Stability of the Couette Flow for a 2D Boussinesq System Without Thermal Diffusivity

Masmoudi, Nader; Said‐Houari, Belkacem; Wei, Zhao

doi:10.1007/s00205-022-01789-x

Cited by 19 publications

(18 citation statements)

References 72 publications

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“…Thus the infinite-in-time growth estimate (3.23) directly implies that any such steady state (with zero density) is nonlinearly unstable, in the sense that for any 0 < k 0 1, an arbitrarily small perturbation ρ 0 = k 0 cos(x 1 ), ω 0 = ω s leads to lim t→∞ ω(t) L 1 = ∞. See [3,5,14,17,41,45,48] for more results on stability/instability of steady states of the inviscid or viscous Boussinesq equations.…”

Section: 2mentioning

confidence: 96%

Small scale formation for the 2D Boussinesq equation

Kiselev¹,

Park²,

Yao³

2022

Preprint

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We study the 2D incompressible Boussinesq equation without thermal diffusion, and aim to construct rigorous examples of small scale formations as time goes to infinity. In the viscous case, we construct examples of global smooth solutions satisfying sup τ ∈[0,t] ∇ρ(τ ) L 2 t α for some α > 0. For the inviscid equation in the strip, we construct examples satisfying ω(t) L ∞ t 3 and sup τ ∈[0,t] ∇ρ(τ ) L ∞ t 2 during the existence of a smooth solution. These growth results hold for a broad class of initial data, where we only require certain symmetry and sign conditions. As an application, we also construct solutions to the 3D axisymmetric Euler equation whose velocity has infinite-in-time growth.

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Section: 2mentioning

confidence: 96%

Small scale formation for the 2D Boussinesq equation

Kiselev¹,

Park²,

Yao³

2022

Preprint

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“…The above result describes the long-time dynamics of the Boussinesq system (1.2) in the perturbative regime near the linearly stratified Couette flow, and it is the first of its kind describing such behavior in a fully inviscid coupled system which has both wave propagation and phase mixing. The works [28,47,64] study nonlinear systems with both phase mixing and wave propagation, but these problems all contain dissipative effects, whereas the works [11,61] are all linear. The inviscid damping due to vorticity mixing is encoded in (1.7)-(1.8).…”

Section: 1mentioning

confidence: 99%

“…The toy model used to build the norm in [47] for the Boussinesq equations with viscosity (but not thermal diffusivity) near Couette flow is more significantly different. However, the derivation and use of the model depend crucially on the presence of viscosity.…”

Section: The Nonlinear Growth Mechanismmentioning

confidence: 99%

“…The system () under investigation models a stably stratified fluid with the additional Boussinesq assumption , according to which density is assumed constant except when it directly causes buoyancy forces [24, 43]. The Boussinesq system gained the interest of the mathematical community thanks to its wide range of applications, especially in oceanography [21, 44], and many mathematical works have been dedicated to it [1, 15–17, 19, 20, 27, 29, 47, 57]. It also holds mathematical interest through a connection with the 3D axisymmetric Euler equations for homogeneous fluids [45], where the term multiplied by β 2 in () plays the role of the vortex stretching.…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear inviscid damping and shear‐buoyancy instability in the two‐dimensional Boussinesq equations

Bedrossian

Bianchini²,

Zelati

et al. 2023

Comm Pure Appl Math

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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 ε. 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 inviscid damping while the vorticity and density gradient grow as . The result holds at least until the natural, nonlinear timescale . Notice that the density behaves very differently from a passive scalar, as can be seen from the inviscid damping and slower gradient growth. The proof relies on several ingredients: (A) a suitable symmetrization that makes the linear terms amenable to energy methods and takes into account the classical Miles‐Howard spectral stability condition; (B) a variation of the Fourier time‐dependent energy method introduced for the inviscid, homogeneous Couette flow problem developed on a toy model adapted to the Boussinesq equations, that is, tracking the potential nonlinear echo chains in the symmetrized variables despite the vorticity growth.

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“…See also [8,45,32] for similar depletion phenomena in various systems. Such damping caused by mixing is a common phenomenon in fluid dynamics, see [45,54,53,70] for the plasma fluid, see [6,66,46] for the stratified fluid, see [3,4,69] for the compressible fluid, and see [59,65] for the geophysical fluid.…”

Section: Enhanced Dissipationmentioning

confidence: 99%

Asymptotic stability of two-dimensional Couette flow in a viscous fluid

Masmoudi

Wei

2022

Preprint

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In this paper, we study the nonlinear asymptotic stability of Couette flow for the two-dimensional Navier-Stokes equation with small viscosity v > 0 in T×R. It's generally known that the nonlinear asymptotic stability of the Couette flow depends closely on the size and regularity of the initial perturbation, which yields the stability threshold problem. This work studies the relationship between the size and the regularity of the initial perturbation that makes the nonlinear asymptotic stability holds. More precisely, we proved that if the initial perturbation is in some Gevrey-1/s class with size εvβ where s ≥ 1-3β/2-3β and β∈[0,1/3], then the nonlinear asymptotic stability holds. We think this index is sharp.

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Stability of the Couette Flow for a 2D Boussinesq System Without Thermal Diffusivity

Cited by 19 publications

References 72 publications

Small scale formation for the 2D Boussinesq equation

Small scale formation for the 2D Boussinesq equation

Nonlinear inviscid damping and shear‐buoyancy instability in the two‐dimensional Boussinesq equations

Asymptotic stability of two-dimensional Couette flow in a viscous fluid

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