Energy estimates and convergence of weak solutions of the porous medium equation. (13th October 2021)

• Record Type: Journal Article
• Authors: De Paula, R; Gonçalves, P; Neumann, A
• Is Part Of: Nonlinearity. Volume 34:Number 11(2021)
• Page Start: 7872
• Page End: 7915
• Publication Date: 2021-10-13
• Subjects: Robin boundary conditions -- porous medium model -- porous medium equation; 60K35 -- 26A24 -- 35K55; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ac2a16
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: We study the convergence of the weak solution of the porous medium equation with a type of Robin boundary conditions, by tuning a parameter either to zero or to infinity. The convergence is in the strong sense, with respect to the L 2 -norm, and the limiting function solves the same equation with Neumann (resp. Dirichlet) boundary conditions when the parameter is taken to zero (resp. infinity). Our approach is to consider an underlying microscopic dynamics whose space–time evolution of the density is ruled by the solution of those equations and from this, we derive sufficiently strong energy estimates which are the keystone to the proof of our convergence result.