An elementary proof of uniqueness of particle trajectories for solutions of a class of shear-thinning non-Newtonian 2D fluids

Abstract

We prove some regularity results for a class of two dimensional non-Newtonian fluids. By applying results from [Dashti and Robinson, Nonlinearity, 22 (2009), 735-746] we can then show uniqueness of particle trajectories.

Existence and regularity results for semi‐linearized compressible 2D fluids with generalized diffusion

“…Here 𝜕D l ∕𝜕x s = 𝜕 s 𝜕 l 𝑦 and 𝜕D i ∕𝜕x s = 𝜕 s 𝜕 i 𝑦. For further details see, for example, previous works 10,16,17 Lemma 5.1. Let T > 0.…”

“…We have now to show that = S(∇𝑦), where (y, 𝜌) is the limiting pair, up to a subsequence, for the approximating sequence {(y k , 𝜌 k )}. This is obtained by an application of the monotonicity trick (see, e.g., Lions, 18 Sections 2-5.2]; see also Berselli and Bisconti 10 ). Testing (5.1) 1 -written in terms of (y, 𝜌)-against y in L 2 , we obtain the following energy inequality, for 0 ≤ t 0 < t, that is, We also have that…”

“…Here and . For further details see, for example, previous works 10,16,17 …”

“…The term F(∇𝑦) mimics-at least in its form-the extra stress-tensor in the case of some types of generalized Newtonian fluids (see, e.g., previous works [8][9][10][11] ). We set…”

“…Also, in this case we proceed formally by providing a number of a priori estimates that, combined with a compactness criterion à la Aubin-Lions, allow us to prove the existence of a weak solution for the considered system supplied with appropriate initial data. As before, calculations can be made rigorous by using a suitable Galerkin scheme (see, e.g., previous works 3,10,11,13 ).…”