Sufficient Conditions For Nonlinear Stability Research Articles

An underactuated drift-free left-invariant control system on the Lie group ISO(3,1)

The purpose of our paper is to study a class of left-invariant, drift-free optimal control problem on the Lie group ISO(3,1). The left-invariant, drift-free optimal control problems involves finding a trajectory-control pair on ISO(3,1), which minimize a cost function and satisfies the given dynamical constrains and boundary conditions in a fixed time. The problem is lifted to the cotangent bundle T*G using the optimal Hamiltonian on G*, where the maximum principle yields the optimal control. We use energy methods (Arnold’s method, in this case) to give sufficient conditions fornonlinear stability of the equilibrium states. Around this equilibrium states we might be able to find the periodical orbits using Moser's theorem, as future work. For the some unstable equilibrium states, a quadratic control is considered in order to stabilize the dynamics.

Collisionless magnetic reconnection dynamics with electron inertia and parallel electron compressibility

AbstractCollisionless magnetic reconnection dynamics is considered by including the effects of electron inertia as well as parallel electron compressibility. A fluid treatment is adopted for both electrons and ions. Collisionless plasma dynamics properties near a two-dimensional X-type magnetic neutral line in the steady state are explored. The effects of electron inertia and parallel electron compressibility on the hyperbolicity (or lack thereof) of the magnetic field lines in the neutral layer are discussed. A unified linear tearing-mode formulation incorporating both electron inertia and parallel electron compressibility is given. The parallel-electron-compressibility branch is shown to couple in general to the electron-inertia branch in the presence of resistivity. A sufficient condition for linear stability in the Lyapunov sense for steady states of this collisionless plasma system signifying current confinement is deduced. Bounds on the equilibrium current gradient are shown to constitute sufficient conditions for nonlinear stability in the Lyapunov sense for steady states via nonlinear bounds for a suitable perturbation norm.

Design and analysis of robust ALE time-integrators for the solution of unsteady flow problems on moving grids

Two methodologies for designing an arbitrary Lagrangian–Eulerian (ALE) time-integrator for the semi-discrete Navier–Stokes equations are reviewed. Each methodology consists of a different mathematical framework for extending to moving grids a numerical scheme originally developed for computational fluid dynamics (CFD) on fixed grids, while preserving its formal order of time-accuracy established on fixed grids. Given a favorite scheme for the solution on fixed grids of the discrete Navier–Stokes equations, each of these two mathematical frameworks can generate multiple ALE extensions that share the same order of time-accuracy on moving grids. Typically, only a subset of these ALE schemes satisfy their respective discrete geometric conservation laws (DGCLs). Next, using a nonlinear scalar conservation law (NSCL) as a model problem, it is proved that satisfying the corresponding DGCL is a necessary condition for any ALE scheme to preserve on moving grids the nonlinear stability properties of its fixed-grid counterpart. Using the same NSCL, it is also proved that for the ALE extension of the second-order time-accurate trapezoidal rule, the DGCL requirement is a necessary as well as sufficient condition for nonlinear stability on moving grids. Hence, each of the two mathematical frameworks described in this paper combined with the DGCL test provides a general methodology for designing a robust ALE time-integrator for the solution of unsteady flow problems on dynamic meshes. As an example, the ALE extension of the popular three-point backward difference scheme is discussed in details. The corresponding theoretical results are illustrated with its application to the solution of various unsteady flow problems arising from the vibrations of the AGARD Wing 445.6 as well as a complete F-16 configuration in transonic airstreams.

On Nonlinear Instability and Stability for Stratified Shear Flow

An example of stratified shear flow is presented in which an explicit construction is given for unstable eigenvalues with smooth eigenfunctions for the Taylor—Goldstein equation. It is proved for any stratified, plane parallel shear flow that the unstable spectrum of the linear operator is purely discrete. A general theorem is then invoked to prove that the specific example is nonlinearly unstable. A sufficient condition for nonlinear stability for stratified shear flow is discussed.

Stability of Viscous Shock Waves for Problems with Nonsymmetric Viscosity Matrices

Sufficient conditions for nonlinear stability of viscous shock wave solutions of systems of conservation laws are given. The analysis applies to strong shocks of Lax type but is restricted to perturbations with zero mass. We use the Laplace transform and reduce the question of stability to a spectral condition for the resolvent equation of the linearized problem.

On the existence of two-dimensional Euler flows satisfying energy-Casimir stability criteria

The energy-Casimir stability method, also known as the Arnold stability method, has been widely used in fluid dynamical applications to derive sufficient conditions for nonlinear stability. The most commonly studied system is two-dimensional Euler flow. It is shown that the set of two-dimensional Euler flows satisfying the energy-Casimir stability criteria is empty for two important cases: (i) domains having the topology of the sphere, and (ii) simply-connected bounded domains with zero net vorticity. The results apply to both the first and the second of Arnold’s stability theorems. In the spirit of Andrews’ theorem, this puts a further limitation on the applicability of the method.

