We study the long-time behaviour of solutions to quasilinear doubly degenerate parabolic problems of fourth order. The equations model for instance the dynamic behaviour of a non-Newtonian thin-film flow on a flat impermeable bottom and with zero contact angle. We consider a shear-rate dependent fluid the rheology of which is described by a constitutive power-law or Ellis-law for the fluid viscosity. In all three cases, positive constants (i.e. positive flat films) are the only positive steady-state solutions. Moreover, we can give a detailed picture of the long-time behaviour of solutions with respect to the H1(Ω)-norm. In the case of shear-thickening power-law fluids, one observes that solutions which are initially close to a steady state, converge to equilibrium in finite time. In the shear-thinning power-law case, we find that steady states are polynomially stable in the sense that, as time tends to infinity, solutions which are initially close to a steady state, converge to equilibrium at rate 1/t1/β for some β>0. Finally, in the case of an Ellis-fluid, steady states are exponentially stable in H1(Ω).
%0 Generic
%1 https://doi.org/10.48550/arxiv.2204.08231
%A Jansen, Jonas
%A Lienstromberg, Christina
%A Nik, Katerina
%D 2022
%I arXiv
%K Lienstromberg equations parabolic quasilinear
%R 10.48550/ARXIV.2204.08231
%T Long-time behaviour and stability for quasilinear doubly degenerate parabolic equations of higher order
%U https://arxiv.org/abs/2204.08231
%X We study the long-time behaviour of solutions to quasilinear doubly degenerate parabolic problems of fourth order. The equations model for instance the dynamic behaviour of a non-Newtonian thin-film flow on a flat impermeable bottom and with zero contact angle. We consider a shear-rate dependent fluid the rheology of which is described by a constitutive power-law or Ellis-law for the fluid viscosity. In all three cases, positive constants (i.e. positive flat films) are the only positive steady-state solutions. Moreover, we can give a detailed picture of the long-time behaviour of solutions with respect to the H1(Ω)-norm. In the case of shear-thickening power-law fluids, one observes that solutions which are initially close to a steady state, converge to equilibrium in finite time. In the shear-thinning power-law case, we find that steady states are polynomially stable in the sense that, as time tends to infinity, solutions which are initially close to a steady state, converge to equilibrium at rate 1/t1/β for some β>0. Finally, in the case of an Ellis-fluid, steady states are exponentially stable in H1(Ω).
@misc{https://doi.org/10.48550/arxiv.2204.08231,
abstract = {We study the long-time behaviour of solutions to quasilinear doubly degenerate parabolic problems of fourth order. The equations model for instance the dynamic behaviour of a non-Newtonian thin-film flow on a flat impermeable bottom and with zero contact angle. We consider a shear-rate dependent fluid the rheology of which is described by a constitutive power-law or Ellis-law for the fluid viscosity. In all three cases, positive constants (i.e. positive flat films) are the only positive steady-state solutions. Moreover, we can give a detailed picture of the long-time behaviour of solutions with respect to the H1(Ω)-norm. In the case of shear-thickening power-law fluids, one observes that solutions which are initially close to a steady state, converge to equilibrium in finite time. In the shear-thinning power-law case, we find that steady states are polynomially stable in the sense that, as time tends to infinity, solutions which are initially close to a steady state, converge to equilibrium at rate 1/t1/β for some β>0. Finally, in the case of an Ellis-fluid, steady states are exponentially stable in H1(Ω).},
added-at = {2023-03-13T12:19:05.000+0100},
author = {Jansen, Jonas and Lienstromberg, Christina and Nik, Katerina},
biburl = {https://puma.ub.uni-stuttgart.de/bibtex/2c32b47833502814b7c75aaaad4c74535/elkepeter},
copyright = {arXiv.org perpetual, non-exclusive license},
doi = {10.48550/ARXIV.2204.08231},
interhash = {72503d5530e6085be17e97b16e653ce7},
intrahash = {c32b47833502814b7c75aaaad4c74535},
keywords = {Lienstromberg equations parabolic quasilinear},
publisher = {arXiv},
timestamp = {2023-04-03T07:25:12.000+0200},
title = {Long-time behaviour and stability for quasilinear doubly degenerate parabolic equations of higher order},
url = {https://arxiv.org/abs/2204.08231},
year = 2022
}