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Linear Hydrodynamic Stability COMMUNICATION Linear Hydrodynamic Stability Y. Charles Li Communicated by Christina Sormani Introduction Hydrodynamic instability and turbulence are ubiquitous in nature, such as the air flow around an airplane of Figure 1. Turbulence was also the subject of paintings by Leonardo da Vinci and Vincent van Gogh (Figure 2). Tur- bulence is the last unsolved problem of classical physics. Mathematically, turbulence is governed by the Navier– Stokes equations, for which even the basic question of well-posedness is not completely solved. Indeed, the global well-posedness of the 3D Navier–Stokes equations is one of the seven Clay millennium prize problems [1]. Hydrodynamic stability goes back to such luminaries as Helmholtz, Kelvin, Rayleigh, O. Reynolds, A. Sommerfeld, Heisenberg, and G. I. Taylor. Many treatises have been written on the subject [5]. Linear hydrodynamic stability theory applies the sim- ple idea of linear approximation of a function through its tangent to function spaces. Hydrodynamicists have been Figure 1. Hydrodynamic instability and turbulence trying to lay down a rigorous mathematical foundation are ubiquitous in nature, such as the turbulent vortex for linear hydrodynamic stability theory ever since its air flow around the tip of an airplane wing. beginning [5]. Recent results on the solution operators of Euler and Navier–Stokes equations [2] [3] [4] imply that the simple idea of tangent line approximation does not apply because the derivative does not exist in the inviscid Linear Approximation in the Simplest Setting case, thus the linear Euler equations fail to provide a linear Let us start with a real-valued function of one variable approximation to inviscid hydrodynamic stability. Even 푦 = 푓(푥). for the linearized Navier–Stokes equations, instabilities If the function is differentiable, then are often dominated by faster than exponential growth, especially when the viscosity is small. Δ푦 = 푓′(푥)Δ푥 + 표(|Δ푥|). Thus in a small neighborhood of Δ푥 = 0, the tangent line Y. Charles Li is professor of mathematics at the University of 푑푦 = 푓′(푥)푑푥 Missouri–Columbia. His email address is [email protected]. For permission to reprint this article, please contact: offers a linear approximation. Sometimes there is a nice [email protected]. change of variables DOI: http://dx.doi.org/10.1090/noti1743 Δ휂 = Δ푦 + 푔(Δ푦), November 2018 Notices of the AMS 1255 COMMUNICATION Mathematical Foundation of Inviscid Linear Hydrodynamic Stability Theory We start with the basic incompressible, inviscid (no viscosity) Euler equations for the velocity 푢(푡, 푥) of a fluid: 휕푡푢 + 푢 ⋅ ∇푢 = −∇푝, ∇ ⋅ 푢 = 0 under various boundary conditions, where the spatial dimension is either 2 or 3. Here 푝 is the pressure. In order to establish linear hydrodynamic stability theory, first of all the (nonlinear) Euler equations must be well posed. The usual well-posedness requires that there is a unique solution which depends on the initial condition contin- uously. Linear hydrodynamic stability theory requires a Figure 2. Vincent van Gogh’s painting “Starry Night” stronger well-posedness: the solution needs to depend exhibits a turbulent fluid pattern. on the initial condition differentiably. The well-posedness of the Euler equations depends critically on the function space in which the Euler equations are posed. First of all, where 푔(0) = 푔′(0) = 0 such that the function space consists of functions that satisfy the boundary condition. Second, a norm is defined on the ′ Δ휂 = 푓 (푥)Δ푥 function space so that every function with a finite norm belongs to the function space. The most natural function in a small neighborhood of Δ푥 = 0. This is called lin- spaces for the well-posedness of Euler equations are the earization. The same idea can be applied to differential 푛 Sobolev spaces 퐻 . A fluid velocity function 푢(푡, 푥1, 푥2) equations. Let us consider a simple first order differential belonging to 퐻푛 must have a finite Sobolev norm which equation, is the square root of the integral over the spatial domain 푑푥 푡ℎ = 푓(푥). of the sum of squares of 푛 order partial derivatives of 푑푡 velocity components 푢푗: 푡 Denote by 푆 (푥0) the solution operator (0.1) 휕 ℓ 휕 푚 2 푡 2 ∫ 푥(푡) = 푆 (푥0), ‖푢‖푛 = ∑ [( ) ( ) 푢푗] 푑푥1푑푥2, ℓ+푚≤푛, 푗=1,2 휕푥1 휕푥2 where 푥(0) = 푥0. The solution operator can be viewed 푡 and similarly for spatial dimension 3. Under the bound- as a family of maps parametrized by 푡. If 푆 (푥0) is ary conditions of either decaying at infinity or spatial differentiable in 푥0, then periodicity, the Euler equations are locally well posed (for a short time) in the Sobolev spaces 퐻푛 when 푛 > 1 + 