In this paper, we consider the stability threshold for the shear flows of the Boussinesq system in a domain $\mathbb{T} \times \mathbb{R}$. The main goal is to prove the nonlinear stability of the shear flow $(U^S,Θ^S)=((e^{νt\partial_{yy}}U(y),0)^{\top},αy)$ with $U(y)$ close to $y$ and $α\geq0$. We separate two cases: one is $α\geq 0$ small scaling with the viscosity coefficients and the case without smallness of $α$ and fixed heat diffusion coefficient. The novelty here is that we don't require $μ=ν$ and only need to assume that $μ$ is scaled with $ν$ or fixed, where $μ$ is the inverse of the Reynolds number and $ν$ is the heat diffusion coefficient.
This paper establishes the nonlinear stability of the Couette flow for the 2D Boussinesq equations with only vertical dissipation. The Boussinesq equations concerned here model buoyancy-driven fluids such as atmospheric and oceanographic flows. Due to the presence of the buoyancy forcing, the energy of the standard Boussinesq equations could grow in time. It is the enhanced dissipation created by the linear non-self-adjoint operator $y\partial_x -ν\partial_{yy}$ in the perturbation equation that makes the nonlinear stability possible. When the initial perturbation from the Couette flow $(y, 0)…
In this paper, we study the nonlinear asymptotic stability of the Couette flow in the stably stratified regime, namely the Richardson number $γ^2>\frac{1}{4}$. Precisely, we prove that if the initial perturbation $(u_{in},\vartheta_{in})$ of the Couette flow $v_s=(y,0)$ and the linear temperature $ρ_s=-γ^2y+1$ satisfies $\|u_{in}\|_{H^{s+1}}+\|\vartheta_{in}\|_{H^{s+2}}\leq ε_0ν^{\frac{1}{2}}$, then the asymptotic stability holds.
In this paper, we study the asymptotic stability for the two-dimensional Navier-Stokes Boussinesq system around the Couette flow with small viscosity $ν$ and small thermal diffusion $μ$ in a finite channel. In particular, we prove that if the initial velocity and initial temperature $(v_{in},ρ_{in})$ satisfies $\|v_{in}-(y,0)\|_{H_{x,y}^2}\leq \e_0 \min\{ν,μ\}^{\f12}$ and $\|ρ_{in}-1\|_{H_x^{1}L_y^2}\leq \e_1 \min\{ν,μ\}^{\f{11}{12}}$ for some small $\e_0,\e_1$ independent of $ν, μ$, then for the solution of the two-dimensional Navier-Stokes Boussinesq system, the velocity remains within $O(\m…
In this paper, we consider the Boussinesq equations with magnetohydrodynamics convection in the domain $\mathbb{T} \times \mathbb{R}$ and establishes the nonlinear stability of the Couette flow $(\mathbf{u}_{sh} = (y,0), \mathbf{b}_{sh} = (1,0), p_{sh} = 0, θ_{sh} = 0$). The novelty in this paper is that we design a new Fourier multiplier operator by using the properties of the enhanced dissipation to overcome the difficult term $\partial_{xy}(-Δ)^{-1}j$ in the linearized and nonlinear system. Then, we prove the asymptotic stability for the linearized system. Finally, we establish the nonlinea…
In this paper, we prove the stability of Couette flow for 2D Navier-Stokes Boussinesq system without thermal diffusivity for the initial perturbation in Gevrey-$\frac{1}{s}$, ($1/3<s\leq 1$). The synergism of density mixing, vorticity mixing and velocity diffusion leads to the stability.
Asymptotic stability of the 2D Boussinesq equations around Couette flow in infinite channel
Institute of Applied Physics and Computational Mathematics（北京应用物理与计算数学研究所）；Graduate School of China Academy of Engineering Physics（中国工程物理研究院研究生院）；Hefei University of Technology（合肥工业大学）；University of Science and Technology Beijing（北京科技大学）
The stability threshold for Boussinesq equations around stratified Couette flow with unequal viscosity and thermal diffusivity on $\mathbb{T}\times\mathbb{R}^2$
School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）
In this paper, we consider the stability threshold of the 2D shear flow $(U(y),0)^{\top}$ of the Navier-Stokes equation at high Reynolds number $Re$. When the shear flow is near in Sobolev norm to the Couette flow $(y,0)^{\top}$ in some sense, we prove that if the initial data $u_0$ satisfies $\|u_0-(U(y),0)^{\top}\|\leq εRe^{-1/3}$, then the solution of the 2D Navier-Stokes equation approaches to some shear flow which is also close to the Couette flow for $t\gg Re^{1/3}$, as $t\to\infty$.
