arXiv subjects

Zhiwu Lin

Publications and source records attributed to Zhiwu Lin.

At least 19 recordsLinked to original sources

arXiv2026

Secular Instability and the Turning-Point Principle for Rigidly Rotating Viscous Stars

We study axisymmetric stability of rigidly rotating viscous stars modeled by the free-boundary Navier--Stokes--Poisson (NSP) system. The unstable index of the linearized NSP generator, counted with Riesz algebraic multiplicity, equals the Morse index of the augmented energy at fixed mass and total angular momentum. We prove an abstract Kelvin--Tait--Chetaev theorem for damped gyroscopic equations with finite negative stiffness index, requiring no compactness assumptions and no spectral gap at zero; the stiffness kernel may be infinite-dimensional. In the NSP application, the viscous dissipation is degenerate: its axisymmetric kernel, generated by rigid axial translation and rigid rotation, is projected out before applying the abstract theorem, while conservation laws and a Routh reduction transfer the resulting instability index back to the full NSP generator. For slowly rotating branches with fixed total angular momentum, viscous instability begins at the continuation of a first nondegenerate spherical mass maximum. By contrast, for every sufficiently small nonzero angular momentum, the corresponding Euler--Poisson star remains axisymmetrically spectrally stable on an interval beyond this maximum. Thus dissipation can destabilize a rotating star before the corresponding inviscid star becomes unstable. This separation of the stability thresholds results from viscous redistribution of angular momentum, which removes the inviscid constraints on its material distribution while preserving its total value.

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arXiv2026

On the nonlinear instability of nonrotating Stars

We study radial nonlinear instability of compactly supported nonrotating equilibria of the three-dimensional Euler--Poisson system with a physical-vacuum boundary. Let $n^u(\mu)$ denote the radial instability index furnished by the turning-point theory of Lin and Zeng. Under general structural assumptions on the pressure law, suppose that $n^u(\mu)>0$ and that the equilibrium is not a mass extremum, $M'(\mu)\neq0$. Conditional on the existence of a sufficiently regular radial solution on the relevant time interval, we establish two nonlinear escape criteria. First, every perturbation with Hamiltonian strictly below that of the equilibrium exits a fixed neighborhood on a logarithmic time scale controlled by the least unstable linear growth rate. For the subclass of data for which the associated Lyapunov functional is initially nonnegative, we also obtain an explicit exponential lower bound in the weighted displacement norm. Second, sufficiently small perturbations whose Riesz projection onto the finite-dimensional unstable subspace is not too small escape on a logarithmic time scale. The second argument uses the exponential trichotomy of the linearized Hamiltonian flow and an invariant-cone estimate. In the polytropic class, the conditional estimates combine with the radial physical-vacuum local theory to yield unconditional nonlinear instability in the corresponding classical-solution topology. The results complement Jang's nonlinear instability theorem for Lane--Emden stars by treating mechanisms that do not require initial alignment with a leading growing eigenmode and by applying, conditionally, to unstable branches for general equations of state.

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arXiv2026

Prandtl--Batchelor and flux-expulsion selection for steady MHD flows in a disk

We study the simultaneous vanishing-viscosity and vanishing-resistivity limit of steady incompressible MHD flows in a disk. The boundary velocity is a small nonaxisymmetric perturbation of a rigid rotation with mean angular speed $\alpha$, while the prescribed tangential magnetic trace has mean $\beta$. Assuming $\alpha\neq0$ and the non-Alfv\'enic condition $|\alpha|\neq|\beta|$, we construct solutions converging on compact interior subdisks to a rigidly rotating ideal MHD core with constant vorticity and out-of-plane current density. A new MHD--Wood law determines the two core rotations from the boundary data: the velocity core is selected by a coupled kinetic--magnetic balance, whereas the magnetic core is fixed by the imposed mean circulation. Consequently, zero circulation gives complete interior magnetic expulsion, while nonzero circulation leaves a uniform magnetic rotation after the nonaxisymmetric modes are confined to a thin boundary layer. This provides a fully coupled realization of Prandtl--Batchelor selection and flux expulsion; unlike classical kinematic models, the magnetic field actively changes the flow and need not be weak. The proof combines a non-Alfv\'enic coercive theory for a periodic MHD boundary layer, global matching of the two fields, and a coupled stability estimate adapted to the magnetic boundary condition. A separate conditional rigidity argument, using exact viscous identities and local convergence but no interior asymptotic expansion, explains the same core structure for a broader single-eddy family.

