Zhiwu Lin

Nonlinear instability of ideal plane flows

We study nonlinear instability of stationary ideal plane flows. For any bounded domain and very general steady flows, we show that if the linearized equation has an exponentially growing solution, then the steady flow is nonlinearly unstable. The nonlinear instability is in the sense that we can find an initial perturbation arbitrarily close to the steady flow such that the L p norm of the velocity perturbation grows exponentially beyond a fixed value. The same result is also proved for the Charney-Hasegawa-Mima equation.

INSTABILITY OF SOME IDEAL PLANE FLOWS

We prove the instability of large classes of steady states of the two-dimensional Euler equation. For an odd shear flow, beginning with the Rayleigh equation, we define a family of operators depending on some positive parameter. Then we use infinite determinants to keep track of the signs of the eigenvalues of these operators. The existence of purely growing modes follows from a continuation argument. Employing a new analysis of neutral modes together with a rigorous justification of Tollmiens classical method, we obtain a sharp condition for linear and hence nonlinear instability of a general class of bounded shear flows. We obtain similar results for bounded rotating flows and unbounded shear flows.

A Resolution of the Sommerfeld Paradox

Sommerfeld paradox roughly says that mathematically Couette linear shear is linearly stable for all Reynolds number, but experimentally arbitrarily small perturbations can induce the transition from the linear shear to turbulence when the Reynolds number is large enough. The main idea of our resolution of this paradox is to show that there is a sequence of linearly unstable shears which approaches the linear shear in the kinetic energy norm but not in the enstrophy (vorticity) norm. These oscillatory shears are single Fourier modes in the Fourier series of all the shears. In experiments, such linear instabilities will manifest themselves as transient nonlinear growth leading to the transition from the linear shear to turbulence no matter how small the intitial perturbations to the linear shear are. Under the Euler dynamics, these oscillatory shears are steady, and cats eye structures bifurcate from them as travelling waves. The 3D shears

Unstable and Stable Galaxy Models

U(y,z)

Metastability of Kolmogorov Flows and Inviscid Damping of Shear Flows

in a neighborhood of these oscillatory shears are linearly unstable too. Under the Navier-Stokes dynamics, these oscillatory shears are not steady rather drifting slowly. When these oscillatory shears are viewed as frozen, the corresponding Orr-Sommerfeld operator has unstable eigenvalues which approach the corresponding inviscid eigenvalues when the Reynolds number tends to infinity. All the linear instabilities mentioned above offer a resolution to the Sommerfeld paradox, and an initiator for the transition from the linear shear to turbulence.

Invariant Manifolds of Traveling Waves of the 3D Gross–Pitaevskii Equation in the Energy Space

To determine the stability and instability of a given steady galaxy configuration is one of the fundamental problems in the Vlasov theory for galaxy dynamics. In this article, we study the stability of isotropic spherical symmetric galaxy models f0(E), for which the distribution function f0 depends on the particle energy E only. In the first part of the article, we derive the first sufficient criterion for linear instability of f0(E) : f0(E) is linearly unstable if the second-order operator

The Existence of Stable BGK Waves

A_{0} \equiv-\Delta+4\pi\int f_{0}^{\prime}(E)\{I-{\mathcal{P}}\}dv

Stability and instability of traveling solitonic bubbles

has a negative direction, where \({\mathcal{P}}\) is the projection onto the function space {g(E, L)}, L being the angular momentum [see the explicit formulae (29) and (28)]. In the second part of the article, we prove that for the important King model, the corresponding A0 is positive definite. Such a positivity leads to the nonlinear stability of the King model under all spherically symmetric perturbations.