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LONG-TIME BEHAVIOR AND STABILITY FOR QUASILINEAR DOUBLY DEGENERATE PARABOLIC EQUATIONS OF HIGHER ORDER

Jansen, Jonas LU ; Lienstromberg, Christina and Nik, Katerina (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).

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author
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organization
publishing date
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
2025-10-14 10:51:07
@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 &gt; 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}},
}