arXiv subjects

Zhiwu Lin

Publications and source records attributed to Zhiwu Lin.

At least 37 records · Page 2Linked to original sources

arXiv2021

Linear instability of Vlasov-Maxwell systems revisited-A Hamiltonian approach

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 Hamiltonian 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.

math.AP↗

arXiv2021

Stability and instability of self-gravitating relativistic matter distributions

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., relativistic models of stars. Such steady states are embedded into one-parameter families parameterized by their central redshift $κ>0$. We prove their linear instability when $κ$ 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 Zel'dovich \& 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.

gr-qc↗

arXiv2021

Instability, index theorem, and exponential trichotomy for Linear 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 $\left\langle L\cdot,\cdot\right\rangle $ on $X$, which has only finitely many negative dimensions $n^{-}(L)$. 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( L\right) $ 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 Schrödinger equations with nonzero condition at infinity, where our general theory applies to yield stability or instability of some coherent states.

math.AP↗

arXiv2021

Separable Hamiltonian PDEs and Turning point principle for stability of gaseous stars

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.

math.AP↗

arXiv2020

Turning point principle for relativistic stars

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 1960's Zel'dovich [50] and Wheeler [22] 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.

math-ph↗

arXiv2019

Barotropic instability of shear flows

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.

math.AP↗

arXiv2018

Dynamics near the solitary waves of the supercritical gKDV Equations

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.

math.AP↗

arXiv2017

Invariant Manifolds of traveling waves of the 3D Gross-Pitaevskii equation in the energy space

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-unstable 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.

math.AP↗

arXiv2017

Linear Inviscid Damping for Couette Flow in Stratified Fluid

We study the inviscid damping of Couette flow with an exponentially stratified density. The optimal decay rates of the velocity field and the density are obtained for general perturbations with minimal regularity. For Boussinesq approximation model, the decay rates we get are consistent with the previous results in the literature. We also study the decay rates for the full Euler equations of stratified fluids, which were not studied before. For both models, the decay rates depend on the Richardson number in a very similar way. Besides, we also study the dispersive decay due to the exponential stratification when there is no shear.

math.AP↗

arXiv2017

Metastability of Kolmogorov flows and inviscid damping of shear flows

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 turbulence. 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.

math.AP↗

arXiv2017

Nonlinear Modulational Instability of Dispersive PDE Models

We prove nonlinear modulational instability for both periodic and localized perturbations of periodic traveling waves for several dispersive PDEs, including the KDV type equations (e.g. the Whitham equation, the generalized KDV equation, the Benjamin-Ono equation), the nonlinear Schrödinger equation and the BBM equation. First, the semigroup estimates required for the nonlinear proof are obtained by using the Hamiltonian structures of the linearized PDEs; Second, for KDV type equations the loss of derivative in the nonlinear term is overcome in two complementary cases: (1) for smooth nonlinear terms and general dispersive operators, we construct higher order approximation solutions and then use energy type estimates; (2) for nonlinear terms of low regularity, with some additional assumption on the dispersive operator, we use a bootstrap argument to overcome the loss of derivative.

math.AP↗

arXiv2017

The existence of stable BGK waves

The 1D Vlasov-Poisson system is the simplest kinetic model for describing an electrostatic collisonless plasma, and the BGK waves are its famous exact steady solutions. They play an important role on the long time dynamics of a collisionless plasma as potential "final states" or "attractors", thanks to many numerical simulations and observations. Despite their importance, the existence of stable BGK waves has been an open problem since their discovery in 1958. In this paper, first linearly stable BGK waves are constructed near homogeneous states.

math.AP↗

arXiv2016

Stability of traveling waves of nonlinear Schrödinger equation with nonzero condition at infinity

