We present results concerning resolvent estimates for the linear operator associated with the system of differential equations governing perturbations of the Couette flow. We prove estimates on the L_2 norm of the resolvent of this operator showing this norm to be proportional to the Reynolds number R for a region of the unstable half plane. For the remaining region, we show that the problem can be reduced to estimating the solution of a homogeneous ordinary differential equation with non-homogeneous boundary conditions. Numerical approximations indicate that the norm of the resolvent is proportional to R in the whole region of interest.
We discuss the application of the resolvent technique to prove stability of plane Couette flow. Using this technique, we derive a threshold amplitude for perturbations that can lead to turbulence in terms of the Reynolds number. Our main objective is to show exactly how much control one should have over the perturbation to assure stability via this technique.
We prove nonlinear stability for finite amplitude perturbations of plane Couette flow. A bound of the solution of the resolvent equation in the unstable complex half-plane is used to estimate the solution of the full nonlinear problem.The result is a lower bound, including Reynolds number dependence,of the threshold amplitude below which all perturbations are stable. Our result is an improvement of the corresponding bound derived in \cite{KLH:1}.
In this paper, we study the transition threshold problem for the 2-D Navier-Stokes equations around the Couette flow $(y,0)$ at large Reynolds number $Re$ in a finite channel. We develop a systematic method to establish the resolvent estimates of the linearized operator and space-time estimates of the linearized Navier-Stokes equations. In particular, three kinds of important effects: enhanced dissipation, inviscid damping and boundary layer, are integrated into the space-time estimates in a sharp form. As an application, we prove that if the initial velocity $v_0$ satisfies $\|v_0-(y, 0)\|_{H…
We prove the existence and stability of smooth solutions to the steady Navier-Stokes equations near plane Poiseuille-Couette flow. Consequently, we also provide the zero viscosity limit of the 2D steady Navier-Stokes equations to the steady Euler equations. First, in the absence of any external force, we prove that there exist smooth solutions to the steady Navier-Stokes equations which are stable under infinitesimal perturbations of plane Poiseuille-Couette flow. In particular, if the basic flow is the Couette flow, then we can prove that the flow is stable for any finite perturbation small e…
This paper concerns with the stability of the plane Couette flow resulted from the motions of boundaries that the top boundary $Σ_1$ and the bottom one $Σ_0$ move with constant velocities $(a,0)$ and $(b,0)$, respectively. If one imposes Dirichlet boundary condition on the top boundary and Navier boundary condition on the bottom boundary with Navier coefficient $α$, there always exists a plane Couette flow which is exponentially stable for nonnegative $α$ and any positive viscosity $μ$ and any $a, b \in \mathbb{R}$, or, for $α<0$ but viscosity $μ$ and the moving velocities of boundaries $(a,0)…
In this paper, we study nonlinear stability of the 3D plane Couette flow $(y,0,0)$ at high Reynolds number ${Re}$ in a finite channel $\mathbb{T}\times [-1,1]\times \mathbb{T}$. It is well known that the plane Couette flow is linearly stable for any Reynolds number. However, it could become nonlinearly unstable and transition to turbulence for small but finite perturbations at high Reynolds number. This is so-called Sommerfeld paradox. One resolution of this paradox is to study the transition threshold problem, which is concerned with how much disturbance will lead to the instability of the fl…
Long-time behaviour of two-dimensional Navier-Stokes equations in the presence of Couette flow on the half plane
New York University Abu Dhabi（阿布扎比纽约大学）；NYUAD Research Institute, New York University Abu Dhabi（阿布扎比纽约大学NYUAD研究院）；Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）
In this paper, we study the nonlinear stability of a steady circular flow created between two rotating concentric cylinders. The dynamics of the viscous fluid are described by 2D Navier-Stokes equations. We adopt scaling variables. For the rescaled equations, we prove that the steady flow (Taylor-Couette flow) is asymptotically stable up to a large perturbation of initial data. Back to the original 2D Navier-Stokes equations, this implies an improved transition threshold for the Taylor-Couette flow. The improvement is due to enhanced dissipation and new observations and constructions of weight…
