Optimal partial boundary condition for degenerate parabolic equations

For the stability of the non-Newtonian fluid equation partial derivative u/partial derivative t - div(a(x)vertical bar del u vertical bar(p-2)del u) - Sigma(N)(i=1) b(i)(x)D(i)u + c(x, t)u = f (x, t), where a(x)vertical bar(x is an element of Omega) > 0, a(x)vertical bar x is an element of partial derivative Omega = 0 and b(i)(x) is an element of C-1((Omega) over bar), we know that the degeneracy of a(x) may make the usual Dirichlet boundary value condition overdetermined and only a partial boundary value condition is expected. How to depict the geometric characteristic of the partial boundary value condition has been a long-time standing open problem. In this study, an optimal partial boundary value condition has been proposed, and the stability of weak solutions based on this partial boundary value condition is established. When the rate of the diffusion coefficient decays to zero, we explore how it affects the stability of weak solutions. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

In this study, an optimal partial boundary value condition has been proposed, and the stability of weak solutions based on this partial boundary value condition is established. The study explores how the stability of weak solutions is affected when the rate of the diffusion coefficient decays to zero.