We consider linear stability of steady states of 1(1/2) and 3D Vlasov-Maxwell systems for collisionless plasmas. The linearized systems can be written as separable Hamiltonian systems with constraints. By using a general theory for separable Hamilton
ian systems, we recover the sharp linear stability criteria obtained previously by different approaches. Moreover, we obtain the exponential trichotomy estimates for the linearized Vlasov-Maxwell systems in both relativistic and nonrelativistic cases.
In this paper, we study the number of traveling wave families near a shear flow $(u,0)$ under the influence of Coriolis force, where the traveling speeds lie outside the range of $u$. Let $beta$ be the Rossby number. If the flow $u$ has at least one
critical point at which $u$ attains its minimal (resp. maximal) value, then a unique transitional $beta$ value exists in the positive (resp. negative) half-line such that the number of traveling wave families near $(u,0)$ changes suddenly from finite one to infinity when $beta$ passes through it. If $u$ has no such critical points, then the number is always finite for positive (resp. negative) $beta$ values. This is true for general shear flows under a technical assumption, and for flows in class $mathcal{K}^+$ unconditionally. The sudden change of the number of traveling wave families indicates that nonlinear dynamics around the shear flow is much richer than the non-rotating case, where no such traveling waves exist.
Upon specifying an equation of state, spherically symmetric steady states of the Einstein-Euler system are embedded in 1-parameter families of solutions, characterized by the value of their central redshift. In the 1960s Zeldovich [50] and Wheeler [2
2] formulated a turning point principle which states that the spectral stability can be exchanged to instability and vice versa only at the extrema of mass along the mass-radius curve. Moreover the bending orientation at the extrema determines whether a growing mode is gained or lost. We prove the turning point principle and provide a detailed description of the linearized dynamics. One of the corollaries of our result is that the number of growing modes grows to infinity as the central redshift increases to infinity.
We consider stability of non-rotating gaseous stars modeled by the Euler-Poisson system. Under general assumptions on the equation of states, we proved a turning point principle (TPP) that the stability of the stars is entirely determined by the mass
-radius curve parameterized by the center density. In particular, the stability can only change at extrema (i.e. local maximum or minimum points) of the total mass. For very general equation of states, TPP implies that for increasing center density the stars are stable up to the first mass maximum and unstable beyond this point until next mass extremum (a minimum). Moreover, we get a precise counting of unstable modes and exponential trichotomy estimates for the linearized Euler-Poisson system. To prove these results, we develop a general framework of separable Hamiltonian PDEs. The general approach is flexible and can be used for many other problems including stability of rotating and magnetic stars, relativistic stars and galaxies.
We consider steady state solutions of the massive, asymptotically flat, spherically symmetric Einstein-Vlasov system, i.e., relativistic models of galaxies or globular clusters, and steady state solutions of the Einstein-Euler system, i.e., relativis
tic models of stars. Such steady states are embedded into one-parameter families parameterized by their central redshift $kappa>0$. We prove their linear instability when $kappa$ is sufficiently large, i.e., when they are strongly relativistic, and that the instability is driven by a growing mode. Our work confirms the scenario of dynamic instability proposed in the 1960s by Zeldovich & Podurets (for the Einstein-Vlasov system) and by Harrison, Thorne, Wakano, & Wheeler (for the Einstein-Euler system). Our results are in sharp contrast to the corresponding non-relativistic, Newtonian setting. We carry out a careful analysis of the linearized dynamics around the above steady states and prove an exponential trichotomy result and the corresponding index theorems for the stable/unstable invariant spaces. Finally, in the case of the Einstein-Euler system we prove a rigorous version of the turning point principle which relates the stability of steady states along the one-parameter family to the winding points of the so-called mass-radius curve.
This work is devoted to study the dynamics of the supercritical gKDV equations near solitary waves in the energy space $H^1$. We construct smooth local center-stable, center-unstable and center manifolds near the manifold of solitary waves and give a
detailed description of the local dynamics near solitary waves. In particular, the instability is characterized as following: any forward flow not starting from the center-stable manifold will leave a neighborhood of the manifold of solitary waves exponentially fast. Moreover, orbital stability is proved on the center manifold, which implies the uniqueness of the center manifold and the global existence of solutions on it.
