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].


P. Degond et al. JMFM
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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