%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 %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 = {2023-12-05T17:17:36.000+0100}, author = {Jansen, Jonas and Lienstromberg, Christina and Nik, Katerina}, biburl = {https://puma.ub.uni-stuttgart.de/bibtex/2c32b47833502814b7c75aaaad4c74535/simtech}, copyright = {arXiv.org perpetual, non-exclusive license}, doi = {10.48550/ARXIV.2204.08231}, interhash = {72503d5530e6085be17e97b16e653ce7}, intrahash = {c32b47833502814b7c75aaaad4c74535}, keywords = {}, publisher = {arXiv}, timestamp = {2023-12-05T17:17:36.000+0100}, title = {Long-time behaviour and stability for quasilinear doubly degenerate parabolic equations of higher order}, url = {https://arxiv.org/abs/2204.08231}, year = 2022 }