On nonlinear stability of baroclinic fronts

We study the nonlinear stability of baroclinic fronts in a channel with variable bottom topography in the framework of the model proposed by Swaters and Flierl (1991). Sufficient conditions for nonlinear stability of steady flows in a channel with arbitrary bottom topography are obtained. We consider quite general perturbations which include non-zero perturbations of the circulations of the velocity along the rigid boundaries. The special case of zonal (x-independent) topography is treated in more detail. Here we make use of an additional invariant of the governing equations (the momentum). This allows us to obtain a less restrictive (than in the general situation) sufficient condition for nonlinear stability.

On Stability of Conservation Laws

We consider the Cauchy problem for systems of PDEs of the general form $$ u_t=P_0u + \eps_1P_1u +\eps_2 Q(u) + \sum_j D_j F_j(x,t), \quad u=u(x,t). $$ Here $P_0$ has constant coefficients. The terms $\eps_1 P_1u$ and $\eps_2 Q(u)$ describe linear and nonlinear perturbations, respectively, and $\textstyle\sum_j D_j F_j(x,t)$ is a forcing term, which decays to zero for $t\to \infty$. The perturbation terms are assumed to have conservation form. We call the system nonlinearly stable if the solution $u(x,t)$ with $u(x,0)=0$ remains smooth for all $t\geq 0$ and the maximum norm of u tends to zero for $t\to \infty$, provided that $\eps_1^2+\eps_2^2$ is sufficiently small. In the paper we give sufficient conditions for nonlinear stability. If the unperturbed system $u_t=P_0u$ is parabolic, then the Laplace transform technique is satisfactory to derive conditions for nonlinear stability. However, if $u_t=P_0u$ is hyperbolic or coupled parabolic-hyperbolic, then the Laplace transform technique fails if the perturb...

Note on the nonlinear stability of stratified fluid equilibria

We study the nonlinear stability of hydrostatic equilibria of an ideal incompressible stratified fluid. We obtain a new a priori estimate for finite-amplitude perturbation of the basic equilibrium state. the main idea of our approach is based on a special decomposition of the density perturbation, namely, we split the density perturbation in two parts, the first part depends on time but has zero initial value, the second one is in some sense time-independent (its L2-norm is time-independent). This decomposition allows us to obtain the a priori estimate for the time-dependent part of the perturbation and hence for the total perturbation. In our approach we avoid the problem of a smooth extension of a locally convex function beyond its initial domian of definition that arises in applications of Arnold's method. Taking advantage of this fact, we consider the nonlinear stability of equilibrium states of stratified fluid endowed with two densities. Such a kind of problem appears, e.g., in atmospheric physics when symmetric flows of a stratified fluid are considered. As a result, we obtain a sufficient condition for nonlinear stability of a general equilibrium state of such a doubly stratified fluid as well as an a priori estimate for perturbations of arbitrary amplitude.

Trkal flows in magnetohydrodynamics

Force free fields can be generalized to Trkal flows in magnetohydrodynamics if a special velocity field parallel to the magnetic field is introduced. Such flows decay exponentially in time in case a constant viscosity and a constant resistivity are added. The stability of generalized Trkal flows is investigated and a sufficient condition for nonlinear stability is derived. For low velocities, this condition reduces essentially to the nonlinear sufficient stability condition for force free fields, previously found by the author.

The conditions for the non-linear stability of plane and helical mhd flows

The stability of the steady flows of an ideal incompressible fluid of uniform density in a magnetic field is investigated. Only those MHD flows are considered which possess one of the types of symmetry (translational, axial, rotational or helical). The sufficient conditions for non-linear stability of the flows in question with respect to perturbations of this symmetry are obtained. These conditions are proved by the method of coupling the integrals of motion [1, 21 in the form [3–81, based on constructing functionals having absolute minima on specified steady solutions. Each of the functionals constructed is the sum of the kinetic energy, the integral of an arbitrary function of the Lagrangian coordinate and another integral, specific for the flows being investigated. The use of Lagrangian ooordinate fields leads to a whole family of new definitions of stability. According to these definitions, deviations of the perturbed flows from the unperturbed ones are measured by the integrals of the squares of the velocity-field and Lagrangiancoordinate perturbations. The stability conditions obtained are extended to existing results [5–7, 9] on new types of flows. These conditions are of an a priori nature since the corresponding theorems of existence of the solutions are not proved.

