Hyperbolic Region Research Articles

Globally hyperbolic moment system by generalized Hermite expansion

In a recent paper, Cai et al. (2014) revealed that a modified 13-moment system taking intrinsic heat fluxes as variables, instead of the heat fluxes along the coordinate vectors which is adopted in the classical Grad 13-moment system, attains some additional advantages than the classical Grad 13-moment system, particularly including that the equilibrium is turned to be the interior point of its hyperbolicity region. The modified 13- moment system was actually derived from the generalized Hermite expansion of the distribution function, where the anisotropy of Hermite expansion is specified by the full temperature tensor. We extend the method therein in this paper to high order of generalized Hermite expansion to derive arbitrary order moment systems, and propose a globally hyperbolic regularization to achieve locally well-posedness similar to the method of Cai et al. (2014). Furthermore, the structure of the eigen-system of the coefficient matrix and all characteristic waves are fully clarified. The obtained systems provide a systematic class of hydrodynamic models as the refined version of Euler equations, which is gradually approaching the Boltzmann equation with increasing order of the expansion.

Hyperbolicity in the Ocean

Some manifestations of hyperbolicity in the ocean, the important concept in dynamical systems theory, are discussed. It is shown how to identify hyperbolic points, hyperbolic trajectories and their stable and unstable manifolds solving advection equations for passive scalars in a satellite-derived AVISO velocity field and computing finite-time Lyapunov exponents by the singular-value decomposition method. To validate our simulation we use available tracks of oceanic drifters following near surface currents in some areas in the Northwestern Pacific Ocean. The tracks illustrate how drifters “feel” the presence of hyperbolic points, hyperbolic trajectories and stable and unstable manifolds and change abruptly their trajectories when approaching a hyperbolicity region.

Mechanism for stickiness suppression during extreme events in Hamiltonian systems

In this paper we study how hyperbolic and nonhyperbolic regions in the neighborhood of a resonant island perform an important role allowing or forbidding stickiness phenomenon around islands in conservative systems. The vicinity of the island is composed of nonhyperbolic areas that almost prevent the trajectory to visit the island edge. For some specific parameters tiny channels are embedded in the nonhyperbolic area that are associated to hyperbolic fixed points localized in the neighborhood of the islands. Such channels allow the trajectory to be injected in the inner portion of the vicinity. When the trajectory crosses the barrier imposed by the nonhyperbolic regions, it spends a long time abandoning the vicinity of the island, since the barrier also prevents the trajectory from escaping from the neighborhood of the island. In this scenario the nonhyperbolic structures are responsible for the stickiness phenomena and, more than that, the strength of the sticky effect. We show that those properties of the phase space allow us to manipulate the existence of extreme events (and the transport associated to it) responsible for the nonequilibrium fluctuation of the system. In fact we demonstrate that by monitoring very small portions of the phase space (namely, ≈1×10(-5)% of it) it is possible to generate a completely diffusive system eliminating long-time recurrences that result from the stickiness phenomenon.

Pell equation and randomness

Finding the integer solutions of a Pell equation is equivalent to finding the integer lattice points in a long and narrow tilted hyperbolic region, located along a straight line passing through the origin with a quadratic irrational slope. The case of inhomogeneous Pell inequalities it is equivalent to finding the integer lattice points in translated copies of a long and narrow tilted hyperbolic region with a quadratic irrational slope. We prove randomness in these natural lattice point counting problems. We have two main results: a central limit theorem (Theorem 2.1) and a law of the iterated logarithm (Theorem 2.2). The classical Circle Problem (counting lattice points in large circles) is a notoriously difficult open problem. What our results show is that the hyperbolic analog of the Circle Problem is solvable with striking precision.

On Singular Closures for the 5-Moment System in Kinetic Gas Theory

AbstractMoment equations provide a flexible framework for the approximation of the Boltzmann equation in kinetic gas theory. While moments up to second order are sufficient for the description of equilibrium processes, the inclusion of higher order moments, such as the heat flux vector, extends the validity of the Euler equations to non-equilibrium gas flows in a natural way.Unfortunately, the classical closure theory proposed by Grad leads to moment equations, which suffer not only from a restricted hyperbolicity region but are also affected by non-physical sub-shocks in the continuous shock-structure problem if the shock velocity exceeds a critical value. Amore recently suggested closure theory based on the maximum entropy principle yields symmetric hyperbolic moment equations. However, if moments higher than second order are included, the computational demand of this closure can be overwhelming. Additionally, it was shown for the 5-moment system that the closing flux becomes singular on a subset of moments including the equilibrium state.Motivated by recent promising results of closed-form, singular closures based on the maximum entropy approach, we study regularized singular closures that become singular on a subset of moments when the regularizing terms are removed. In order to study some implications of singular closures, we use a recently proposed explicit closure for the 5-moment equations. We show that this closure theory results in a hyperbolic system that can mitigate the problem of sub-shocks independent of the shock wave velocity and handle strongly non-equilibrium gas flows.