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arXiv2026

Unstable Manifolds of Stratified Euler Equations

We consider a spectrally unstable steady state $(\rho_0,v_0)$ of the incompressible stratified Euler equations on a class of $d$-dimensional domains. Assuming that the linearized equation admits an exponential dichotomy with a reasonably large spectral gap relative to the maximal Lyapunov exponent of the background steady flow $v_0$, we construct the local stable and unstable manifolds of $(\rho_0,v_0)$. The proof is based on the Lyapunov--Perron method after reformulating the Euler equation as an ODE on the infinite-dimensional manifold of volume-preserving Lagrangian maps, with the density treated as a frozen Lagrangian parameter as well as the weight in the $L^2$ metric. We also discuss some applications to two-dimensional steady flows.

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arXiv2025

Classification of atmospheric traveling waves at cloud level

We classify within the quasi-geostrophic framework all types of traveling waves in zonal bands of the planetary atmosphere at cloud level according to their wave speeds. This classification pertains to waves of all amplitudes, going beyond the small-amplitude perturbative regime. It provides a structurally robust criterion for determining which traveling-wave profiles are dynamically possible and we show that each wave classification type was observed on Jupiter or Saturn. Building on this classification, we also investigate the related rigidity issue for large-amplitude traveling waves and waves propagating near shear flows. Our study offers a unified quantitative characterization of the intrinsic constraints for traveling waves in the quasi-geostrophic regime of planetary atmospheric flow.

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arXiv2025

The onset of instability for zonal stratospheric flows

We investigate some qualitative aspects of the dynamics of the Euler equation on a rotating sphere that are relevant or stratospheric flows. Zonal flow dominates the dynamics of the stratosphere and for most known planetary stratospheres the observed flow pattern is a small perturbation of an n-jet, which corresponds to choosing the Legendre polynomial of degree n as the stream function. Since the 1-jet and the 2-jet are stable, the main interest is the onset of instability for the 3-jet. We confirm long standing conjectures based on numerical simulations by proving that the 3-jet is linearly unstable if and only if the rotation rate belongs to a critical interval. Turning to the nonlinear problem, we prove that linear instability implies nonlinear instability and that, as the rotation rate goes to infinity, nearby traveling waves change gradually from a cat's eyes streamline pattern to a profile with no stagnation points.

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arXiv2024

Prandtl-Batchelor flows with a point vortex on a disk

We study steady two-dimensional incompressible Navier-Stokes flow in the unit disk in a high Reynolds number regime, driven by a localized central forcing and a nearly rigid rotating boundary. The forcing models a compact source of circulation that sustains a vortex core in a viscous flow. For sufficiently small viscosity and boundary perturbation, we construct a very weak steady solution on the whole disk and identify its inviscid limit: a point vortex embedded in a constant vorticity background and joined to the boundary motion by a thin boundary layer. The background vorticity is selected by the Batchelor-Wood formula. The singular core introduces non-coercive interactions in the linearized problem that are absent for regular Prandtl-Batchelor flows. To overcome this difficulty, we map the punctured disk to a semi-infinite cylinder by a logarithmic radial transformation and develop a Fourier-mode analysis that isolates the delicate first Fourier mode, yielding uniform stability and vorticity estimates in the presence of the point-vortex singularity.