푑/2, 푑 푡 Δ푥(푡) = 푆 (푥0)Δ푥0 + 표(|Δ푥0|). where 푑 is the spatial dimension. In two spatial dimen- 푑푥0 sions, the local well-posedness can be extended to global In a small neighborhood of Δ푥0 = 0, the tangent line (all time) well-posedness. There are also well-posedness 푑 results for other boundary conditions, and boundary 푑푥(푡) = 푆푡(푥 )푑푥 conditions usually do not pose any substantial issue for 푑푥 0 0 0 well-posedness. offers a linear approximation, and 푑푥(푡) satisfies the Once we know well-posedness, we can define a solution linear equation map 푆푡 (a flow, see Figure 3) in the function space: 푑 푆푡(푢(0)) = 푢(푡), for any 푢(0) ∈ 퐻푛, (푑푥) = 푓′(푥)푑푥, 푑푡 where 푢(푡) is the solution starting at the initial condition whereas Δ푥 satisfies 푢(0). The well-posedness of the Euler equations in 퐻푛 푡 푑 implies that 푆 (푢(0)) is continuous in 푢(0). To study the (Δ푥) = 푓′(푥)Δ푥 + 표(|Δ푥|). 푑푡 stability of any solution 푢(푡), one needs to introduce a perturbation in its initial condition Sometimes there is a nice change of variables 푢(0) + Δ푢(0), Δ휂 = Δ푥 + 푔(Δ푥) which generates a new solution to the Euler equations where 푔(0) = 푔′(0) = 0 such that ̂푢(푡)= 푆푡(푢(0) + Δ푢(0)). 푑 The stability of the solution 푢(푡) is studied via investigat- (Δ휂) = 푓′(푥)Δ휂 푑푡 ing the growth or decay of the difference in a small neighborhood of Δ휂 = 0. This is called Δ푢(푡) = ̂푢(푡)− 푢(푡) linearization of the differential equation. = 푆푡(푢(0) + Δ푢(0)) − 푆푡(푢(0)). 1256 Notices of the AMS Volume 65, Number 10 COMMUNICATION There are different ways for the derivative (0.2) to fail to exist. The most common way is that the norm of the 푡 derivative ∇푢(0)푆 (푢(0)) is infinite. In such a case, the amplification from Δ푢(0) to Δ푢(푡) can be expected to be much faster than the exponential growth associated with any unstable eigenvalue of the linearized Euler equations (0.4). This is the phenomenon that we call “rough dependence upon initial data” [4]. Such superfast growth can reach substantial magnitude in very short time during which the exponential growth is very small as in Figure 6 (see p. 1258). Thus the classical inviscid instability due to unstable eigenvalues does not capture the true inviscid instability of superfast growth. Rayleigh Equation as an Equation for a Directional Differential Figure 3. A solution of the Euler equation may be viewed as an orbit in the function space, and the solution map 푆푡 is the flow map. The linear hydrodynamic stability theory must be based on the linear approximation to Δ푢(푡) as a function of Δ푢(0), which requires that 푆푡(푢(0)) is differentiable in 푢(0). If 푆푡(푢(0)) were differentiable in 푢(0), then we would have 푡 (0.2) Δ푢(푡) = [∇푢(0)푆 (푢(0))] Δ푢(0) + 표 (‖Δ푢(0)‖푛), Figure 4. Simple channel flows for which the where ∇푢(0) represents the derivative in 푢(0) and ‖‖푛 is the 퐻푛 norm (0.1). In a small neighborhood of Δ푢(0) = 0, approximating Rayleigh equation cannot capture the we would have the linear approximation dominant instability of superfast growth. (0.3) 푑푢(푡) = [∇ 푆푡(푢(0))] 푑푢(0). 푢(0) Now we focus on a 2D channel flow as in Figure 4, where Indeed, 푑푢(푡) would satisfy the linearized Euler equations 푥1 is the streamwise direction and 푥2 is the transverse direction with the boundaries at 푥 = 푎, 푏. The classical (0.4) 휕푡푑푢(푡) + 푑푢 ⋅ ∇푢 + 푢 ⋅ ∇푑푢 = −∇푑푝, ∇ ⋅ 푑푢 = 0. 2 interest is to study the linear stability of the steady state Unfortunately, H. Inci ([2], 2015) proved that 푆푡(푢(0)) is nowhere differentiable in either 2 or 3 spatial dimensions 푢1 = 푈(푥2), 푢2 = 0, under decaying boundary condition at infinity. That is, at any initial condition in 퐻푛, the solution operator 푆푡(푢(0)) which satisfies the boundary condition is not differentiable in 푢(0). Recently, Inci and the au- 푢2 = 0 at 푥2 = 푎, 푏. thor proved that differentiability fails for other boundary conditions as well. The derivative (0.2) never exists, the The solution operator is not differentiable at the steady differential (0.3) never exists, and the linearized Euler state, so the linearized Euler equations at the steady state equations (0.4) always fail to provide a linear approxima- cannot provide a linear approximation. Nevertheless, the tion for the inviscid hydrodynamic stability. In conclusion, classical inviscid linear hydrodynamic stability theory the inviscid linear hydrodynamic stability theory always assumed that the solution operator was differentiable. fails to have a rigorous mathematical foundation! Formally starting from the linearized Euler equations, November 2018 Notices of the AMS 1257 COMMUNICATION Rayleigh derived the so-called Rayleigh equation as follows. Let 푖푘(푥1−휎푡) (0.5) 푑푢 = (휕푥2 휓, −휕푥1 휓), 휓 = 휙(푥2)푒 .
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