Nonlinear Analysis: Real World Applications 45 (2019) 704–720 Contents lists available at ScienceDirect Nonlinear Analysis: Real World Applications www.elsevier.com/locate/nonrwa Structure of the set of stationary solutions to the equations of motion of a class of generalized Newtonian fluids Jiří Neustupa a, * , Dennis A. Siginer b, c a Czech Academy of Sciences, Institute of Mathematics, Žitná 25, 115 67 Praha 1, Czech Republic b Department of Mathematics and Statistical Sciences & Department of Mechanical, Energy and Industrial Engineering, Botswana International University of Science and Technology, Palapye, Botswana c Centro de Investigación en Creatividad y Educación Superior, Universidad de Santiago de Chile, Santiago, Chile article info Article history: Received 26 February 2018 Received in revised form 19 July 2018 Accepted 25 July 2018 Keywords: Generalized Newtonian fluid Equations of motion Stationary solutions 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. © 2018 Elsevier Ltd. All rights reserved. 1. Introduction 1.1. The dynamic stress tensor We denote by v the velocity of a moving fluid and by Dv the rate of deformation tensor, which coincides with the symmetric gradient of velocity: Dv := 1 2 [∇v +(∇v) T ]. It is well known that the dynamic stress tensor S in the so called Stokesian fluid generally depends on the rate of deformation tensor through the formula S(Dv)= α I + β Dv + γ (Dv) 2 , (1.1) where the coefficients α, β and γ may further depend on the state variables (pressure q, density ρ, temperature ϑ) and on the principal invariants of tensor D. (See e.g. [1, Sec. 5.21, 5.22].) If the density is constant (which implies that the fluid is incompressible and the first principal invariant of Dv, i.e. div v, is equal to zero) * Corresponding author. E-mail addresses: neustupa@math.cas.cz (J. Neustupa), siginerd@biust.ac.bw, dennis.siginer@usach.cl (D.A. Siginer). https://doi.org/10.1016/j.nonrwa.2018.07.029 1468-1218/© 2018 Elsevier Ltd. All rights reserved.