Magnetostatic Modes in Samples With Inhomogeneous Internal Fields

Magnetostatic modes in ferromagnetic samples have been studied systematically since the 1950s. They are very well characterized and understood in samples with uniform internal magnetic fields. More recently, interest has shifted to the study of magnetization modes in ferromagnetic samples with inhomogeneous internal fields. This paper shows that under the magnetostatic approximation and for samples of arbitrary shape and/or arbitrary inhomogeneous internal magnetic fields, the modes can be classified as elliptic or hyperbolic, and their associated frequency spectrum and spatial range can be delimited. This results from the analysis of the character of the second-order partial differential equation satisfied by the magnetostatic potential: i.e., if it is hyperbolic or elliptic (parabolic is a limiting case). In elliptic regions, the magnetostatic modes have a smooth monotonic character (with generally decaying or “tunneling” behavior from the surfaces) and in hyperbolic regions an oscillatory wave-like character. A simple local criterion distinguishes hyperbolic from elliptic regions: the sign of a frequency-dependent susceptibility parameter. This study shows that one may control to some extent magnetostatic modes via external fields or geometry. For example, one may imagine propagation along interior regions avoiding surfaces, as is suggested in one case presented here.

Amplification of pulses in nonlinear geometric optics

In this companion paper to our study of amplification of wavetrains J.-F. Coulombel, O. Guès and M. Williams, Semilinear geometric optics with boundary amplification, Anal. PDE7(3) (2014) 551–625, we study weakly stable semilinear hyperbolic boundary value problems with pulse data. Here weak stability means that exponentially growing modes are absent, but the so-called uniform Lopatinskii condition fails at some boundary frequency in the hyperbolic region. As a consequence of this degeneracy there is again an amplification phenomenon: outgoing pulses of amplitude O(ε2) and wavelength ε give rise to reflected pulses of amplitude O(ε), so the overall solution has amplitude O(ε). Moreover, the reflecting pulses emanate from a radiating pulse that propagates in the boundary along a characteristic of the Lopatinskii determinant. In the case of N × N systems considered here, a single outgoing pulse produces on reflection a family of incoming pulses traveling at different group velocities. Unlike wavetrains, pulses do not interact to produce resonances that affect the leading order profiles. However, pulse interactions do affect lower-order profiles and so these interactions have to be estimated carefully in the error analysis. Whereas the error analysis in the wavetrain case dealt with small divisor problems by approximating periodic profiles by trigonometric polynomials (which amounts to using a high frequency cutoff), in the pulse case we approximate decaying profiles with nonzero moments by profiles with zero moments (a low frequency cutoff). Unlike the wavetrain case, we are now able to obtain a rate of convergence in the limit describing convergence of approximate to exact solutions.

Local well-posedness of a class of hyperbolic PDE–ODE systems on a bounded interval

The well-posedness theory for hyperbolic systems of first-order quasilinear PDE's with ODE's boundary conditions (on a bounded interval) is discussed. Such systems occur in multi-scale blood flow models, as well as valveless pumping and fluid mechanics. The theory is presented in the setting of Sobolev spaces Hm (m ≥ 3 being an integer), which is an appropriate set-up when it comes to proving existence of smooth solutions using energy estimates. A blow-up criterion is also derived, stating that if the maximal time of existence is finite, then the state leaves every compact subset of the hyperbolicity region, or its first-order derivatives blow-up. Finally, we discuss physical examples which fit in the general framework presented.

Trace Formula for Linear Hamiltonian Systems with its Applications to Elliptic Lagrangian Solutions

In the present paper, we build up trace formulas for both the linear Hamiltonian systems and Sturm-Liouville systems. The formula connects the monodromy matrix of a symmetric periodic orbit with the infinite sum of eigenvalues of the Hessian of the action functional. A natural application is to study the non-degeneracy of linear Hamiltonian systems. Precisely, by the trace formula, we can give an estimation for the upper bound such that the non-degeneracy preserves. Moreover, we could estimate the relative Morse index by the trace formula. Consequently, a series of new stability criteria for the symmetric periodic orbits is given. As a concrete application, the trace formula is used to study the linear stability of elliptic Lagrangian solutions of the classical planar three-body problem. It is well known that the linear stability of elliptic Lagrangian solutions depends on the mass parameter $\bb=27(m_1m_2+m_2m_3+m_3m_1)/(m_1+m_2+m_3)^2\in [0,9]$ and the eccentricity $e\in [0,1)$. Based on the trace formula, we estimate the stable region and hyperbolic region of the elliptic Lagranian solutions.