We consider barotropic instability of shear flows for incompressible fluids with Coriolis effects. For a class of shear flows, we develop a new method to find the sharp stability conditions. We study the flow with Sinus profile in details and obtain
the sharp stability boundary in the whole parameter space, which corrects previous results in the fluid literature. Our new results are confirmed by more accurate numerical computation. The addition of the Coriolis force is found to bring fundamental changes to the stability of shear flows. Moreover, we study dynamical behaviors near the shear flows, including the bifurcation of nontrivial traveling wave solutions and the linear inviscid damping. The first ingredient of our proof is a careful classification of the neutral modes. The second one is to write the linearized fluid equation in a Hamiltonian form and then use an instability index theory for general Hamiltonian PDEs. The last one is to study the singular and non-resonant neutral modes using Sturm-Liouville theory and hypergeometric functions.
We study the local dynamics near general unstable traveling waves of the 3D Gross-Pitaevskii equation in the energy space by constructing smooth local invariant center-stable, center-unstable and center manifolds. We also prove that (i) the center-un
stable manifold attracts nearby orbits exponentially before they get away from the traveling waves along the center directions and (ii) if an initial data is not on the center-stable manifolds, then the forward flow will be ejected away from traveling waves exponentially fast. Furthermore, under a non-degenerate assumption, we show the orbital stability of the traveling waves on the center manifolds, which also implies the local uniqueness of the local invariant manifolds. Our approach based on a geometric bundle coordinates should work for a general class of Hamiltonian PDEs.
First, we consider Kolmogorov flow (a shear flow with a sinusoidal velocity profile) for 2D Navier-Stokes equation on a torus. Such flows, also called bar states, have been numerically observed as one type of metastable states in the study of 2D turb
ulence. For both rectangular and square tori, we prove that the non-shear part of perturbations near Kolmogorov flow decays in a time scale much shorter than the viscous time scale. The results are obtained for both the linearized NS equations with any initial vorticity in L^2, and the nonlinear NS equation with initial L^2 norm of vorticity of the size of viscosity. In the proof, we use the Hamiltonian structure of the linearized Euler equation and RAGE theorem to control the low frequency part of the perturbation. Second, we consider two classes of shear flows for which a sharp stability criterion is known. We show the inviscid damping in a time average sense for non-shear perturbations with initial vorticity in L^2. For the unstable case, the inviscid damping is proved on the center space. Our proof again uses the Hamiltonian structure of the linearized Euler equation and an instability index theory recently developed by Lin and Zeng for Hamiltonian PDEs.
Consider a general linear Hamiltonian system $partial_{t}u=JLu$ in a Hilbert space $X$. We assume that$ L: X to X^{*}$ induces a bounded and symmetric bi-linear form $leftlangle Lcdot,cdotrightrangle $ on $X$, which has only finitely many negative di
mensions $n^{-}(L)$. There is no restriction on the anti-self-dual operator $J: X^{*} supset D(J) to X$. We first obtain a structural decomposition of $X$ into the direct sum of several closed subspaces so that $L$ is blockwise diagonalized and $JL$ is of upper triangular form, where the blocks are easier to handle. Based on this structure, we first prove the linear exponential trichotomy of $e^{tJL}$. In particular, $e^{tJL}$ has at most algebraic growth in the finite co-dimensional center subspace. Next we prove an instability index theorem to relate $n^{-}left( Lright) $ and the dimensions of generalized eigenspaces of eigenvalues of$ JL$, some of which may be embedded in the continuous spectrum. This generalizes and refines previous results, where mostly $J$ was assumed to have a bounded inverse. More explicit information for the indexes with pure imaginary eigenvalues are obtained as well. Moreover, when Hamiltonian perturbations are considered, we give a sharp condition for the structural instability regarding the generation of unstable spectrum from the imaginary axis. Finally, we discuss Hamiltonian PDEs including dispersive long wave models (BBM, KDV and good Boussinesq equations), 2D Euler equation for ideal fluids, and 2D nonlinear Schr{o}dinger equations with nonzero condition at infinity, where our general theory applies to yield stability or instability of some coherent states.
