Stability of the Couette flow at high Reynolds numbers in two dimensions and three dimensions

Abstract

We review works on the asymptotic stability of the Couette flow. The majority of the paper is aimed towards a wide range of applied mathematicians with an additional section aimed towards experts in the mathematical analysis of PDEs. Contents Hydrodynamic stability at high Reynolds number1 2. Linear dynamics in dimension d = 2 8 3. Nonlinear dynamics in dimension d = 2 11 4. Linear dynamics in dimension d = 3 1 As remarked above, in order to get non-trivial equilibria, we usually need to impose boundary conditions or an external force field. However, let us ignore this for the moment, as it is not directly relevant to our discussions. STABILITY OF THE COUETTE FLOW AT HIGH REYNOLDS NUMBERS IN 2D AND 3D3 simply by dropping the quadratic nonlinearity in (1.1):( 1.3)The simplest notion of linear stability is spectral stability:Many classical theories of hydrodynamic stability are focused on studies of spectral stability (see [122] and the references therein). In the context of shear flows at high Reynolds number, the most famous results are Rayleigh's inflection point theorem [96,40] and its refinement due to Fjørtoft [48], regarding the stability of shear flows in the 2D Euler equations (ν = 0). These results (unfortunately) extend to planar 3D shear flows via Squire's theorem [107,39,40]. Squire's theorem for inviscid flows states: If the 3D Euler equations linearized around the shear flow u E = (f (y), 0, 0) have unstable eigenvalues, then so do the 2D Euler equations linearized around u E = (f (y), 0) (the converse implication being obvious). Therefore, for any shear flow of this type, spectral stability in 2D implies spectral stability in 3D. This is 'unfortunate' because it gives the false impression that most of the interesting aspects of hydrodynamic stability can be found in the 2D equations. As we will see, hydrodynamic stability in 2D and 3D, at high Reynolds numbers, are quite distinct.Spectral stability is not the only relevant definition of linear stability. In particular, we have the following distinct definition, which was suggested as more natural in fluid mechanics by Kelvin [68], Orr [92], and later Case [29]and Dikii [38].Definition 1.2 (Lyapunov linear stability). Given two norms X and Y (often assumed the same), the equilibrium u E is called linearly stable (from X to Y ) if

On the Stability of Laminar Flows Between Plates

“…Consider the case of Couette flow U = x in (−1, 1), which is one of the most popular examples of uniaxial flows ( [35,30,7]. In this case…”

On well-posedness of the Cauchy problem for 3D MHD system in critical Sobolev–Gevrey space

“…where m [ 0 is the viscosity, g [ 0 denotes the magnetic diffusivity, u(t, x) and b(t, x) are two unknown three-dimensional vectors representing respectively the velocity and the magnetic field, p(t, x) is an unknown scalar which stands for pressure, and the system is subject to periodic boundary conditions with basic domain T 3 ¼ ½0; 2pL 3 .…”

“…5, Theorem 5.3). This problematic was also addressed in [3]. To beat the odds, we estimate the cubic terms such as hðf Á rgÞ; Khi L 2 a;r ðT 3 Þ ;…”

Pseudospectral Bound and Transition Threshold for the 3D Kolmogorov Flow

“…In the viscous case, the problem is more difficult, since it involves solving a fourth-order ODE (Orr-Sommerfeld equation). We refer to the survey paper [5] for more recent results and open questions. To our knowledge, there are few results in 3D except for the Couette flow.…”

“…To our knowledge, there are few results in 3D except for the Couette flow. We refer to the survey paper [5] for more recent results and open questions.…”