On the nonlinear stability of dissipative fluids

A general sufficient condition for nonlinear stability of steady and unsteady flows in hydrodynamics and magnetohydrodynamics is derived. It leads to strong limitations in the Reynolds and magnetic Reynolds numbers. It is applied to the stability of generalized time-dependent planar Couette flows in magneto-hydrodynamics. Reynolds and magnetic Reynolds numbers have to be typically less than 2π2 for stability.

Nonlinear stability theorem for rotating coherent structures in a non-neutral plasma column

A nonlinear stability theorem is developed for arbitrary-amplitude, two-dimensional coherent structures, nR(θ − ωr t) and φR(θ − ωr t), in a strongly magnetized, low-density (ω2pe/ω2ce ≪ 1) non-neutral plasma column confined radially by a uniform axial magnetic field B0êz. Here, a grounded, perfectly conducting, cylindrical wall is located at radius r = rw, and ωr = const is the angular rotation velocity of the coherent structures about the cylinder axis (r=0). A cold-fluid guiding-center model based on the continuity-Poisson equations is used to describe the nonlinear evolution of the electron density ne(r,θ,t) = nR + δne and electrostatic potential φ(r,θ,t) = φR + δφ. Making use of global (spatially averaged) nonlinear conservation constraints, it is shown that ∂nR(ψR)/∂ψR ≤ 0 is a sufficient condition for nonlinear stability of the rotating equilibrium state (φR,nR) to arbitrary-amplitude perturbations δne and δφ. Here, ψR(r,θ − ωr t) = −eφR(r,θ − ωr t) + ωr(eB0/2c)r 2 is an effective streamfunction, and the stability theorem is valid no matter how complex the radial and azimuthal structure of φR(r,θ − ωr t) and nR(r,θ − ωr t).

Nonlinear energy stability in a compressible atmosphere

Abstract We provide sufficient conditions for nonlinear exponential stability of the compressible Benard problem. In particular, by using a generalized energy analysis we prove stability whenever the Rayleigh number does not exceed a computable critical number Rc . The value of Rc is given for finite amplitude depth and for thin layers as well, and such values are compared with those already computed in the linear theory. In the limit of depth which goes to zero a necessary and sufficient condition for nonlinear stability of the Benard problem is proved. The principle of exchange of stabilities is not required to hold.

Nonlinear stability of a laminar flow past a two-dimensional grid

Sufficient conditions for nonlinear stability of an exact solution to the Navier–Stokes equations (Kovasznay flow) are given by use of the energy method. These results markedly improve on earlier ones obtained by Lin and Tobak [Phys. Fluids 30, 3388 (1987)].

Conditions for nonlinear stability of flows of an ideal incompressible liquid

Stability of steady state flows of an ideal incompressible liquid with homogeneous density with some type of symmetry (translational, axial, rotational, or helical) is considered. Two types of sufficient conditions for nonlinear stability are obtained, which can be proven by constructing two types of functionals which have absolute minima at the given steady state solutions. Each of the functionals used is the sum of the kinetic energy and some other integral, specific to the given class of motion. The first type of stability conditions are a generalization to the case of finite perturbations and a new class of flows of the well known Rayleigh criterion [1] for “centrifugal” stability of rotating flows relative to perturbations with rotational symmetry. In the same sense the second type of stability conditions generalize another result, also originally proposed by Rayleigh, according to which plane-parallel flow of a liquid is stable in the absence of an inflection point in the velocity profile [1]. A nonlinear variant of the latter condition for the class of planar motions was first obtained in [2]. To systematize the results extensive use is made of the analogy between the effects of density stratification and rotation in the form of [3], The results to be presented relate to stability of a wide class of hydrodynamic flows having the required symmetry. For example, they relate to flows in tubes and channels which rotate or are at rest, and flows with concentrated annular or spiral vortices.

Lyapunov stability conditions for relativistic multifluid plasma

The Hamiltonian structure is given for ideal relativistic multifluid plasma dynamics in the laboratory frame with the Hamiltonian functional equaling the relativistic energy minus the mass energy. The noncanonical Poisson bracket for this system turns out to be the same as for the nonrelativistic multifluid plasma, but with dynamical variables replaced by their relativistic counterparts. New constants of the motion are then derived from the Hamiltonian structure and used as Lyapunov functionals for proving sufficient conditions for nonlinear stability of relativistic multifluid plasma equilibria. The nonrelativistic limit of the formulation is uniformly regular, and nonlinear Lyapunov stability conditions derived previously for a nonrelativistic multifluid plasma reemerge in that limit.

Nonlinear stability analysis in multilayer quasigeostrophic systems

A sufficient condition for nonlinear stability of steady solutions of the quasigeostrophic equations in a multilayer system is found. It proves the stability of given shear flows in a two-layer model that were known to be neutrally stable only.