Time-asymptotic States Research Articles

Asymptotic linear - nonlinear duality, indeterminism and mathematical intelligence

The formalism of asymptotic duality structure with applications to late time asymptotic states of linear and nonlinear systems is presented. A scale invariant linear differential system is shown to admit higher derivative discontinuous, nonsmooth solutions under the action of asymptotic duality transformations. The relevant asymptotic quantities are self dual and finitely differentiable. The geometrical meaning and applications to higher order nonlinear systems are presented on the space of differentiable functions. A novel duality relationship between linear and nonlinear systems is exposed. A linear system is shown to extend self similarly over nonlinear, classically inaccessible scales thereby extending classical smooth solution space to a nonsmooth one. A nonlinear system, on the other hand, is shown to proliferate into a system of self similar linear equations that could be solved exactly to have a very efficient high precision computation of periodic orbits. An extension of self dual asymptotics to critically self dual case in the space of continuous functions reveals a connection to prime number theorem and the associated correction term respecting the Riemann hypothesis.

Simulations of axisymmetric, inviscid swirling flows in circular pipes with various geometries

The numerical simulations of the dynamics of high Reynolds number ($$Re>100{,}000$$) swirling flows in pipes with varying geometries of engineering applications continues to be a challenging computational problem, specifically when vortex-breakdown zones or wall-separation regions naturally evolve in the flows. To tackle this challenge, the present paper describes a simulation scheme of the evolution of inviscid, axisymmetric and incompressible swirling flows in expanding or contracting pipes. The integration in time of the circulation together with azimuthal vorticity uses an explicit, first-order accurate finite-difference scheme with a second-order accurate upwind difference formulation in the axial and radial directions. The Poisson solver for advancing in time the spatial distribution of the stream function as a function of azimuthal vorticity uses a second-order accurate over-relaxation difference scheme. No additional numerical steps are needed for computing the natural evolution of flows including the dynamics to states with slow-speed recirculation zones along the pipe centerline or attached to pipe wall. This numerical method shows convergence of the computed results with mesh refinement for various swirl levels and pipe geometry variations. The computed results of time-asymptotic states also present agreement with available theoretical predictions of steady vortex flows in diverging or contracting pipes. In addition, comparison with available experimental data demonstrates that the present algorithm accurately predicts instability processes and long-term mean-flow dynamics of vortex flows in pipes at high Re. The inviscid-flow simulation results support the theoretical predictions and clarify the nature of high-Re flow evolution.

Swirling flow states in finite-length diverging or contracting circular pipes

The dynamics of inviscid-limit, incompressible and axisymmetric swirling flows in finite-length, diverging or contracting, long circular pipes is studied through global analysis techniques and numerical simulations. The inlet flow is described by the profiles of the circumferential and axial velocity together with a fixed azimuthal vorticity while the outlet flow is characterized by a state with zero radial velocity. A mathematical model that is based on the Squire–Long equation (SLE) is formulated to identify steady-state solutions of the problem with special conditions to describe states with separation zones. The problem is then reduced to the columnar (axially-independent) SLE, with centreline and wall conditions for the solution of the outlet flow streamfunction. The solution of the columnar SLE problem gives rise to the existence of four types of solutions. The SLE problem is then solved numerically using a special procedure to capture states with vortex-breakdown or wall-separation zones. Numerical simulations based on the unsteady vorticity circulation equations are also conducted and show correlation between time-asymptotic states and steady states according to the SLE and the columnar SLE problems. The simulations also shed light on the stability of the various steady states. The uniqueness of steady-state solutions in a certain range of swirl is proven analytically and demonstrated numerically. The computed results provide the bifurcation diagrams of steady states in terms of the incoming swirl ratio and size of pipe divergence or contraction. Critical swirls for the first appearance of the various types of states are identified. The results show that pipe divergence promotes the appearance of vortex-breakdown states at lower levels of the incoming swirl while pipe contraction delays the appearance of vortex breakdown to higher levels of swirl and promotes the formation of wall-separation states.