In this paper, we study stationary structures near the planar Couette flow in Sobolev spaces on a channel $\mathbb{T}\times[-1,1]$, and asymptotic behavior of Couette flow in Gevrey spaces on $\mathbb{T}\times\mathbb{R}$ for the $β$-plane equation. Let $T>0$ be the horizontal period of the channel and $α={2π\over T}$ be the wave number. We obtain a sharp region $O$ in the whole $(α,β)$ half-plane such that non-parallel steadily traveling waves do not exist for $(α,β)\in O$ and such traveling waves exist for $(α,β)$ in the remaining regions, near Couette flow for $H^{\geq5}$ velocity perturbati…
In this paper, we study the transition threshold problem for the 2-D Navier-Stokes equations around the Poiseuille flow $(1-y^2,0)$ in a finite channel with Navier-slip boundary condition. Based on the resolvent estimates for the linearized operator around the Poiseuille flow, we first establish the enhanced dissipation estimates for the linearized Navier-Stokes equations with a sharp decay rate $e^{-c\sqrtνt}$. As an application, we prove that if the initial perturbation of vortiticy satisfies $$\|ω_0\|_{L^2}\leq c_0ν^{\frac{3}{4}},$$ for some small constant $c_0>0$ independent of the viscosi…
We investigate and solve a special class of integrals involving associated Legendre functions, which can be regarded as generalized Mehler-Fock transformations. Some of the integrals appear naturally when dealing with the heat or resolvent equation of a wedge in the hyperbolic plane. As an application, we derive explicit formulas for the Green's function and the heat kernel of a wedge.
In the paper, we study the plane Couette flow of a rarefied gas between two parallel infinite plates at $y=\pm L$ moving relative to each other with opposite velocities $(\pm αL,0,0)$ along the $x$-direction. Assuming that the stationary state takes the specific form of $F(y,v_x-αy,v_y,v_z)$ with the $x$-component of the molecular velocity sheared linearly along the $y$-direction, such steady flow is governed by a boundary value problem on a steady nonlinear Boltzmann equation driven by an external shear force under the homogeneous non-moving diffuse reflection boundary condition. In case of t…
We study Sobolev regularity disturbances to the periodic, plane Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number $\textbf{Re}$. Our goal is to estimate how the stability threshold scales in $\textbf{Re}$: the largest the initial perturbation can be while still resulting in a solution that does not transition away from Couette flow. In this work we prove that initial data which satisfies $\| u_{in} \|_{H^σ} \leq δ\textbf{Re}^{-3/2}$ for any $σ> 9/2$ and some $δ= δ(σ) > 0$ depending only on $σ$, is global in time, remains within $O(\textbf{Re}^{-1/2})$ of the…
We identify a class of time-periodic linear symmetric hyperbolic equations that are finite codimension stable, because an associated operator has compact resolvent, sufficiently far to the right in the complex plane. This paper is an attempt to capture abstractly the observation in numerical general relativity that some discretely self-similar spacetimes, such as Choptuik's critical spacetime, are finite codimension stable.
We study the high frequency limit for a non-dissipative Helmholtz equation. We first prove the absence of eigenvalue on the upper half-plane and close to an energy which satisfies a weak damping assumption on trapped trajectories. Then we generalize to this setting the resolvent estimates of Robert-Tamura and prove the limiting absorption principle. We finally study the semiclassical measures of the solution when the source term concentrates on a bounded submanifold of R^n.
This is the second in a pair of works which study small disturbances to the plane, periodic 3D Couette flow in the incompressible Navier-Stokes equations at high Reynolds number $\textbf{Re}$. In this work, we show that there is constant $0 < c_0 \ll 1$, independent of $\textbf{Re}$, such that sufficiently regular disturbances of size $ε\lesssim \textbf{Re}^{-2/3-δ}$ for any $δ> 0$ exist at least until $t = c_0ε^{-1}$ and in general evolve to be $O(c_0)$ due to the lift-up effect. Further, after times $t \gtrsim \textbf{Re}^{1/3}$, the streamwise dependence of the solution is rapidly diminishe…
We study small disturbances to the periodic, plane Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number $\textbf{Re}$. We prove that for sufficiently regular initial data of size $ε\leq c_0\textbf{Re}^{-1}$ for some universal $c_0 > 0$, the solution is global, remains within $O(c_0)$ of the Couette flow in $L^2$, and returns to the Couette flow as $t \rightarrow \infty$. For times $t \gtrsim \textbf{Re}^{1/3}$, the streamwise dependence is damped by a mixing-enhanced dissipation effect and the solution is rapidly attracted to the class of "2.5 dimensional" stre…
