On Existence analysis of steady flows of generalized Newtonian fluids with concentration dependent power-law index

Existence Analysis for a Model Describing Flow of an Incompressible Chemically Reacting Non-Newtonian Fluid

“…The mathematical analysis of the model (1.1)-(1.8), where the power-law index is also unknown, starts in [6]. The existence theory is established with the help of generalized monotone operator theory for p − > 3d d+2 ; compare the work of Antontsev and Rodrigues in [2].…”

“…First, the validity of (4.5) directly follows from (4.2)-(4.4) and from the weak lower semicontinuity. (See [6] for more details.) Next, we extend each v n outside Ω by zero and each p n as in Lemma 3.2.…”

“…(5.2) Due to the boundedness of G n (|v n | 2 ), one can adapt the technique from [6] (a generalization of the monotone operator theory) and establish the existence of a weak solution to (5.1)-(5.2). Moreover, one can set ψ := v n in (5.1) and ϕ := c n − c d in (5.2), and with the help of the assumptions (1.4)-(1.8) and the fact that div v n = 0 deduce the following uniform estimate:…”

“…Moreover, knowing that the concentration sequence is essentially bounded, see [6] for the proof of the weak maximum/minimum principle, we can apply Lemma 3.4 with g = c n v n . Then, for some α > 0 we obtain following uniform bound: …”

Thermo-visco-elasticity for Norton–Hoff-type models

“…This is however difficult to combine with a classical Galerkin method as the truncation of a function may no longer be a linear combination of the functions from the Galerkin basis. Therefore we appeal to non-standard energy methods, such as two-level Galerkin approximation, see also [10][11][12]. The new difficulty which arises here is the construction of the appropriate basis for approximation of the strain tensor ε p , for details see Appendix B.…”