered by Constantin and Ignatova in the framework of the SQG equation [P. Constantin and M. Ignatova, Critical SQG in bounded domains, Ann. PDE 2 2016, 2, Article ID 8] in bounded domains, and by two of the authors [Q.-H. Nguyen and J. L. Vázquez, Porous medium equation with nonlocal pressure in a bounded domain, Comm. Partial Differential Equations 43 2018, 10, 1502–1539] in the framework of the porous medium with nonlocal pressure in bounded domains. We will use the estimates in this work in a forthcoming paper on the study of porous medium equations with pressure given by Riesz-type potentials.

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