Correction to: Coupled Self-Organized Hydrodynamics and Stokes Models for Suspensions of Active Particles

This note provides a list of errata and their correction for Reference [1].

To obtain these results we modify the proof of Th. 5.1 Case B, which we rewrite fully next (notice that the number of the equations that do not start by 0 refer to the number of the equations as they appear in the original article).
Proof. Case (B) Suppose that k ⊥ = 0. Doing the inner product of Eq. (0.1) with k and using that k ⊥ · k = |k| 2 − k 2 0 , the dispersion relation is given by (using Eq. (5.11)): and from Eq. (5.10b) we have the relation Next we distinguish between the casesk = 0 andk = 0. Suppose thatk = 0, then from Eq. (0.5) we have that −α + U 0 · k = 0 and multiplying Eq. (0.4) by −α + U 0 · k = 0 and using Eq. (0.5), we get the dispersion relation in Eq. (0.2), after simplifyingk. If, on the contrary,k = 0, then one can check that k ⊥ ·Ω = 0 (thanks to Eq. (5.10a)) and, therefore, they are normal. Since by assumption k ⊥ = 0 from Eq. (0.1) we conclude that the coefficient in front of k ⊥ on the right hand side must be zero. Rewriting this term using thatk = 0 and Eq. (5.11) we have thatρ From here we deduce thatρ = 0 because otherwise we should have that I = 0 but the imaginary part of I is non-zero, soρ = 0. In particular this implies that Eq. (0.5) is fulfilled and thatΩ = 0 (otherwise we would have null perturbation). SinceΩ = 0, from Eq. (0.1) again (remembering that k ⊥ ⊥Ω) we must have that the coefficient in front ofΩ is equal to zero. This gives the dispersion relation (0.3). Now, we go back to the casek = 0. To simplify the analysis we will restrict ourselves to the case where k ⊥ = k, i.e. k 0 = k · Ω 0 = 0 andk = 0. This implies, in particular, that U 0 · k = V 0 · k = v 0 · k. With these considerations one can simplify the dispersion relation (0.2) intõ D(α, k) = 0, whereD(α, k) is given in Eq. (5.7).
• There are two typos at the end of page 21: in the third line of Case (A) part b) should read Im(α) instead of Im(ω); and in the fifth line should read "Ω is arbitrary withΩ · Ω 0 = 0" rather than "Ω is arbitrary withΩ, Ω 0 = 0".

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