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In this work, we prove a threshold theorem for the 2D Navier-Stokes equations posed on the periodic channel, $$\mathbb{T} \times [-1,1]$$, supplemented with Navier boundary conditions $$\omega|_{y = \pm 1} = 0$$. Initial datum is taken to be a perturbation of Couette in the following sense: the shear component of the perturbation is assumed small (in an appropriate Sobolev space) but importantly is independent of $$\nu$$. On the other hand, the nonzero modes are assumed size $$O(\nu^{\frac12})$$ in an anisotropic Sobolev space. For such datum, we prove nonlinear enhanced dissipation and inviscid damping for the resulting solution. The principal innovation is to capture quantitatively the \textit{inviscid damping}, for which we introduce a new Singular Integral Operator which is a physical space analogue of the usual Fourier multipliers which are used to prove damping. We then include this SIO in the context of a nonlinear hypocoercivity framework. 

The interaction between radio jets and quasar host galaxies plays a paramount role in quasar and galaxy co-evolution. However, very little is known at present about this interaction at very high−z. Here, we present new Atacama Large Millimeter/submillimeter Array (ALMA) observations in Bands 7 and 3 of six radio-loud (RL) quasar host galaxies atz > 5. We recovered [C II] 158 μm line and underlying dust continuum emission at > 2σfor five sources, while we obtained upper limits for the CO(6-5) emission line and continuum for the remaining source. At the spatial resolution of our observations (∼1″​​_.0–1″​​_.4), we did not recover any perturbed or extended morphologies or kinematics, which are known signatures of potential mergers. These galaxies already host large quantities of gas (∼10¹⁰M_⊙), with [C II] luminosities ofL_[C II] ∼ 10^8 − 9 L_⊙and [C II]-based star formation rates of 30 − 400 M_⊙yr⁻¹. In building their radio/submillimeter (radio/submm) spectral energy distributions (SEDs), we found that in at least four cases, the 1 mm continuum intensity arises from a combination of synchrotron and dust emission. The initial estimation of synchrotron contribution at 300 GHz in these cases is of ≳10%. Assuming a scenario where the continuum emission is solely due to cold dust as an upper limit, we obtained infrared (IR) luminosities ofL_IR ∼ 10^{11 − 12} L_⊙. We compared the properties of the sources inspected here with a large collection of radio-quiet sources from the literature, as well as a sample of RL quasars from previous studies at comparable redshifts. We recovered a mild potential decrease inL_[C II]for the RL sources, which might be due to a suppression of the cool gas emission due to the radio jets. We did not find any [C II] emitting companion galaxy candidate around the five RL quasars observed in Band 7. Given the depth of our dataset, this result is still consistent with what has been observed around radio-quiet quasars. Future higher spatial-resolution observations, over a broader frequency range, of high−zRL quasars hosts will allow us to further improve our understanding of the physics of these sources.