The Shock Reflection Phenomenon for Scalar Conservation Law with Dirac Measure Source Term

This paper is mainly concerned with the shock reflection phenomenon for convex scalar conservation law with Dirac measure source term. The Riemann solutions are constructed completely and then the impact of the strength of source term on the Riemann solutions is considered in detail. In order to illustrate the shock reflection phenomenon, the initial data with three pieces of constant states are considered and the interactions between a backward shock wave plus a stationary wave discontinuity and a rarefaction wave are displayed in all kinds of situations. Furthermore, the global solutions are constructed completely and the large time asymptotic states are obtained. In some certain situations, the shock reflection phenomenon is captured when a rarefaction wave interacts with a stationary wave discontinuity and then is divided into a transmitted rarefaction wave and a reflected shock wave at the critical point. In addition, it is shown that the Riemann solutions are stable with respect to the particular small perturbations of the initial data.

On nonlinear asymptotic stability of the Lane–Emden solutions for the viscous gaseous star problem

This paper proves the nonlinear asymptotic stability of the Lane–Emden solutions for spherically symmetric motions of viscous gaseous stars if the adiabatic constant γ lies in the stability range (4/3,2). It is shown that for small perturbations of a Lane–Emden solution with same mass, there exists a unique global (in time) strong solution to the vacuum free boundary problem of the compressible Navier–Stokes–Poisson system with spherical symmetry for viscous stars, and the solution captures the precise physical behavior that the sound speed is C1/2-Hölder continuous across the vacuum boundary provided that γ lies in (4/3,2). The key is to establish the global-in-time regularity uniformly up to the vacuum boundary, which ensures the large time asymptotic uniform convergence of the evolving vacuum boundary, density and velocity to those of the Lane–Emden solution with detailed convergence rates, and detailed large time behaviors of solutions near the vacuum boundary. In particular, it is shown that every spherical surface moving with the fluid converges to the sphere enclosing the same mass inside the domain of the Lane–Emden solution with a uniform convergence rate and the large time asymptotic states for the vacuum free boundary problem (1.1.2) are determined by the initial mass distribution and the total mass. To overcome the difficulty caused by the degeneracy and singular behavior near the vacuum free boundary and coordinates singularity at the symmetry center, the main ingredients of the analysis consist of combinations of some new weighted nonlinear functionals (involving both lower-order and higher-order derivatives) and space–time weighted energy estimates. The constructions of these weighted nonlinear functionals and space–time weights depend crucially on the structures of the Lane–Emden solution, the balance of pressure and gravitation, and the dissipation. Finally, the uniform boundedness of the acceleration of the vacuum boundary is also proved.

A well-balanced numerical scheme for a one-dimensional quasilinear hyperbolic model of chemotaxis

We introduce a numerical scheme to approximate a quasi-linear hyperbolic system which models the movement of cells under the influence of chemotaxis. Since we expect to find solutions which contain vacuum parts, we propose an upwinding scheme which handles properly the presence of vacuum and, besides, which gives a good approximation of the time asymptotic states of the system. For this scheme we prove some basic analytical properties and study its stability near some of the steady states of the system. Finally, we present some numerical simulations which show the dependence of the asymptotic behavior of the solutions upon the parameters of the system.

Out-of-Equilibrium Phase Transitions and Time-Asymptotic One-Particle Dynamics in the Vlasov Limit of The Hamiltonian Mean Field Model

When starting from specific initial conditions, the ferromagnetic-like XY Hamiltonian mean field (HMF) model evolves toward quasistationary states, with lifetimes diverging with the number N of degrees of freedom that violate equilibrium statistical mechanics predictions. Phase transitions have been reported between low-energy magnetized quasistationary states and large energy unexpected, antiferromagnetic-like ones with low, but nonvanishing, magnetization. This issue is addressed here in the Vlasov N→∞ limit. It is argued that the time asymptotic states emerging in the Vlasov limit can be related to simple generic time asymptotic forms for the force field. The proposed picture unveils the nature of the out-of-equilibrium phase transitions reported for the ferromagnetic HMF in the second order regime: This is a bifurcation point connecting an effective integrable Vlasov one-particle time-asymptotic dynamic to a partly ergodic one, which means an abrupt open-up of the Vlasov one-particle phase space. This is proposed as a mechanism for second-order phase transitions compatible with nonvanishing time-asymptotic values of the order parameter in mean-field long-range systems.