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arXiv2023

Dynamical magneto-rotational instability

Magneto-rotational instability (MRI) is an important instability mechanism for rotating flows with magnetic fields. In particular, when the strength of the magnetic field tends to zero, the stability criterion for rotating flows is generally different from the classical Rayleigh criterion for rotating flows without a magnetic field. MRI has wide applications in astrophysics, particularly to the turbulence and enhanced angular momentum transport in accretion disks. For the case of vertical magnetic fields, we give rigorous proof of linear MRI and a complete description of the spectra and semigroup growth of the linearized operator. Moreover, we prove nonlinear stability and instability from the sharp linear stability/instability criteria.

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arXiv2023

Expanding solutions near unstable Lane-Emden stars

We consider the compressible Euler-Poisson equations for polytropes $P(\rho)=K\rho^{\gamma}$ with $\gamma\in \left(\frac{6}{5},\frac{4}{3} \right]$ and the white dwarf stars. For $\gamma=\frac{4}{3},$ we establish the existence of a global weak solution for the spherically symmetric initial data with mass less than the mass of the Lane-Emden stars (i.e. non-rotating polytropes). For $\gamma\in \left(\frac{6}{5},\frac{4}{3} \right)$, we show the existence of global weak solution for spherical symmetric initial data in an invariant set containing a neighborhood of Lane-Emden stars. Moreover, the support of these solution expands to infinity. As a corollary, this proves the strong instability of the Lane-Emden stars for $\gamma\in \left( \frac{6}{5},\frac{4}{3}\right] $. For $\gamma\in \left(\frac{6}{5},\frac{4}{3} \right),$ our results provide the first example of expanding solutions near the Lane-Emden stars. For white dwarf stars, we prove that the solution cannot collapse if the mass of initial data is less than the Chandrasekhar limit mass, which is the supremum of the mass of the non-rotating white dwarf stars. Our proof strongly uses the variational characterization of the Lane-Emden stars. First, we relate the best constant of a Hardy-Littlewood type inequality with the mass of the Lane-Emden stars with $\gamma=\frac{4}{3}$, which is further shown to equal the Chandrasekhar limit mass. For $\gamma\in\left( \frac{6}{5},\frac{4}{3}\right) $, we show that the Lane-Emden stars are minimizers of an energy-mass functional subject to a Pohozaev type constraint. This is crucial in the construction of the invariant set of expanding solutions.

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arXiv2023

Nonlinear stability of non-rotating gaseous stars

For the non-rotating gaseous stars modeled by the compressible Euler-Poisson system with general pressure law, Lin and Zeng [18] proved a turning point principle, which gives the sharp linear stability/instability criteria for the non-rotating gaseous stars. In this paper, we prove that the sharp linear stability criterion for the non-rotating stars also implies nonlinear orbital stability against general perturbations provided the global weak solutions exist. If the perturbations are further restricted to be spherically symmetric, then nonlinear stability holds true unconditionally in the sense that the existence of global weak solutions near the non-rotating star can be proved.

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arXiv2023

On the stability and instability of Kelvin-Stuart cat's-eye flows

Kelvin-Stuart vortices are classical mixing layer flows with many applications in fluid mechanics, plasma physics and astrophysics. We prove that the whole family of Kelvin-Stuart vortices is nonlinearly orbitally stable for co-periodic perturbations, and linearly unstable for multi-periodic and modulational perturbations. This verifies a long-standing conjecture since the discovery of the Kelvin-Stuart cat's-eye flows in the 1960s. Kelvin-Stuart cat's eyes also appear as magnetic islands which are magnetostatic equilibria for the planar ideal MHD equations in plasmas. We prove nonlinear orbital stability of Kelvin-Stuart magnetic islands for co-periodic perturbations, and give the first rigorous proof of coalescence instability for the whole family, which is important for magnetic reconnection.