Time machines with the compactly determined Cauchy horizon

The building of a time machine, if possible at all, requires the relevant regions of spacetime to be compact (that is, physically speaking, free from sources of unpredictability such as infinities and singularities). Motivated by this argument we consider the spacetimes with the compactly determined Cauchy horizons (CDCHs), the defining property of which is the compactness of $\overline{J^-(\EuScript U)}\cap J^+(\EuScript S_0)$, where $\EuScript U$ is an open subset of the Cauchy horizon and $\EuScript S_0$ is a Cauchy surface of the initial globally hyperbolic region $\ingh$. The following two facts are established: 1) $\ingh$ has no globally hyperbolic maximal extension. This means that by shaping appropriately a precompact portion of a globally hyperbolic region one can \emph{force} the Universe to produce either a closed causal curve, or a quasiregular singularity, whichever it abhors less; 2) Before a CDCH is formed a null geodesic appears which infinitely approaches the horizon returning again and again in the same --- arbitrarily small --- region. The energy of the photon moving on such a geodesic increases with each passage, or at least falls insufficiently fast. As a result, an observer located in the mentioned region would see a bunch of photons passing through his laboratory with the arbitrarily large total energy. We speculate that this phenomenon may have observable consequences.

Semilinear geometric optics with boundary amplification

We study weakly stable semilinear hyperbolic boundary value problems with highly oscillatory data. Here weak stability means that exponentially growing modes are absent, but the so-called uniform Lopatinskii condition fails at some boundary frequency $\beta$ in the hyperbolic region. As a consequence of this degeneracy there is an amplification phenomenon: outgoing waves of amplitude $O(\eps^2)$ and wavelength $\eps$ give rise to reflected waves of amplitude $O(\eps)$, so the overall solution has amplitude $O(\eps)$. Moreover, the reflecting waves emanate from a radiating wave that propagates in the boundary along a characteristic of the Lopatinskii determinant. An approximate solution that displays the qualitative behavior just described is constructed by solving suitable profile equations that exhibit a loss of derivatives, so we solve the profile equations by a Nash-Moser iteration. The exact solution is constructed by solving an associated singular problem involving singular derivatives of the form $\partial_{x'}+\beta\frac{\partial_{\theta_0}}{\eps}$, $x'$ being the tangential variables with respect to the boundary. Tame estimates for the linearization of that problem are proved using a first-order calculus of singular pseudodifferential operators constructed in the companion article \cite{CGW2}. These estimates exhibit a loss of one singular derivative and force us to construct the exact solution by a separate Nash-Moser iteration. The same estimates are used in the error analysis, which shows that the exact and approximate solutions are close in $L^\infty$ on a fixed time interval independent of the (small) wavelength $\eps$. The approach using singular systems allows us to avoid constructing high order expansions and making small divisor assumptions.

Lattice-Based WOM Codes for Multilevel Flash Memories

We consider t-write codes for write-once memories with n cells that can store multiple levels. Assuming an underlying lattice-based construction and using the continuous approximation, we derive upper bounds on the worst-case sum-rate optimal and fixed-rate optimal n-cell t-write write-regions for the asymptotic case of continuous levels. These are achieved using hyperbolic shaping regions that have a gain of 1 bit/cell over cubic shaping regions. Motivated by these hyperbolic write-regions, we discuss construction and encoding of codebooks for cells with discrete support. We present a polynomial-time algorithm to assign messages to the codebooks and show that it achieves the optimal sum-rate for any given codebook when n = 2. Using this approach, we construct codes that achieve high sum-rate. We describe an alternative formulation of the message assignment problem for n≥ 3, a problem which remains open.

An exact smooth Gowdy-symmetric generalized Taub–NUT solution

In a recent paper (Beyer and Hennig 2012 Class. Quantum Grav. 29 245017), we have introduced a class of inhomogeneous cosmological models: the smooth Gowdy-symmetric generalized Taub–NUT solutions. Here we derive a three-parametric family of exact solutions within this class, which contains the two-parametric Taub solution as a special case. We also study properties of this solution. In particular, we show that for a special choice of the parameters, the spacetime contains a curvature singularity with directional behaviour that can be interpreted as a ‘true spike’ in analogy to previously known Gowdy-symmetric solutions with spatial -topology. For other parameter choices, the maximal globally hyperbolic region is singularity-free, but may contain ‘false spikes’.