Nonlinear Landau Damping

In this article, we report the mathematical details of a systematic analysis of the nonlinear Landau damping of longitudinal electrostatic waves propagating in a collisionless plasma. Many of the main results have been reported previously; unfortunately, major parts of the essential mathematical developments had to be omitted for various reasons, making it almost impossible for even the most well-prepared reader to follow the analysis. Sufficient details are provided here to remedy this situation. Some important results that have not been reported previously are also are included here. Most notably among these is the distinction between strong branches and weak branches of nonzero time-asymptotic electric field amplitudes that bifurcate from the zero-amplitude solution for the time-asymptotic electric field, and the results on the weak branches that had to be omitted in the earlier report of this research. Based on the decomposition of the electric field E into a transient part T and a time-asymptotic part A, we show that A is given by a finite superposition of wave modes, whose frequencies obey a Vlasov dispersion relation and whose amplitudes satisfy a set of nonlinear algebraic equations. These time-asymptotic mode amplitudes are calculated explicitly based on approximate solutions for the particle distribution functions obtained by linearizing only the term that contains T in the Vlasov equation for each particle species and then integrating the resulting equation along the nonlinear characteristics associated with A, which are obtained via Hamiltonian perturbation theory. For “linearly stable” initial Vlasov equilibria, we obtain a critical initial amplitude, separating the initial conditions that Landau damp to zero from those that lead to nonzero multiple-traveling-wave time-asymptotic states via nonlinear particle trapping. These theoretical results explain why in some cases experiments and large-scale numerical simulations have resulted in zero-field final states; whereas in other cases they have yielded nonzero multiple-traveling-wave final states because the theoretical results establish the existence of a “threshold” in the initial electric field below which the field damps to zero and above which it evolves to a finite-amplitude multiple-traveling-wave final state.

Optimal convergence rates of Landau equation with external forcing in the whole space

In this paper, we combine the method of constructing the compensating function introduced by Kawashima and the standard energy method for the study on the Landau equation with external forcing. Both the global existence of solutions near the time asymptotic states which are local Maxwellians and the optimal convergence rates are obtained. The method used here has its own advantage for this kind of studies because it does not involve the spectrum analysis of the corresponding linearized operator.

Asymptotic Stability of Rarefaction Wave for the Navier–Stokes Equations for a Compressible Fluid in the Half Space

This paper is concerned with the asymptotic stability towards a rarefaction wave of the solution to an outflow problem for the Navier–Stokes equations in a compressible fluid in the Eulerian coordinate in the half space. This is the second one of our series of papers on this subject. In this paper, firstly we classify completely the time-asymptotic states, according to some parameters, that is the spatial-asymptotic states and boundary conditions, for this initial boundary value problem, and some pictures for the classification of time-asymptotic states are drawn in the state space. In order to prove the stability of the rarefaction wave, we use the solution to Burgers’ equation to construct a suitably smooth approximation of the rarefaction wave and establish some time-decay estimates in Lp-norm for the smoothed rarefaction wave. We then employ the L2-energy method to prove that the rarefaction wave is non-linearly stable under a small perturbation, as time goes to infinity.

Dynamics of two-sign point vortices in positive and negative temperature states

The characteristic features of a two-sign point vortex system in positive and negative temperature states are examined by massive numerical simulations using MDGRAPE-2. The temperature is determined by a density of states for a microcanonical ensemble consisting of randomly generated 10 7 states. Since the density of states lias a single peak. the system has negative temperature states. The distributions of vortices in time-asymptotic equilibrium states in positive and negative temperature are obtained by time-development simulations. In positive temperature, both-sign vortices mix with each other and neutralize. In negative temperature, part of the vortices condense and form clumps exclusively consisting of the same-sign vortices, while the other part of the vortices distribute uniformly outside the clumps. It is found that the vortices inside the clumps gain energy and the vortices outside the clumps lose energy to keep the total energy constant. This suggests the common and essential role of the background vortices in the energy-conserving system that assists the formation of the clumps as well as the crystallization and generation of the symmetric configuration observed in the non-neutral plasma experiments.

