Related papers: Linear Inviscid Damping for Monotone Shear Flows
In this note we revisit the proof of Bedrossian and Masmoudi [arXiv:1306.5028] about the inviscid damping of planar shear flows in the 2D Euler equations under the assumption of zero mean perturbation. We prove that a small perturbation to…
In this article we establish linear inviscid damping with optimal decay rates around 2D Taylor-Couette flow and similar monotone flows in an annular domain $B_{r_{2}}(0) \setminus B_{r_{1}}(0) \subset \mathbb{R}^{2}$. Following recent…
We consider a two-dimensional, two-layer, incompressible, steady flow, with vorticity which is constant in each layer, in an infinite channel with rigid walls. The velocity is continuous across the interface, there is no surface tension or…
We study the large time behavior of solutions to two-dimensional Euler and Navier-Stokes equations linearized about shear flows of the mixing layer type in the unbounded channel $\mathbb{T} \times \mathbb{R}$. Under a simple spectral…
First, we consider Kolmogorov flow (a shear flow with a sinusoidal velocity profile) for 2D Navier-Stokes equation on a torus. Such flows, also called bar states, have been numerically observed as one type of metastable states in the study…
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…
We study the dynamics of the two dimensional Navier-Stokes equations linearized around a shear flow on a (non-square) torus which possesses exactly two non-degenerate critical points. We obtain linear inviscid damping and vorticity…
In this paper, we investigate the asymptotic stability threshold problem for the 2-D Navier-Stokes equations in a finite channel with no-slip boundary conditions, around monotone shear flow $(U(t,y),0)$. We establish that the flow is…
This paper extends the mathematical theory of axisymmetrization and vorticity depletion within the two-dimensional (2D) Euler equations, with an emphasis on the dynamics of radially symmetric, monotonic vorticity profiles. By analyzing…
We investigate the stability of the 2-D Navier-Stokes equations in the infinite channel $\mathbb{R}\times [-1,1]$ with the Navier-slip boundary condition. We show that if the initial perturbations $\omega^{in}$ around the Couette flow…
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.…
In this expository note we discuss our recent work [arXiv:1306.5028] on the nonlinear asymptotic stability of shear flows in the 2D Euler equations of ideal, incompressible flow. In that work it is proved that perturbations to the Couette…
We prove the asymptotic stability of shear flows close to the Couette flow for the 2-D inhomogeneous incompressible Euler equations on $\mathbb{T}\times \mathbb{R}$. More precisely, if the initial velocity is close to the Couette flow and…
We consider the long-time behavior of solutions to the two dimensional non-homogeneous Euler equations under the Boussinesq approximation posed on a periodic channel. We study the linearized system near a linearly stratified Couette flow…
In this paper, we study the generation of eigenvalues of a stable monotonic shear flow under perturbations in $C^s$ with $s<2$. More precisely, we study the Rayleigh operator $\mathcal{L}_{U_{m,\gamma}}=…
In this article we prove a linear inviscid damping result with optimal decay rates of the 2D irrotational circulation flow around an elliptical cylinder. In our result, all components of the asymptotic velocity field do not vanish and the…
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 $\nu \to 0$ limit. We split the vorticity…
We consider the problem of asymptotic stability and linear inviscid damping for perturbations of a point vortex and similar degenerate circular flows. Here, key challenges include the lack of strict monotonicity and the necessity of working…
We prove asymptotic stability of shear flows close to the planar Couette flow in the 2D inviscid Euler equations on $\Torus \times \Real$. That is, given an initial perturbation of the Couette flow small in a suitable regularity class,…
This note is devoted to the linear stability of the Couette flow for the non-isentropic compressible Euler equations in a domain $\mathbb{T}\times \mathbb{R}$. Exploiting the several conservation laws originated from the special structure…