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Register a new userInviscid damping and enhanced dissipation of the boundary layer for 2D Navier-Stokes linearized around Couette flow in a channel
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We study the 2D Navier-Stokes equations linearized around the Couette flow $(y,0)^t$ in the periodic channel $mathbb T times [-1,1]$ with no-slip boundary conditions in the vanishing viscosity $ u to 0$ limit. We split the vorticity evolution into the free evolution (without a boundary) and a boundary corrector that is exponentially localized to at most an $O( u^{1/3})$ boundary layer. If the initial vorticity perturbation is supported away from the boundary, we show inviscid damping of both the velocity and the vorticity associated to the boundary layer. For example, our $L^2_t L^1_y$ estimate of the boundary layer vorticity is independent of $ u$, provided the initial data is $H^1$. For $L^2$ data, the loss is only logarithmic in $ u$. Note both such estimates are false for the vorticity in the interior. To the authors knowledge, this inviscid decay of the boundary layer vorticity seems to be a new observation not previously isolated in the literature. Both velocity and vorticity satisfy the expected $O(exp(-delta u^{1/3}alpha^{2/3}t))$ enhanced dissipation in addition to the inviscid damping. Similar, but slightly weaker, results are obtained also for $H^1$ data that is against the boundary initially. For $L^2$ data against the boundary, we at least obtain the boundary layer localization and enhanced dissipation.
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We consider solutions to the 2d Navier-Stokes equations on $mathbb{T}timesmathbb{R}$ close to the Poiseuille flow, with small viscosity $ u>0$. Our first result concerns a semigroup estimate for the linearized problem. Here we show that the $x$-dependent modes of linear solutions decay on a time-scale proportional to $ u^{-1/2}|log u|$. This effect is often referred to as emph{enhanced dissipation} or emph{metastability} since it gives a much faster decay than the regular dissipative time-scale $ u^{-1}$ (this is also the time-scale on which the $x$-independent mode naturally decays). We achieve this using an adaptation of the method of hypocoercivity. Our second result concerns the full nonlinear equations. We show that when the perturbation from the Poiseuille flow is initially of size at most $ u^{3/4+}$, then it remains so for all time. Moreover, the enhanced dissipation also persists in this scenario, so that the $x$-dependent modes of the solution are dissipated on a time scale of order $ u^{-1/2}|log u|$. This transition threshold is established by a bootstrap argument using the semigroup estimate and a careful analysis of the nonlinear term in order to deal with the unboundedness of the domain and the Poiseuille flow itself.
We study the weak boundary layer phenomenon of the Navier-Stokes equations in a 3D bounded domain with viscosity, $epsilon > 0$, under generalized Navier friction boundary conditions, in which we allow the friction coefficient to be a (1, 1) tensor on the boundary. When the tensor is a multiple of the identity we obtain Navier boundary conditions, and when the tensor is the shape operator we obtain conditions in which the vorticity vanishes on the boundary. By constructing an explicit corrector, we prove the convergence of the Navier-Stokes solutions to the Euler solution as the viscosity vanishes. We do this both in the natural energy norm with a rate of order $epsilon^{3/4}$ as well as uniformly in time and space with a rate of order $epsilon^{3/8 - delta}$ near the boundary and $epsilon^{3/4 - delta}$ in the interior, where $delta, delta$ decrease to 0 as the regularity of the initial velocity increases. This work simplifies an earlier work of Iftimie and Sueur, as we use a simple and explicit corrector (which is more easily implemented in numerical applications). It also improves a result of Masmoudi and Rousset, who obtain convergence uniformly in time and space via a method that does not yield a convergence rate.
In this paper, we study the decay rates for the global small smooth solutions to 3D anisotropic incompressible Navier-Stokes equations. In particular, we prove that the horizontal components of the velocity field decay like the solutions of 2D classical Navier-Stokes equations. While the third component of the velocity field decays as the solutions of 3D Navier-Stokes equations. We remark that such enhanced decay rate for the third component is caused by the interplay between the divergence free condition of the velocity field and the horizontal Laplacian in the anisotropic Navier-Stokes equations.
Consider inviscid fluids in a channel {-1<y<1}. For the Couette flow v_0=(y,0), the vertical velocity of solutions to the linearized Euler equation at v_0 decays in time. At the nonlinear level, such inviscid damping has not been proved. First, we show that in any (vorticity) H^{s}(s<(3/2)) neighborhood of Couette flow, there exist non-parallel steady flows with arbitrary minimal horizontal period. This implies that nonlinear inviscid damping is not true in any (vorticity) H^{s}(s<(3/2)) neighborhood of Couette flow and for any horizontal period. Indeed, the long time behavior in such neighborhoods are very rich, including nontrivial steady flows, stable and unstable manifolds of nearby unstable shears. Second, in the (vorticity) H^{s}(s>(3/2)) neighborhood of Couette, we show that there exist no non-parallel steadily travelling flows v(x-ct,y), and no unstable shears. This suggests that the long time dynamics in H^{s}(s>(3/2)) neighborhoods of Couette might be much simpler. Such contrasting dynamics in H^{s} spaces with the critical power s=(3/2) is a truly nonlinear phenomena, since the linear inviscid damping near Couette is true for any initial vorticity in L^2.
In this paper we study the dynamics of an incompressible viscous fluid evolving in an open-top container in two dimensions. The fluid mechanics are dictated by the Navier-Stokes equations. The upper boundary of the fluid is free and evolves within the container. The fluid is acted upon by a uniform gravitational field, and capillary forces are accounted for along the free boundary. The triple-phase interfaces where the fluid, air above the vessel, and solid vessel wall come in contact are called contact points, and the angles formed at the contact point are called contact angles. The model that we consider integrates boundary conditions that allow for full motion of the contact points and angles. Equilibrium configurations consist of quiescent fluid within a domain whose upper boundary is given as the graph of a function minimizing a gravity-capillary energy functional, subject to a fixed mass constraint. The equilibrium contact angles can take on any values between $0$ and $pi$ depending on the choice of capillary parameters. The main thrust of the paper is the development of a scheme of a priori estimates that show that solutions emanating from data sufficiently close to the equilibrium exist globally in time and decay to equilibrium at an exponential rate.
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