Equilibrium statistical mechanics for single waves and wave spectra in Langmuir wave-particle interaction

Under the conditions of weak Langmuir turbulence, a self-consistent wave-particle Hamiltonian models the effective nonlinear interaction of a spectrum of M waves with N resonant out-of-equilibrium tail electrons. In order to address its intrinsically nonlinear time-asymptotic behavior, a Monte Carlo code was built to estimate its equilibrium statistical mechanics in both the canonical and microcanonical ensembles. First, the single wave model is considered in the cold beam-plasma instability and in the O’Neil setting for nonlinear Landau damping. O’Neil’s threshold, which separates nonzero time-asymptotic wave amplitude states from zero ones, is associated with a second-order phase transition. These two studies provide both a testbed for the Monte Carlo canonical and microcanonical codes, with the comparison with exact canonical results, and an opportunity to propose quantitative results to longstanding issues in basic nonlinear plasma physics. Then, the properly speaking weak turbulence framework is considered through the case of a large spectrum of waves. Focusing on the small coupling limit as a benchmark for the statistical mechanics of weak Langmuir turbulence, it is shown that Monte Carlo microcanonical results fully agree with an exact microcanonical derivation. The wave spectrum is predicted to collapse towards small wavelengths together with the escape of initially resonant particles towards low bulk plasma thermal speeds. This study reveals the fundamental discrepancy between the long-time dynamics of single waves, which can support finite amplitude steady states, and of wave spectra, which cannot.

Propagation of a plane-strain hydraulic fracture with a fluid lag: Early-time solution

This paper studies the propagation of a plane-strain fluid-driven fracture with a fluid lag in an elastic solid. The fracture is driven by a constant rate of injection of an incompressible viscous fluid at the fracture inlet. The leak-off of the fracturing fluid into the host solid is considered negligible. The viscous fluid flow is lagging behind an advancing fracture tip, and the resulting tip cavity is assumed to be filled at some specified low pressure with either fluid vapor (impermeable host solid) or pore-fluids infiltrating from the permeable host solid. The scaling analysis allows to reduce problem parametric space to two lumped dimensionless parameters with the meaning of the solid toughness and of the tip underpressure (difference between the specified pressure in the tip cavity and the far field confining stress). A constant lumped toughness parameter uniquely defines solution trajectory in the parametric space, while time-varying lumped tip underpressure parameter describes evolution along the trajectory. Further analysis identifies the early and large time asymptotic states of the fracture evolution as corresponding to the small and large tip underpressure solutions, respectively. The former solution is obtained numerically herein and is characterized by a maximum fluid lag (as a fraction of the crack length), while the latter corresponds to the zero-lag solution of Spence and Sharp [Spence, D.A., Sharp, P.W., 1985. Self-similar solution for elastohydrodynamic cavity flow. Proc. Roy. Soc. London, Ser. A (400), 289–313]. The self-similarity at small/large tip underpressure implies that the solution for crack length, crack opening and net fluid pressure in the fluid-filled part of the crack is a given power-law of time, while the fluid lag is a constant fraction of the increasing fracture length. Evolution of a fluid-driven fracture between the two limit states corresponds to gradual expansion of the fluid-filled region and disappearance of the fluid lag. For small solid toughness and small tip underpressure, the fracture is practically devoid of fluid, which is localized into a narrow region near the fracture inlet. Corresponding asymptotic solution on the fracture lengthscale corresponds to that of a crack loaded by a pair of point forces which magnitude is determined from the coupled hydromechanical solution in the fluid-filled region near the crack inlet. For large solid toughness, the fluid lag is vanishingly small at any underpressure and the solution is adequately approximated by the zero-lag self-similar large toughness solution at any stage of fracture evolution. The small underpressure asymptotic solutions obtained in this work are sought to provide initial condition for the propagation of fractures which are initially under plane-strain conditions.