In this article we consider the stability and damping problem for the 2D Boussinesq equations with partial dissipation near a two parameter family of stationary solutions which includes Couette flow and hydrostatic balance. In the first part we show that for the linearized problem in an infinite periodic channel the evolution is asymptotically stable if any diffusion coefficient is non-zero. In particular, this imposes weaker conditions than for example vertical diffusion. Furthermore, we study the interaction of shear flow, hydrostatic balance and partial dissipation. In a second part we adap…
In this paper, we investigate linear stability properties of the 2D isentropic compressible Euler equations linearized around a shear flow given by a monotone profile, close to the Couette flow, with constant density, in the domain $\mathbb{T}\times \mathbb{R}$. We begin by directly investigating the Couette shear flow, where we characterize the linear growth of the compressible part of the fluid while proving time decay for the incompressible part (inviscid damping with slower rates). Then we extend the analysis to monotone shear flows near Couette, where we are able to give an upper bound, s…
Quantitative stability of the 2D Monotone shear flow for Boussinesq equation in a finite channel
Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）；School of Mathematical Sciences, Key Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing Normal University（北京师范大学数学科学学院，教育部数学与复杂系统重点实验室）
We prove the existence of steady \emph{space quasi-periodic} stream functions, solutions for the Euler equation in vorticity-stream function formulation in the two dimensional channel ${\mathbb R}\times [-1,1]$. These solutions bifurcate from a prescribed shear equilibrium near the Couette flow, whose profile induces finitely many modes of oscillations in the horizontal direction for the linearized problem. Using a Nash-Moser implicit function iterative scheme, near such equilibrium we construct small amplitude, space reversible stream functions slightly deforming the linear solutions and reta…
We investigate the long-time properties of the two-dimensional inviscid Boussinesq equations near a stably stratified Couette flow, for an initial Gevrey perturbation of size $\varepsilon$. Under the classical Miles-Howard stability condition on the Richardson number, we prove that the system experiences a shear-buoyancy instability: the density variation and velocity undergo an $O(t^{-1/2})$ inviscid damping while the vorticity and density gradient grow as $O(t^{1/2})$. The result holds at least until the natural, nonlinear timescale $t \approx \varepsilon^{-2}$. Notice that the density behav…
We investigate the linear stability of shears near the Couette flow for a class of 2D incompressible stably stratified fluids. Our main result consists of nearly optimal decay rates for perturbations of stationary states whose velocities are monotone shear flows $(U(y),0)$ and have an exponential density profile. In the case of the Couette flow $U(y)=y$, we recover the rates predicted by Hartman in 1975, by adopting an explicit point-wise approach in frequency space. As a by-product, this implies optimal decay rates as well as Lyapunov instability in $L^2$ for the vorticity. For the previously…
We study the stability threshold of the 2D Couette flow in Sobolev spaces at high Reynolds number $Re$. We prove that if the initial vorticity $Ω_{in}$ satisfies $\|Ω_{in}-(-1)\|_{H^σ}\leq εRe^{-1/3}$, then the solution of the 2D Navier-Stokes equation approaches to some shear flow which is also close to Couette flow for time $t\gg Re^{1/3}$ by a mixing-enhanced dissipation effect and then converges back to Couette flow when $t\to +\infty$.
We study the inviscid damping of Couette flow with an exponentially stratified density. The optimal decay rates of the velocity field and the density are obtained for general perturbations with minimal regularity. For Boussinesq approximation model, the decay rates we get are consistent with the previous results in the literature. We also study the decay rates for the full Euler equations of stratified fluids, which were not studied before. For both models, the decay rates depend on the Richardson number in a very similar way. Besides, we also study the dispersive decay due to the exponential …
We consider the 2D Navier-Stokes equation on $\mathbb T \times \mathbb R$, with initial datum that is $\varepsilon$-close in $H^N$ to a shear flow $(U(y),0)$, where $\| U(y) - y\|_{H^{N+4}} \ll 1$ and $N>1$. We prove that if $\varepsilon \ll ν^{1/2}$, where $ν$ denotes the inverse Reynolds number, then the solution of the Navier-Stokes equation remains $\varepsilon$-close in $H^1$ to $(e^{t ν\partial_{yy}}U(y),0)$ for all $t>0$. Moreover, the solution converges to a decaying shear flow for times $t \gg ν^{-1/3}$ by a mixing-enhanced dissipation effect, and experiences a transient growth of gra…
On the baroclinic instability of inviscid non-conducting Boussinesq equations with rotation in 3-D
School of Science, Zhejiang University of Science and Technology（浙江科技学院理学院）；School of Mathematical Sciences, Zhejiang University of Technology（浙江工业大学数学科学学院）
In this note, we investigate the stability property of shear flows under the 2D stationary Navier-Stokes equations, and we obtain that the Couette flow $(y,0)$ is stable under the space of $\mathcal{D}^{1,q}(\mathbb{R}^2)$ for any $1<q<\infty$ and unstable in the space of $\mathcal{D}^{1,\infty}(\mathbb{R}^2)$. A key observation is the anisotropic cut-off function. We also consider the Poiseuille flow $(y^2,0)$, which is stable in $\mathcal{D}^{1,q}(\mathbb{R}^2)$ with $\frac43<q\leq4.$