The non-topological fluxes of a two-particle system in the Chern–Simons theory

In the paper, a planar relativistic self-dual Chern–Simons model, with two Higgs particles and two gauge fields is considered. The main purpose is to locate all the possible values of the magnetic fluxes for the radially symmetric non-topological solitons. As has been known, the non-topological fluxes are not quantized. We further show that the value set of the non-topological fluxes exactly form a planar continuum portrayed by a certain hyperbolic region.

On the connectivity of the Julia sets of meromorphic functions

We prove that every transcendental meromorphic map f with a disconnected Julia set has a weakly repelling fixed point. This implies that the Julia set of Newton's method for finding zeroes of an entire map is connected. Moreover, extending a result of Cowen for holomorphic self-maps of the disc, we show the existence of absorbing domains for holomorphic self-maps of hyperbolic regions whose iterates tend to a boundary point. In particular, the results imply that periodic Baker domains of Newton's method for entire maps are simply connected, which solves a well-known open question.

A model for transonic plasma flow

A linear, two-dimensional model of a transonic plasma flow in equilibrium is constructed and given an explicit solution in the form of a complex Laplace integral. The solution indicates that the transonic state can be solved as an elliptic boundary value problem, as is done in the numerical code FLOW [Guazzotto et al., Phys. Plasmas 11, 604 (2004)]. Moreover, the presence of a hyperbolic region does not necessarily imply the presence of a discontinuity or any other singularity of the solution.

On hyperbolicity of 13-moment system

We point out that the thermodynamic equilibrium is not an interior point of the hyperbolicity region of Grad's 13-moment system. With a compact expansion of the phase density, which is more compact than Grad's expansion, we derive a modified 13-moment system. The new 13-moment system admits the thermodynamic equilibrium as an interior point of its hyperbolicity region. We deduce a concise criterion to ensure the hyperbolicity, and thus the hyperbolicity region can be quantitatively depicted.

Effects of pressure anisotropy on magnetospheric magnetohydrodynamics equilibrium of an internal ring current system

Effects of pressure anisotropy on magnetospheric magnetohydrodynamics equilibrium are studied analytically and numerically, where the plasma is confined by only poloidal magnetic field generated by an internal ring current. The plasma current due to finite pressure can be divided into two components; one remains at isotropic pressure and the other arises from pressure anisotropy. When p⊥, the pressure perpendicular to the magnetic field, is larger than p∥, the pressure parallel to the magnetic field, those two components of plasma current tend to cancel each other to reduce the total amount of plasma current. Equilibrium beta limit is also examined, where the beta is a ratio of the plasma pressure to the magnetic pressure. The equilibrium beta limit decreases as the pressure anisotropy becomes strong. The beta value is strictly limited by ellipticity of the equilibrium equation when p∥>p⊥. On the other hand, when p⊥>p∥, although the tendency of the beta limit agrees with the ellipticity condition of the equilibrium equation, equilibria with a hyperbolic region can be obtained by iterative procedure with practically reasonable convergence criteria.

Hyperbolic regions in flows through three-dimensional pore structures

Finite time Lyapunov exponents are used to determine expanding, contracting, and hyperbolic regions in computational simulations of laminar steady-state fluid flows within realistic three dimensional pore structures embedded within an impermeable matrix. These regions correspond approximately to pores where flow converges (contraction) or diverges (expansion), and to throats between pores where the flow mixes (hyperbolic). The regions are sparse and disjoint from one another, occupying only a small percentage of the pore space. Nonetheless, nearly every percolating fluid particle trajectory passes through several hyperbolic regions indicating that the effects of in-pore mixing are distributed throughout an entire pore structure. Furthermore, the observed range of fluid dynamics evidences two scales of heterogeneity within each of these flow fields. There is a larger scale that affects dispersion of fluid particle trajectories across the connected network of pores and a relatively small scale of nonuniform distributions of velocities within an individual pore.

An estimation for the hyperbolic region of elliptic Lagrangian solutions in the planar three-body problem

It is well known that the linear stability of elliptic Lagrangian solutions depends on the mass parameter β = 27(m1m2 + m2m3 + m3m1)/(m1 + m2 + m3)2 ∈ [0, 9] and the eccentricity e ∈ [0, 1). Based on new techniques for evaluating the hyperbolicity and the recently developed trace formula for Hamiltonian systems [9], we identify regions for (β, e) such that elliptic Lagrangian solutions are hyperbolic. Consequently, we have proven that the elliptic relative equilibrium of square central configurations is hyperbolic with any eccentricity.