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 $(y, 0)$ is no more than the viscosity to a suitable power (in the Sobolev space $H^b$ with $b>\frac43$), we prove that the solution of the 2D Boussnesq system with only vertical dissipation on $\mathbb T\times \mathbb R$ remains close to the Couette at the same order. A special consequence of this result is the stability of the Couette for the 2D Navier-Stokes equations with only vertical dissipation.
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）
This paper establishes the global in time existence of classical solutions to the 2D anisotropic Boussinesq equations with vertical dissipation. When only the vertical dissipation is present, there is no direct control on the horizontal derivatives and the global regularity problem is very challenging. To solve this problem, we bound the derivatives in terms of the $L^\infty$-norm of the vertical velocity $v$ and prove that $\|v\|_{L^{r}}$ with $2\le r<\infty$ at any time does not grow faster than $\sqrt{r \log r}$ as $r$ increases. A delicate interpolation inequality connecting $\|v\|_{L^\inf…
In this article we consider the asymptotic stability of the two-dimensional Boussinesq equations with partial dissipation near a combination of Couette flow and temperature profiles $T(y)$. As a first main result we show that if $T'$ is of size at most $ν^{1/3}$ in a suitable norm, then the linearized Boussinesq equations with only vertical dissipation of the velocity but not of the temperature are stable. Thus, mixing enhanced dissipation can suppress Rayleigh-Bénard instability in this linearized case. We further show that these results extend to the (forced) nonlinear equations with vertica…
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…
We prove the global well-posedness of the two-dimensional Boussinesq equations with only vertical dissipation. The initial data $(u_0,θ_0)$ are required to be only in the space $X=\{f\in L^2(\mathbb R^2)\,|\,\partial_xf\in L^2(\mathbb R^2)\}$, and thus our result generalizes that in [C. Cao, J. Wu, Global regularity for the two-dimensional anisotropic Boussinesq equations with vertical dissipation, Arch. Rational Mech. Anal., Vol. 208 (2013), 985-1004], where the initial data are assumed to be in $H^2(\mathbb R^2)$. The assumption on the initial data is at the minimal level that is required to…
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…
Using a perturbation approach, we make rigorous the formal boundary layer asymptotic analysis of Turcotte, Spence and Bau from the early eighties for the vertical flow of an internally heated Boussinesq fluid in a vertical channel with viscous dissipation and pressure work. A key point in our proof is to establish the non-degeneracy of a special solution of the Painlevé-I transcendent. To this end, we relate this problem to recent studies for the ground states of the focusing nonlinear Schrödinger equation in an annulus. We also relate our result to a particular case of the well known Lazer-Mc…
In this paper, we investigate the Cauchy problem for the tridimensional Boussinesq equations with horizontal dissipation. Under the assumption that the initial data is an axisymmetric without swirl, we prove the global well-posedness for this system. In the absence of vertical dissipation, there is no smoothing effect on the vertical derivatives. To make up this shortcoming, we first establish a magic relationship between $\frac{u^{r}}{r}$ and $\frac{ω_θ}{r}$ by taking full advantage of the structure of the axisymmetric fluid without swirl and some tricks in harmonic analysis. This together wi…
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 …
Linear Growth and Nonlinear Stability of Two-Dimensional MHD Couette Flow with Vertical Dissipation
School of Mathematics, South China University of Technology（华南理工大学数学学院）；Department of Mathematics, University of Notre Dame（圣母大学数学系）；School of Mathematics and Statistics, Guangdong University of Technology（广东工业大学数学与统计学院）
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 investigate here the interactions of waves governed by a Boussinesq system with a partially immersed body allowed to move freely in the vertical direction. We show that the whole system of equations can be reduced to a transmission problem for the Boussinesq equations with transmission conditions given in terms of the vertical displacement of the object and of the average horizontal discharge beneath it; these two quantities are in turn determined by two nonlinear ODEs with forcing terms coming from the exterior wave-field. Understanding the dispersive contribution to the added mass phenome…
In this paper, for the 3-D Navier-Stokes-Boussinesq system with horizontal dissipation, where there is no smoothing effect on the vertical derivatives, we prove a uniqueness result of solutions $ (u,ρ)\in L^{\infty}_T\big( H^{0,s}\times H^{0,1-s}\big)$ with $ (\nabla_h u,\nabla_hρ)\in L^{2}_T\big( H^{0,s}\times H^{0,1-s}\big)$ and $s\in [1/2,1]$. As a consequence, we improve the conditions stated in the paper \cite{Miao} in order to obtain a global well-posedness result in the case of axisymmetric initial data.
