Existence, uniqueness and stability of regular steady motions of a second-grade fluid

On steady third grade fluids equations

“…Taking the scalar product of (11) with E and using relations (7), (9), (12) and (14) we obtain after a few calculations that…”

Asymptotic behaviour at infinity of three-dimensional steady viscoelastic flows

“…On the other hand, since v ∈ V l+1,q γ−1 (Ω) 3 it follows that r −3 v ∈ V l+1,q γ+2 (Ω) 3 . In particular, it holds r −3 v ∈ W l+1,q (Ω) 3 . Recalling that for l ≥ 2 and q > 3 2 the spaces W l,q (Ω) 3 form a Banach algebra, one obtains…”

“…Finally, in order to estimate the L q γ (Ω)-norm of D l x z, one first extends z, v andĝ continuously to R 3 , then applies the operator D l x to the equation in R 3 and finally mollifies the resulting equation in order to integrate by parts in the term Ω R q(γ−l) z | z| q−2 v · ∇D l x z dx. The standard properties of the mollifier then yield the estimate 3 is chosen small enough. From (4.4), (4.5) and (4.6), one concludes that…”

“…[7,8,22,23,32,26]. These studies, as most of the other related works on viscoelastic fluids of differential and rate type, see e.g., [2,28,9,3,25] and all the references quoted therein, are crucially based on the mixed structure of the equations of motion. In fact, the equations form a system of nonlinear PDE's of elliptichyperbolic type which can most naturally be studied by suitably decoupling the elliptic and hyperbolic parts of the equations.…”

On the 3D Steady Flow of a Second Grade Fluid Past an Obstacle

“…The question of signs and values of the stress moduli α 1 , α 2 and especially of α 1 + α 2 in this model is not clear. In [1] the authors show that the constraint α 1 + α 2 = 0 is not necessary for the mathematical problem being well set. In [5] the authors show that for α 1 < 0 the rest state of flow of second grade fluid in exterior domain is instable.…”