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arXiv2023

Turning point principle for stability of viscous gaseous stars

We consider stability of non-rotating viscous gaseous stars modeled by the Navier-Stokes-Poisson system. Under general assumptions on the equations of states, we proved that the number of unstable modes for the linearized Navier-Stokes-Poisson system equals that of the linearized Euler-Poisson system modeling inviscid gaseous stars. In particular, the turning point principle holds true for non-rotating stars with or without viscosity. That is, the transition of stability only occurs at the extrema of the total mass and the number of unstable modes is determined by the mass-radius curve. For the proof, we establish an infinite dimensional Kelvin-Tait-Chetaev theorem for a class of linear second order PDEs with dissipation. Moreover, we prove that linear stability implies nonlinear asymptotic stability and linear instability implies nonlinear instability for Navier-Stokes-Poisson system.

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arXiv2022

Stability of rotating gaseous stars

We consider stability of rotating gaseous stars modeled by the Euler-Poisson system with general equation of states. When the angular velocity of the star is Rayleigh stable, we proved a sharp stability criterion for axi-symmetric perturbations. We also obtained estimates for the number of unstable modes and exponential trichotomy for the linearized Euler-Poisson system. By using this stability criterion, we proved that for a family of slowly rotating stars parameterized by the center density with fixed angular velocity profile, the turning point principle is not true. That is, unlike the case of non-rotating stars, the change of stability of the rotating stars does not occur at extrema points of the total mass. By contrast, we proved that the turning point principle is true for the family of slowly rotating stars with fixed angular momentum distribution. When the angular velocity is Rayleigh unstable, we proved linear instability of rotating stars. Moreover, we gave a complete description of the spectra and sharp growth estimates for the linearized Euler-Poisson equation.

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arXiv2021

Prandtl-Batchelor flows on an annulus

For steady two-dimensional Navier-Stokes flows with a single eddy (i.e. nested closed streamlines) in a simply connected domain, Prandtl (1905) and Batchelor (1956) found that in the inviscid limit, the vorticity is constant inside the eddy. In this paper, we consider the generalized Prandtl-Batchelor theory for the forced steady Navier-Stokes equations on an annulus. First, we observe that in the limit of infinite Reynolds number, if forced steady Navier-Stokes solutions has nested closed streamlines on an annulus, then the inviscid limit is a rotating shear flow uniquely determined by the external force and boundary conditions. We call solutions of steady Navier-Stokes equations with the above property Prandtl-Batchelor flows. Then, by constructing higher order approximate solutions of the forced steady Navier-Stokes equations and establishing the validity of Prandtl boundary layer expansion, we give a rigorous proof of the existence of Prandtl-Batchelor flows on an annulus with the wall velocities slightly different from the rigid-rotations along the same direction.

math.AP↗

arXiv2021

Prandtl-Batchelor flows on a disk

For steady two-dimensional flows with a single eddy (i.e. nested closed streamlines), Prandtl (1905) and Batchelor (1956) proposed that in the limit of vanishing viscosity the vorticity is constant in an inner region separated from the boundary layer. In this paper, by constructing higher order approximate solutions of the Navier-Stokes equations and establishing the validity of Prandtl boundary layer expansion, we give a rigorous proof of the existence of Prandtl-Batchelor flows on a disk with the wall velocity slightly different from the rigid-rotation. The leading order term of the flow is the constant vorticity solution (i.e. rigid rotation) satisfying Batchelor-Wood formula.

math.AP↗

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.

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arXiv2020

The number of traveling wave families in a running water with Coriolis force

In this paper, we study the number of traveling wave families near a shear flow under the influence of Coriolis force, where the traveling speeds lie outside the range of the flow $u$. Under the $\beta$-plane approximation, if the flow $u$ has a 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 the shear flow changes suddenly from finite to infinite when $\beta$ passes through it. On the other hand, 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 mildly technical assumptions, and for a large class of shear flows including a cosine jet $u(y) = {1+\cos(\pi y)\over 2}$ (i.e. the sinus profile) and analytic monotone flows unconditionally. The sudden change of the number of traveling wave families indicates that long time dynamics around the shear flow is much richer than the non-rotating case, where no such traveling wave families exist.

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.

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