We consider the 2D incompressible Navier-Stokes equations on $\mathbb{T}\times \mathbf{R}$, with initial vorticity that is $δ$ close in $H^{log}_xL^2_{y}$ to $-1$(the vorticity of the Couette flow $(y,0)$). We prove that if $δ\ll ν^{1/2}$, where $ν$ denotes the viscosity, then the solution of the Navier-Stokes equation approaches some shear flow which is also close to Couette flow for time $t\gg ν^{-1/3}$ by a mixing-enhanced dissipation effect and then converges back to Couette flow when $t\to +\infty$. In particular, we show the nonlinear enhanced dissipation and the inviscid damping results…
We are concerned with the linear stability of the Couette flow for the non-isentropic compressible Navier-Stokes equations with vanished shear viscosity in a domain $\mathbb{T}\times \mathbb{R}$. For a general initial data settled in Sobolev spaces, we obtain a Lyapunov type instability of the density, the temperature, the compressible part of the velocity field, and also obtain an inviscid damping for the incompressible part of the velocity field. Moreover, if the initial density, the initial temperature and the incompressible part of the initial velocity field satisfy some quality relation, …
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 the initial density is close to a positive constant in the Gevrey class 2, then 2-D inhomogeneous incompressible Euler equations are globally well-posed and the velocity converges strongly to a shear flow close to the Couette flow, and the vorticity will be driven to small scales by a linear evolution and weakly converges as $t\to \infty$. To our knowledge, t…
In this paper we reveal the existence of a large family of new, nontrivial and smooth traveling waves for the 2D Euler equation at an arbitrarily small distance from the Couette flow in $H^s$, with $s<3/2$, at the level of the vorticity. The speed of these waves is of order 1 with respect to this distance. This result strongly contrasts with the setting of very high regularity in Gevrey spaces (see arXiv:1306.5028), where the problem exhibits an inviscid damping mechanism that leads to relaxation of perturbations back to nearby shear flows. It also complements the fact that there not exist non…
We prove asymptotic stability of shear flows in a neighborhood of the Couette flow for the 2D Euler equations in the domain $\T\times[0,1]$. More precisely we prove that if we start with a small and smooth perturbation (in a suitable Gevrey space) of the Couette flow, then the velocity field converges strongly to a nearby shear flow. The vorticity, which is initially assumed to be supported in the interior of the channel, will remain supported in the interior of the channel, will be driven to higher frequencies by the linear flow, and will converge weakly to $0$ as $t\to\infty$, modulo the she…
We study the dynamic behaviour of two viscous fluid films confined between two concentric cylinders rotating at a small relative velocity. It is assumed that the fluids are immiscible and that the volume of the outer fluid film is large compared to the volume of the inner one. Moreover, while the outer fluid is considered to have constant viscosity, the rheological behaviour of the inner thin film is determined by a strain-dependent power-law. Starting from a Navier--Stokes system, we formally derive evolution equations for the interface separating the two fluids. Two competing effects drive t…
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 neighbor…
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 the 2D Euler Couette flow in $\mathbb{T}\times \mathbb{R}$ strongly converge to zero, under the additional assumption that the average in $x$ is always zero. In general the mean is not a conserved quantity for the nonlinear dynamics, for this reason this is a particular case. Nevertheless our assumption allow the presence of echoes in the problem, which we c…
Sommerfeld paradox (turbulence paradox) roughly says that mathematically the Couette linear shear flow is linearly stable for all values of the Reynolds number, but experimentally transition from the linear shear to turbulence occurs under perturbations of any size when the Reynolds number is large enough. In [Li, Lin 2011], we offered a resolution of this paradox. The aim of this paper is to provide a numerical implementation of the resolution. The main idea of the resolution is that even though the linear shear is linearly stable, slow orbits (also called quasi-steady states) in arbitrarily …
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 flow which are small in a suitable regularity class converge strongly in $L^2$ to a shear flow which is close to the Couette flow. Enstrophy is mixed to small scales by an almost linear evolution and is generally lost in the weak limit as t -> +/- infinity. In this note we discuss the most important physical and mathematical aspects of the result and the key …
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, specifically Gevrey space of class smaller than 2, the velocity converges strongly in L^2 to a shear flow which is also close to the Couette flow. The vorticity is asymptotically driven to small scales by a linear evolution and weakly converges as $t \rightarrow \pm\infty$. The strong convergence of the velocity field is sometimes referred to as inviscid damping…
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…
Inverse Energy Cascade in Turbulent Taylor-Couette Flows
Faculty of Mechanical Engineering, Zhejiang Sci-Tech University（浙江理工大学机械工程学院）；School of Mechanical Engineering, Zhejiang University of Water Resources and Electric Power（浙江水利水电学院机械工程学院）；School of Mechanical Engineering, Hangzhou Dianzi University（杭州电子科技大学机械工程学院）
We consider the (in)stability problem of the inviscid 2D Boussinesq equations near a combination of a shear flow $v=(y,0)$ and a stratified temperature $θ=αy$ with $α>\frac{1}{4}$. We show that for any $ε>0$ there exist non-trivial explicit solutions, which are initially perturbations of size $ε$, and grow to size $1$ on a time scale $ε^{-2}$. Moreover, the (simplified) linearized problem around these non-trivial states exhibits improved upper bounds on the possible size of norm inflation for frequencies larger and smaller than $ε^{-4}$.
