LONG-TIME BEHAVIOR AND STABILITY FOR QUASILINEAR DOUBLY DEGENERATE PARABOLIC EQUATIONS OF HIGHER ORDER
(2023) In SIAM Journal on Mathematical Analysis 55(2). p.674-700- Abstract
We study the long-time behavior of solutions to quasilinear doubly degenerate parabolic problems of fourth order. The equations model, for instance, the dynamic behavior 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 behavior of solutions with respect to the H1(\Omega)-norm. In the case of shear-thickening power-law fluids, one observes that solutions which are initially... (More)
We study the long-time behavior of solutions to quasilinear doubly degenerate parabolic problems of fourth order. The equations model, for instance, the dynamic behavior 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 behavior of solutions with respect to the H1(\Omega)-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/\beta for some \beta > 0. Finally, in the case of an Ellis fluid, steady states are exponentially stable in H1(\Omega).
(Less)
- author
- Jansen, Jonas LU ; Lienstromberg, Christina and Nik, Katerina
- organization
- publishing date
- 2023
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- degenerate parabolic equation, Ellis fluid, long-time asymptotics, non-Newtonian fluid, power-law fluid, thin-film equation, weak solution
- in
- SIAM Journal on Mathematical Analysis
- volume
- 55
- issue
- 2
- pages
- 27 pages
- publisher
- Society for Industrial and Applied Mathematics
- external identifiers
-
- scopus:85153860438
- ISSN
- 0036-1410
- DOI
- 10.1137/22M1491137
- language
- English
- LU publication?
- yes
- id
- 1ba8e934-3618-4be4-977e-927c619e0919
- date added to LUP
- 2023-07-14 11:19:36
- date last changed
- 2023-07-14 11:19:36
@article{1ba8e934-3618-4be4-977e-927c619e0919, abstract = {{<p>We study the long-time behavior of solutions to quasilinear doubly degenerate parabolic problems of fourth order. The equations model, for instance, the dynamic behavior 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 behavior of solutions with respect to the H<sup>1</sup>(\Omega)-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/t<sup>1/\beta</sup> for some \beta > 0. Finally, in the case of an Ellis fluid, steady states are exponentially stable in H<sup>1</sup>(\Omega).</p>}}, author = {{Jansen, Jonas and Lienstromberg, Christina and Nik, Katerina}}, issn = {{0036-1410}}, keywords = {{degenerate parabolic equation; Ellis fluid; long-time asymptotics; non-Newtonian fluid; power-law fluid; thin-film equation; weak solution}}, language = {{eng}}, number = {{2}}, pages = {{674--700}}, publisher = {{Society for Industrial and Applied Mathematics}}, series = {{SIAM Journal on Mathematical Analysis}}, title = {{LONG-TIME BEHAVIOR AND STABILITY FOR QUASILINEAR DOUBLY DEGENERATE PARABOLIC EQUATIONS OF HIGHER ORDER}}, url = {{http://dx.doi.org/10.1137/22M1491137}}, doi = {{10.1137/22M1491137}}, volume = {{55}}, year = {{2023}}, }