We consider an initial boundary value problem of the multi-dimensional Boussinesq equations in the absence of thermal diffusion with velocity damping or velocity diffusion under the stress free boundary condition in horizontally periodic strip domain. We prove the global-in-time existence of classical solutions in high order Sobolev spaces satisfying high order compatibility conditions around the linearly stratified equilibrium, the convergence of the temperature to the asymptotic profile, and sharp decay rates of the velocity field and temperature fluctuation in all intermediate norms based o…
We study the global existence issue for the two-dimensional Boussinesq system with horizontal viscosity in only one equation. We first examine the case where the Navier-Stokes equation with no vertical viscosity is coupled with a transport equation. Second, we consider a coupling between the classical two-dimensional incompressible Euler equation and a transportdiffusion equation with diffusion in the horizontal direction only. For the both systems and for arbitrarily large data, we construct global weak solutions `a la Leray. Next, we state global wellposedness results for more regular data. …
The Boussinesq system for buoyancy driven fluids couples the momentum equation forced by the buoyancy with the convection-diffusion equation for the temperature. One fundamental issue on the Boussinesq system is the stability problem on perturbations near the hydrostatic balance. This problem can be extremely difficult when the system lacks full dissipation. This paper solves the stability problem for a two-dimensional Boussinesq system with only vertical dissipation and horizontal thermal diffusion. We establish the stability for the nonlinear system and derive precise large-time behavior for…
We consider the 2D Boussinesq equations with viscous but without thermal dissipation and observe that in any neighborhood of Couette flow and hydrostatic balance (with respect to local norms) there are time-dependent traveling wave solutions of the form $ω=-1+f(t)\cos(x-ty)$, $θ=αy + g(t)\sin(x-ty)$. As our main result we show that the linearized equations around these waves for $α=0$ exhibit echo chains and norm inflation despite viscous dissipation of the velocity. Furthermore, we construct initial data in a critical Gevrey 3 class, for which temperature and vorticity diverge to infinity in …
Freely floating cylinder on a 3D fluid governed by the Boussinesq equations in the axisymmetric without swirl case
Univ Rennes, IRMAR UMR 6625（雷恩大学，IRMAR UMR 6625）；Centre Inria de l’Université de Rennes (MINGuS)（雷恩大学Inria中心（MINGuS））；ENS Rennes（雷恩高等师范学院）；ENS de Physique de Lyon（里昂物理高等学院）
On the Profile of Singularity Formation for the Incompressible Hydrostatic Boussinesq system
University of Victoria（维多利亚大学）；Pacific Institute for Mathematical Sciences（太平洋数学科学研究所）；Clemson University（克莱姆森大学）；University of Cambridge（剑桥大学）；Texas A&M University（德克萨斯农工大学）；Weizmann Institute of Science（魏茨曼科学研究所）
The hydrostatic equilibrium is a prominent topic in fluid dynamics and astrophysics. Understanding the stability of perturbations near the hydrostatic equilibrium of the Boussinesq systems helps gain insight into certain weather phenomena. The 2D Boussinesq system focused here is anisotropic and involves only horizontal dissipation and horizontal thermal diffusion. Due to the lack of the vertical dissipation, the stability and precise large-time behavior problem is difficult. When the spatial domain is $\mathbb R^2$, the stability problem in a Sobolev setting remains open. When the spatial dom…
We study the Boussinesq approximation for rapidly rotating stably-stratified fluids in a three dimensional infinite layer with either stress-free or periodic boundary conditions in the vertical direction. For initial conditions satisfying a certain quasi-geostrophic smallness condition, we use dispersive estimates and the large rotation limit to prove global-in-time existence of solutions. We then use self-similar variable techniques to show that the barotropic vorticity converges to an Oseen vortex, while other components decay to zero. We finally use algebraically weighted spaces to determin…
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…
The three--dimensional incompressible viscous Boussinesq equations, under the assumption of hydrostatic balance, govern the large scale dynamics of atmospheric and oceanic motion, and are commonly called the primitive equations. To overcome the turbulence mixing a partial vertical diffusion is usually added to the temperature advection (or density stratification) equation. In this paper we prove the global regularity of strong solutions to this model in a three-dimensional infinite horizontal channel, subject to periodic boundary conditions in the horizontal directions, and with no-penetration…
The water wave theory traditionally assumes the fluid to be perfect, thus neglecting all effects of the viscosity. However, the explanation of several experimental data sets requires the explicit inclusion of dissipative effects. In order to meet these practical problems, the theory of visco-potential flows has been developed (see P.-F. Liu & A. Orfila (2004) and D. Dutykh & F. Dias (2007)). Then, usually this formulation is further simplified by developing the potential in an entire series in the vertical coordinate and by introducing thus, the long wave approximation. In the present study we…
Considered here is a class of Boussinesq systems of Nwogu type. Such systems describe propagation of nonlinear and dispersive water waves of significant interest such as solitary and tsunami waves. The initial-boundary value problem on a finite interval for this family of systems is studied both theoretically and numerically. First, the linearization of a certain generalized Nwogu system is solved analytically via the unified transform of Fokas. The corresponding analysis reveals two types of admissible boundary conditions, thereby suggesting appropriate boundary conditions for the nonlinear N…
In this article we consider the 2D Navier-Stokes equations with variable viscosity depending on the vertical position. As our main result we establish linear enhanced dissipation near the non-affine stationary states replacing Couette flow. Moreover it turns out that the shear flow overcompensates for weakening viscosity: decreasing viscosity leads to stronger enhanced dissipation and increasing viscosity leads to weaker dissipation than in the constant viscosity case.
We study here the propagation of long waves in the presence of vorticity. In the irrotational framework, the Green-Naghdi equations (also called Serre or fully nonlinear Boussinesq equations) are the standard model for the propagation of such waves. These equations couple the surface elevation to the vertically averaged horizontal velocity and are therefore independent of the vertical variable. In the presence of vorticity, the dependence on the vertical variable cannot be removed from the vorticity equation but it was however shown in [?] that the motion of the waves could be described using …
