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This paper is the continuation of a program, initiated in Grenier and Nguyen [SIAM J.Math. Anal. 51 (2019); J. Differential Equations 269 (2020)], to derive pointwise estimates on theGreen function of Orr–Sommerfeld equations. In this paper we focus on long wavelength perturbations, more precisely horizontal wave numbers\alphaof order\nu^{1/4}, which correspond to the lower boundary of the instability area for monotonic profiles. 

Consider the Vlasov–Poisson–Landau system with Coulomb potential in the weakly collisional regime on a $3$-torus, i.e. $\begin{array}{c}{\mathrm{\partial <\#comment/>}}_{t}F(t,x,v)+v_i{\mathrm{\partial <\#comment/>}}_{x_i}F(t,x,v)+E_i(t,x){\mathrm{\partial <\#comment/>}}_{v_i}F(t,x,v)=\mathrm{\nu <\#comment/>}Q(F,F)(t,x,v),\\ E(t,x)=\mathrm{\nabla <\#comment/>}{\mathrm{\Delta <\#comment/>}}^{-<\#comment/>1}\left({\int <\#comment/>}_{{\mathbb{R}}^3}F\right(t,x,v)\mathrm{d}v-<\#comment/>{\int <\#comment/>-<\#comment/>}_{{\mathbb{T}}^3}{\int <\#comment/>}_{{\mathbb{R}}^3}F(t,x,v\left)\mathrm{d}v\mathrm{d}x\right),\end{array}$with $\mathrm{\nu <\#comment/>}\ll <\#comment/>1$. We prove that for $\mathrm{ϵ<\#comment/>}>0$sufficiently small (but independent of $\mathrm{\nu <\#comment/>}$), initial data which are $O\left(\mathrm{ϵ<\#comment/>}{\mathrm{\nu <\#comment/>}}^{1/3}\right)$-Sobolev space perturbations from the global Maxwellians lead to global-in-time solutions which converge to the global Maxwellians as $t\to <\#comment/>\mathrm{\infty <\#comment/>}$. The solutions exhibit uniform-in- $\mathrm{\nu <\#comment/>}$Landau damping and enhanced dissipation. Our main result is analogous to an earlier result of Bedrossian for the Vlasov–Poisson–Fokker–Planck equation with the same threshold. However, unlike in the Fokker–Planck case, the linear operator cannot be inverted explicitly due to the complexity of the Landau collision operator. For this reason, we develop an energy-based framework, which combines Guo’s weighted energy method with the hypocoercive energy method and the commuting vector field method. The proof also relies on pointwise resolvent estimates for the linearized density equation. 

We prove the inviscid limit for the incompressible Navier-Stokes equations for data that are analytic only near the boundary in a general two-dimensional bounded domain. Our proof is direct, using the vorticity formulation with a nonlocal boundary condition, the explicit semigroup of the linear Stokes problem near the flatten boundary, and the standard wellposedness theory of Navier-Stokes equations in Sobolev spaces away from the boundary.