Search Results for « Linear dynamical stability »

We give meaning to linear and semi-linear (possibly degenerate) parabolic partial differential equations with (affine) linear rough path noise and establish stability in a rough path metric. In the case of enhanced Brownian motion (Brownian motion with its Lévy area) as rough path noise the solution coincides with the standard variational solution of the SPDE.

We show that the dynamical stability under linear perturbations of interacting systems in the hydrodynamic regime follows from the first and the second laws of thermodynamics. Our argument extends to systems with spontaneously or softly broken symmetries and in the presence of magnetic fields.

This work analyzes the non-linear dynamical stability of ideal-gas polytropes under homologous flow. A non-constant density profile requires the inclusion of magnetic fields, which is done by introducing a mean-field model that treats the spherically-averaged radial Lorentz force self-consistently and has the following properties: 1) The only essential simplifications are the Cowling approximation and a dominant radial flow. 2) The average radial Lorentz force due to an isotropic field is $-\frac13 dP_B/dr$, not $-dP_B/dr$ as is typically assumed. 3) A central peak in the magnetic field requires isotropy there; all other configurations are zero at the origin due to magnetic tension. 4) Solutions with negligible surface fields require $\gtrsim1/2$ of the magnetic energy to be in the radial component. 5) Solutions that resemble Lane-Emden solutions are restricted to $γ= 4/3$, where $γ$ is the material adiabatic index, and exhibit either collapse or escape. 6) Solutions for general $γ$ have a harmonic enthalpy profile and allow for non-linear radial pulsations. 7) A harmonic-enthalpy homologous flow becomes unbound when an overpressure satisfies $δ= ΔP_0/P_{\rm eq} > \frac{3γ- 4}{1 + 3(γ-1)α_0}$, where $P$ is the total pressure, $P_{\rm eq}$ is its equilibrium value, $α$ is the ratio of radiation to material pressure, and a zero subscript denotes minimum volume. This indicates that radiation pressure can unbind a linearly-stable polytrope in the presence of small but finite radial perturbations. The condition to unbind a fully-ionized $n = 3$ polytrope with $2/3$ of its magnetic energy in the radial component is $δ\gtrsim 0.15μ^{-1}\left(300M_\odot/M\right)^{1/2}$, where $μ$ is the mean molecular weight. This non-linear dynamical instability threshold may have some relevance for mass loss in and dispersal of evolved high-mass stars.

As a step toward understanding the analytic behavior of Type-III Ricci flow singularities, i.e. immortal solutions that exhibit |Rm|<C/t curvature decay, we examine the linearization of an equivalent flow at fixed points discovered recently by Baird--Danielo and Lott: nongradient homogeneous expanding Ricci solitons on nilpotent or solvable Lie groups. For all explicitly known nonproduct examples, we demonstrate linear stability of the flow at these fixed points and prove that the linearizations generate strongly continuous semigroups.

A numerical method devised for the solution of the Vlasov–Poisson coupled equations in three-dimensional phase space (radial distance, absolute value of the velocity and µ ≡ cos (v, r) is described. The method is based on the Lax–Wendroff integration scheme with a specially adapted Shuman filter. The method provides a second order accurate and conservative (in mass) direct integration of the above coupled system without making additional assumptions, like expansion into moments, local equilibrium etc. The method is used to analyse the non-linear dynamic stability of spherical self-gravitating systems. We conclude that some Newtonian stellar clusters which are found to be thermodynamically unstable are dynamically stable. The method allows us to study the details of the relaxation towards a steady state. We find that the long spatial wavelengths relax first and the relaxation proceeds towards shorter wavelengths.

We study genericity of dynamical properties in the space of homeomorphisms of the Cantor set and in the space of subshifts of a suitably large shift space. These rather different settings are related by a Glasner-King type correspondence: genericity in one is equivalent to genericity in the other. By applying symbolic techniques in the shift-space model we derive new results about genericity of dynamical properties for transitive and totally transitive homeomorphisms of the Cantor set. We show that the isomorphism class of the universal odometer is generic in the space of transitive systems. On the other hand, the space of totally transitive systems displays much more varied dynamics. In particular, we show that in this space the isomorphism class of every Cantor system without periodic points is dense, and the following properties are generic: minimality, zero entropy, disjointness from a fixed totally transitive system, weak mixing, strong mixing, and minimal self joinings. The last two stand in striking contrast to the situation in the measure-preserving category. We also prove a correspondence between genericity of dynamical properties in the measure-preserving category and genericity of systems supporting an invariant measure with the same property.

We consider the problem of dynamical stability for the $n$-vortex of the Ginzburg-Landau model. Vortices are one of the main examples of topological solitons, and their dynamic stability is the basic assumption of the asymptotic ``particle plus field'' description of interacting vortices. In this paper we focus on co-rotational perturbations of vortices and establish decay estimates for their linearized evolution in the relativistic case. One of the main ingredients is a construction of the distorted Fourier basis associated to the linearized operator at the vortex. The general approach follows that of Krieger-Schlag-Tataru and Krieger-Miao-Schlag and relies on the spectral analysis of Schrödinger operators with strongly singular potentials. Since one of the operators appearing in the linearization has zero energy solutions that oscillate at infinity, additional work is needed for our construction and to control the spectral measure. The decay estimates that we obtain are of both wave and Klein-Gordon type, and are consistent with the general theory for $2$d Schrödinger operators, including those that have an $s$-wave resonance, as in the present case, but faster decaying potentials. Finally, we give a new proof of the absence of unstable spectrum and provide an estimate on the location of embedded eigenvalues by using a suitable Lieb-Thirring inequality due to Ekholm and Frank. In particular, we show that eigenvalues must lie in the interval $(1.332,2)$, where $2$ represents the effective mass of one of the two scalar operators appearing in the linearization.

Recent experiments in hypersonic vehicle technology have had some successful outcomes (ex. NASA X-43, Boeing X-51) but also have seen some unexpected failures (DARPA Hypersonic Test Vehicle 2). One important failure mode is the problem of engine unstart. Morgan, et al (2012) has discussed and analyzed a Scramjet isolator using two unit problems, viz. that of an oblique shock impinging on a turbulent boundary layer and that of a normal shock train in a constant-area duct. In order to analyze unstart, we use the numerical data obtained from the LES calculations of the normal shock train in a constant area duct and perform temporal linear stability analysis on it under the assumptions of inviscid semi-parallel flow. This paper describes the methodology and the details of this linear stability analysis. The results obtained from the investigations have been documented and interpreted with their relevance to the physical problem. The report concludes that temporal linear stability analysis is insufficient to analyze the problem of unstart. Finally, the need for a new approach (namely, dynamic mode decomposition (DMD)) has been discussed.