Continuous dependence for the Brinkman–Darcy–Kelvin–Voigt equations backward in time

Abstract

We show that the solution to the Brinkman–Darcy–Kelvin–Voigt equations backward in time depends Hölder continuously upon the final data. A logarithmic convexity technique is employed, and uniqueness of the solution is simultaneously achieved.

Stability for the Kelvin–Voigt variable order equations backward in time

“…The method employed here also works for the Brinkman–Darcy–Kelvin–Voigt equations backward in time studied in Straughan 37 . In Straughan, 37 continuous dependence was achieved by appealing to a logarithmic convexity argument.…”

“…The method employed here also works for the Brinkman–Darcy–Kelvin–Voigt equations backward in time studied in Straughan 37 . In Straughan, 37 continuous dependence was achieved by appealing to a logarithmic convexity argument. The presence of the Kelvin–Voigt term λ Δ v i , t allows one to prove continuous dependence for the Brinkman–Darcy–Kelvin–Voigt equations backward in time using a similar technique to that of the present article.…”

“…The topic of continuous dependence of a solution to a boundary–initial value problem is very important. This subject, including continuous dependence upon the models themselves is analysed by previous works 23–39 . An increasingly important class of continuous dependence analyses are those pertaining to improperly posed problems, or non‐well‐posed problems; see previous literature 12,13,24,40–53 .…”

“…This subject, including continuous dependence upon the models themselves is analysed by previous works. [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39] An increasingly important class of continuous dependence analyses are those pertaining to improperly posed problems, or non-well-posed problems; see previous literature. 12,13,24,[40][41][42][43][44][45][46][47][48][49][50][51][52][53] Improperly posed problems have been amenable to analyse especially with the aid of the famous paper of John.…”

Proof of Straughan’s claim on Payne–Song’s and modified Guyer–Krumhansl’s equations

“…In an elegant analysis of stationary and oscillatory convection in a Brinkman-Darcy-Kelvin-Voigt fluid, Straughan [1] (see also [2][3][4][5]) considered a system of evolution laws including-beyond a regularized balance of momentum-Payne-Song's equation [6] in Eulerian representation, that is an energy balance given by…”