We study the stability of traveling waves of nonlinear Schrödinger equation with nonzero condition at infinity obtained via a constrained variational approach. Two important physical models are Gross-Pitaevskii (GP) equation and cubic-quintic equation. First, under a non-degeneracy condition we prove a sharp instability criterion for 3D traveling waves of (GP), which had been conjectured in the physical literature. This result is also extended for general nonlinearity and higher dimensions, including 4D (GP) and 3D cubic-quintic equations. Second, for cubic-quintic type sub-critical or critical nonlinearity, we construct slow traveling waves and prove their nonlinear instability in any dimension. For traveling waves without vortices (i.e. nonvanishing) of general nonlinearity in any dimension, we find the sharp condition for linear instability. Third, we prove that any 2D traveling wave of (GP) is transversally unstable and find the sharp interval of unstable transversal wave numbers. Near unstable traveling waves of above cases, we construct unstable and stable invariant manifolds.

math.AP↗

arXiv2013

Unstable Galaxy Models

The dynamics of collisionless galaxy can be described by the Vlasov-Poisson system. By the Jean's theorem, all the spherically symmetric steady galaxy models are given by a distribution of Φ(E,L), where E is the particle energy and L the angular momentum. In a celebrated Doremus-Feix-Baumann Theorem, the galaxy model Φ(E,L) is stable if the distribution Φ is monotonically decreasing with respect to the particle energy E. On the other hand, the stability of Φ(E,L) remains largely open otherwise. Based on a recent abstract instability criterion of Guo-Lin, we constuct examples of unstable galaxy models of f(E,L) and f(E) in which f fails to be monotone in E.

astro-ph.GA↗

arXiv2011

Unstable manifolds of Euler equations

We consider a steady state $v_{0}$ of the Euler equation in a fixed bounded domain in $\mathbf{R}^{n}$. Suppose the linearized Euler equation has an exponential dichotomy of unstable and center-stable subspaces. By rewriting the Euler equation as an ODE on an infinite dimensional manifold of volume preserving maps in $W^{k, q}$, $(k>1+\frac{n}{q})$, the unstable (and stable) manifolds of $v_{0}$ are constructed under certain spectral gap condition which is verified for both 2D and 3D examples. In particular, when the unstable subspace is finite dimensional, this implies the nonlinear instability of $v_{0}$ in the sense that arbitrarily small $W^{k, q}$ perturbations can lead to $L^{2}$ growth of the nonlinear solutions.

math.AP↗

arXiv2011

Samll BGK waves and nonlinear Landau damping (higher dimensions)

Consider Vlasov-Poisson system with a fixed ion background and periodic condition on the space variables, in any dimension d\geq2. First, we show that for general homogeneous equilibrium and any periodic x-box, within any small neighborhood in the Sobolev space W_{x,v}^{s,p} (p>1,s<1+(1/p)) of the steady distribution function, there exist nontrivial travelling wave solutions (BGK waves) with arbitrary traveling speed. This implies that nonlinear Landau damping is not true in W^{s,p}(s<1+(1/p)) space for any homogeneous equilibria and in any period box. The BGK waves constructed are one dimensional, that is, depending only on one space variable. Higher dimensional BGK waves are shown to not exist. Second, for homogeneous equilibria satisfying Penrose's linear stability condition, we prove that there exist no nontrivial invariant structures in the (1+|v|^{2})^{b}-weighted H_{x,v}^{s} (b>((d-1)/4), s>(3/2)) neighborhood. Since arbitrarilly small BGK waves can also be constructed near any homogeneous equilibria in such weighted H_{x,v}^{s} (s<(3/2)) norm, this shows that s=(3/2) is the critical regularity for the existence of nontrivial invariant structures near stable homogeneous equilibria. These generalize our previous results in the one dimensional case.

math.AP↗

arXiv2010

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 cat's eye structures bifurcate from them as travelling waves. The 3D shears $U(y,z)$ 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.

math.AP↗

arXiv2010

Inviscid dynamical structures near Couette flow

Consider inviscid fluids in a channel {-1 (3/2)) neighborhood of Couette, we show that there exist no non-parallel steadily travelling flows v(x-ct,y), and no unstable shears. This suggests that the long time dynamics in H^{s}(s>(3/2)) neighborhoods of Couette might be much simpler. Such contrasting dynamics in H^{s} spaces with the critical power s=(3/2) is a truly nonlinear phenomena, since the linear inviscid damping near Couette is true for any initial vorticity in L^2.

math.AP↗