### Abstract

We investigate the steady-state equations of motion of the generalized Newtonian fluid in a bounded domain Ω⊂R^{N}, when N=2 or N=3. Applying the tools of nonlinear analysis (Smale's theorem, theory of Fredholm operators, etc.), we show that if the dynamic stress tensor has the 2-structure then the solution set is finite and the solutions are C^{1}-functions of the external volume force f for generic f. We also derive a series of properties of related operators in the case of a more general p-structure, show that the solution set is compact if p>3N∕(N+2) and explain why the same approach as in the case p=2 cannot be applied if p≠2.

Original language | English |
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Pages (from-to) | 704-720 |

Number of pages | 17 |

Journal | Nonlinear Analysis: Real World Applications |

Volume | 45 |

DOIs | |

Publication status | Published - Feb 2019 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Engineering(all)
- Economics, Econometrics and Finance(all)
- Computational Mathematics
- Applied Mathematics

### Cite this

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**Structure of the set of stationary solutions to the equations of motion of a class of generalized Newtonian fluids.** / Neustupa, Jiří; Dennis, Siginer.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Structure of the set of stationary solutions to the equations of motion of a class of generalized Newtonian fluids

AU - Neustupa, Jiří

AU - Dennis, Siginer

PY - 2019/2

Y1 - 2019/2

N2 - We investigate the steady-state equations of motion of the generalized Newtonian fluid in a bounded domain Ω⊂RN, when N=2 or N=3. Applying the tools of nonlinear analysis (Smale's theorem, theory of Fredholm operators, etc.), we show that if the dynamic stress tensor has the 2-structure then the solution set is finite and the solutions are C1-functions of the external volume force f for generic f. We also derive a series of properties of related operators in the case of a more general p-structure, show that the solution set is compact if p>3N∕(N+2) and explain why the same approach as in the case p=2 cannot be applied if p≠2.

AB - We investigate the steady-state equations of motion of the generalized Newtonian fluid in a bounded domain Ω⊂RN, when N=2 or N=3. Applying the tools of nonlinear analysis (Smale's theorem, theory of Fredholm operators, etc.), we show that if the dynamic stress tensor has the 2-structure then the solution set is finite and the solutions are C1-functions of the external volume force f for generic f. We also derive a series of properties of related operators in the case of a more general p-structure, show that the solution set is compact if p>3N∕(N+2) and explain why the same approach as in the case p=2 cannot be applied if p≠2.

UR - http://www.scopus.com/inward/record.url?scp=85051400003&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85051400003&partnerID=8YFLogxK

U2 - 10.1016/j.nonrwa.2018.07.029

DO - 10.1016/j.nonrwa.2018.07.029

M3 - Article

VL - 45

SP - 704

EP - 720

JO - Nonlinear Analysis: Real World Applications

JF - Nonlinear Analysis: Real World Applications

SN - 1468-1218

ER -