Efficiency of Nonlinear Particle Acceleration at Cosmic Structure Shocks

We have calculated the evolution of cosmic-ray (CR) modified astrophysical shocks for a wide range of shock Mach numbers and shock speeds through numerical simulations of diffusive shock acceleration (DSA) in one-dimensional quasi-parallel plane shocks. The simulations include thermal leakage injection of seed CRs, as well as preexisting, upstream CR populations. Bohm-like diffusion is assumed. We model shocks similar to those expected around cosmic structure pancakes, as well as other accretion shocks driven by flows with upstream gas temperatures in the range T0 = 104-107.6 K and shock Mach numbers spanning Ms = 2.4-133. We show that CR-modified shocks evolve to time-asymptotic states by the time injected particles are accelerated to moderately relativistic energies (p/mc ≳ 1), and that two shocks with the same Mach number, but with different shock speeds, evolve qualitatively similarly when the results are presented in terms of a characteristic diffusion length and diffusion time. We determine and compare the "efficiencies" of CR acceleration in our simulated shocks by calculating the ratio of the total CR energy generated at the shock to the total kinetic energy that would pass through the shock over time in its initial frame of reference. For these models the time-asymptotic value for this efficiency ratio is controlled mainly by shock Mach number, as expected from the aforementioned similarity in CR shocks. In the presence of a preexisting CR population, shock evolution proceeds similarly to that for higher thermal injection rates compared to thermal leakage CR sources alone. This added contribution has little or no impact on the postshock or CR properties of the high Mach number shocks simulated. The modeled high Mach number shocks all evolve toward efficiencies ~50%, regardless of the upstream CR pressure. On the other hand, the upstream CR pressure increases the overall CR energy in moderate strength shocks (Ms ~ a few), since it is a significant fraction of the shock ram pressure. These shocks have been shown to dominate dissipation during cosmic structure formation, so such enhanced efficiency could significantly increase their potential importance as sources of cosmic rays.

NUMERICAL STUDIES OF COSMIC RAY ACCELERATION AT COSMIC SHOCKS

Shocks are ubiquitous in astrophysical environments and cosmic-rays (CRs) are known to be accelerated at collisionless shocks via diffusive shock acceleration. It is believed that the CR pressure is important in the evolution of the interstellar medium of our galaxy and most of galactic CRs with energies up to eV are accelerated by supernova remnant shocks. In this contribution we have studied the CR acceleration at shocks through numerical simulation of 1D, quasi-parallel shocks for a wide range of shock Mach numbers and shock speeds. We show that CR modified shocks evolve to time-asymptotic states by the time injected particles are accelerated to moderately relativistic energies, and that two shocks with the same Mach number, but with different shock speeds, evolve qualitatively similarly when the results are presented in terms of a characteristic diffusion length and diffusion time. We find that of the particles passed through the shock are accelerated to form the CR population, and the injection rate is higher for shocks with higher Mach number. The time asymptotic value for the CR acceleration efficiency is controlled mainly by shock Mach number, and high Mach number shocks all evolve towards efficiencies , regardless of the injection rate and upstream CR pressure. We conclude that the injection rates in strong quasi-parallel shocks are sufficient to lead to significant nonlinear modifications to the shock structures, implying the importance of the CR acceleration at astrophysical shocks.

COSMIC RAY ACCELERATION AT COSMOLOGICAL SHOCKS

Cosmological shocks form as an inevitable consequence of gravitational collapse during the large scale structure formation and cosmic-rays (CRs) are known to be accelerated at collisionless shocks via diffusive shock acceleration (DSA). We have calculated the evolution of CR modified shocks for a wide range of shock Mach numbers and shock speeds through numerical simulations of DSA in 1D quasi-parallel plane shocks. The simulations include thermal leakage injection of seed CRs, as well as pre-existing, upstream CR populations. Bohm-like diffusion is assumed. We show that CR modified shocks evolve to time-asymptotic states by the time injected particles are accelerated to moderately relativistic energies (p/mc <TEX>$\ge$</TEX> 1), and that two shocks with the same Mach number, but with different shock speeds, evolve qualitatively similarly when the results are presented in terms of a characteristic diffusion length and diffusion time. We find that <TEX>$10^{-4} - 10^{-3}$</TEX> of the particles passed through the shock are accelerated to form the CR population, and the injection rate is higher for shocks with higher Mach number. The CR acceleration efficiency increases with shock Mach number, but it asymptotes to <TEX>${\~}50\%$</TEX> in high Mach number shocks, regardless of the injection rate and upstream CR pressure. On the other hand, in moderate strength shocks (<TEX>$M_s {\le} 5$</TEX>), the pre-existing CRs increase the overall CR energy. We conclude that the CR acceleration at cosmological shocks is efficient enough to lead to significant nonlinear modifications to the shock structures.

