Pablo Braz e Silva
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.
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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 propo…
Linear enhanced dissipation for the 2D Taylor-Couette flow in the exterior region: A supplementary example for Gearhart-Prüss type lemma
本文研究外部区域二维Taylor-Couette流的线性化系统，证明其解呈时空耦合多项式衰减而非指数衰减，说明Gearhart-Prüss型引理不适用于退化剪切流，并给出相关推广。
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}.
Nonlinear asymptotic stability of 2D Taylor-Couette flow in the exterior disk
本文通过引入时间相关预解估计，证明了外部圆盘中二维泰勒-库埃特流在时空耦合多项式衰减下的非线性渐近稳定性，并得到与有界区域相同的1/2过渡阈值。
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…
Stability threshold of the two-dimensional Couette flow in the whole plane
本文研究全平面二维Couette流的稳定性阈值，得出带低水平频率控制的Sobolev扰动的渐近稳定性阈值上限，还证明了弱Sobolev正则性扰动的渐近稳定性。
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…
Global existence and low mach number limit of strong solutions to the full compressible Navier-Stokes equations around the plane Couette flow
本文证明二维完全可压缩 Navier-Stokes 方程在平面 Couette 流附近小扰动强解的全局存在性与渐近稳定性，并在壁面等温时建立其向不可压缩稳态解收敛的低马赫数极限。
Nonlinear stability of 3-D plane Couette flow for Boussinesq system in a finite channel
本文证明了在有限通道内，满足特定初始条件的Boussinesq系统三维平面Couette流具有非线性稳定性，这是该领域首个相关结果，并表明强边界层效应不会导致强不稳定性。
Long-time behaviour of two-dimensional Navier-Stokes equations in the presence of Couette flow on the half plane
本文研究了在半平面存在科氏流条件下二维纳维-斯托克斯方程解的长期行为，通过证明总涡量趋近于一个包含科氏流涡量和福克-计划类型算子核的表达式，揭示了涡量的渐进行为。
Linear decay of the beta-plane equation near Couette flow on the plane
本文证明了平面上 Couette 流附近线性化 β-平面方程结合无粘阻尼与 Rossby 波色散的时间衰减估计，通过 Van der Corput 型估计与边界层及多尺度渐近分析处理了低流向频率下的奇异极限。
Global stability of Oldroyd-B fluids in plane Couette flow
本文通过引入扰动熵这一新泛函，证明了Oldroyd-B粘弹性流体在平面库埃特流中的全局非线性稳定性条件，并将牛顿流体的经典稳定性结果自然推广到粘弹性情形。
Quantitative stability for the 2D Couette flow on the infinite channel with non-slip boundary condition
针对无限通道二维 Couette 流，论文借助精确预解式与 Airy 函数估计划分低、中频并建立时空估计，得到非线性转捩阈值和增强耗散条件。
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…
Stability of 3D Stratified Plane Couette Flow: A New Lift-up Mechanism and Potential Vorticity
本文研究垂直分层背景下三维平面Couette流的稳定性，发现非平行几何产生新的升力机制，通过位势涡度重构建立无粘阻尼，并证明强分层可抑制瞬态增长，实现非线性稳定性。
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…
Enhanced Dissipation, Taylor Dispersion, and Inviscid Damping of Couette flow in the Boussinesq system on the Plane
本文在平面全无界区域上证明了具有大 Richardson 数的稳定分层 Couette 流在二维全耗散 Boussinesq 系统中的定量渐近稳定性，首次给出显式增强耗散、Taylor 色散和无粘阻尼估计。
Inviscid damping without derivatives near Couette flow
该研究针对无界几何中二维欧拉方程的Couette流，在Yudovich正则性下识别出非线性无粘阻尼机制，对不同几何的扰动动力学进行分类，通过Lyapunov泛函与哈密顿守恒的相互作用完成分类。
Quantitative Hydrodynamic Stability for Couette Flow on Unbounded Domains with Navier Boundary Conditions
本文在三个无界区域上证明了二维 Navier-Stokes 方程 Couette 流的稳定性阈值定理，通过新的低频与中频非线性计算，首次给出无界域上完全非线性剪切流的增强耗散结果，并将结论推广到扰动 β-平面方程。
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.
Nonlinear asymptotic stability and transition threshold for 2D Taylor-Couette flows in Sobolev spaces
本文证明二维Taylor-Couette流在Sobolev空间中具有非线性渐近稳定性，给出最优指数衰减率和过渡阈值，并构造新正交基揭示增强耗散与无粘阻尼机制。
Monotone energy stability of magnetohydrodynamics Couette and Hartmann flows
本文证明磁流体动力学 Couette 流和 Hartmann 流在能量范数下最危险的扰动是二维展向扰动，给出临界 Reynolds 数，解决了已有猜想并建立了非线性能量 Squire 定理。
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…
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