Time-asymptotic wave propagation in collisionless plasmas.

We report the results of a new, systematic study of nonlinear longitudinal wave propagation in a collisionless plasma. Based on the decomposition of the electric field E into a transient part T and a time-asymptotic part A, we show that A is given by a finite superposition of wave modes, whose frequencies obey a Vlasov dispersion relation, and whose amplitudes satisfy a set of nonlinear algebraic equations. These time-asymptotic mode amplitudes are calculated explicitly, based on approximate solutions for the particle distribution functions obtained by linearizing only the term that contains T in the Vlasov equation for each particle species, and then integrating the resulting equation along the nonlinear characteristics associated with A, which are obtained via Hamiltonian perturbation theory. For "linearly stable" initial Vlasov equilibria, we obtain a critical initial amplitude (or threshold), separating the initial conditions that produce Landau damping to zero (A [triple bond] 0) from those that lead to nonzero multiple-traveling-wave time-asymptotic states via nonlinear particle trapping (A not identical with 0). These theoretical results have important implications about the stability of spatially uniform plasma equilibria, and they also explain why large-scale numerical simulations in some cases lead to zero-field final states whereas in others they yield nonzero multiple-traveling-wave final states.

Initial Boundary Value Problem for the System of Compressible Adiabatic Flow Through Porous Media

We study the initial value problem for the system of compressible adiabatic flow through porous media in the one space dimension with fixed boundary condition. Under the restriction on the oscillations in the initial data, we establish the global existence and large time behavior for the classical solutions via the combination of characteristic analysis and energy estimate methods. The time-asymptotic states for the solution are found and the exponential convergence to the asymptotic states is proved. It is also shown that this problem can be approximated very well time-asymptotically by an initial boundary problem for the related diffusion equations obtained from the original hyperbolic system by Darcy's law, provided that the initial total mass is the same. The difference between these two solutions tends to zero exponentially fast as time goes to infinity in the sense of H1. The diffusive phenomena caused by damping mechanism with boundary effects are thus observed.

Bifurcation phenomena of the magnetofluid equations

Abstract We report on bifurcation studies for the incompressible magnetohydrodynamic equations in three space dimensions with periodic boundary conditions and a temporally constant external forcing. Fourier representations of velocity, pressure and magnetic field have been used to transform the original partial differential equations into a system of ordinary differential equations (ODE), to which then special numerical methods for the qualitative analysis of systems of ODE have been applied, supplemented by the simulative calculation of solutions for selected initial conditions. In a part of the calculations, in order to reduce the number of modes to be retained, the concept of approximate inertial manifolds has been applied. For varying (increasing from zero) strength of the imposed forcing, or varying Reynolds number, respectively, time-asymptotic states, notably stable stationary solutions, have been traced. A primary non-magnetic steady state loses, in a Hopf bifurcation, stability to a periodic state with a non-vanishing magnetic field, showing the appearance of a generic dynamo effect. From now on the magnetic field is present for all values of the forcing. The Hopf bifurcation is followed by further, symmetry-breaking, bifurcations, leading finally to chaos. We pay particular attention to kinetic and magnetic helicities. The dynamo effect is only observed if the forcing is chosen such that a mean kinetic helicity is generated; otherwise the magnetic field diffuses away, and the time-asymptotic states are non-magnetic, in accordance with traditional kinematic dynamo theory.

The weakly coupled infinite Dicke model

For the Dicke model including all modes in R 3, the infinite atom limit within a weak coupling to the photon field is performed. Hereby, the infinite atomic system is treated as a mean field quantum lattice system. The global limiting dynamics is given by cocycle equations depending on the classical part of the atomic part and the one-photon structure. It is shown how the collective atomic behaviour influences the emitted quantized radiation in different photonic subsectors. The growth of expectation values, especially of the photon mean number, and the spatial distribution of the emitted photons in the time asymptotic photon states are calculated. In the photonic Fock sector the (un)necessity of the rotating-wave-approximation of the Dicke model and the quantum optical coherence of the time asymptotic photon states are discussed.